A posteriori estimates for the two-step backward differentiation formula and discrete regularity for the time-dependent Stokes equations

A posteriori estimates for the two-step backward differentiation formula and discrete regularity... Abstract We derive optimal order residual-based a posteriori error estimates for fully discrete finite element approximations to the time-dependent Stokes equations. The time discretization uses the two-step backward differentiation formula, and the space discretization is based on inf–sup stable pairs of finite elements, which are allowed to change with time. We show that the time error estimators are of optimal order. This proof of optimality uses time regularity of the semidiscrete (discrete in space) time-dependent Stokes equations. Computational examples are given to confirm the theoretical findings. For completeness, a priori estimates are also presented. 1. Introduction Adaptive algorithms have become very popular and powerful tools for finite element approximations of partial differential equations. They are naturally linked to error control, which analysis is provided by a posteriori estimates. While for elliptic problems the theory and practice of adaptive methods has reached sound maturity, cf. Verfürth (1996) and Ainsworth & Oden (2011), and the references therein, error control for time-dependent problems is still a challenging area. We mention Makridakis & Nochetto (2003), Verfürth (2003), Akrivis et al. (2006), Lakkis & Makridakis (2006), Akrivis et al. (2009) and Lozinski et al. (2009) for some recent contributions to parabolic problems. For time-dependent problems involving elliptic constraints like the incompressible Navier–Stokes equations, there is even less known. In this paper we present a posteriori estimates for the time-dependent Stokes equations discretized in time by the two-step backward differentiation formula (BDF2) and in space by mixed finite elements, which are allowed to change in time. For linear parabolic problems discretized in time by BDF2 (and still continuous in space) we refer to the study by Akrivis & Chatzipantelidis (2010). To the best of our knowledge, however, there is no rigorous a posteriori analysis for the fully discrete Stokes equations using BDF2 in time so far. We round out our analysis by proving the optimality of our temporal error estimators (in case of nonchanging meshes). Here, optimality is understood in the sense that the a posteriori estimators are asymptotically of the same order of convergence as the corresponding a priori error estimates. Our main tool in deriving a posteriori estimates is an application of the elliptic reconstruction technique introduced by Makridakis & Nochetto (2003) for parabolic problems. A main advantage of this approach is that (in our case) any available a posteriori estimate for the stationary Stokes equations can be used to control the main part of the spatial error. The reason behind is that in deriving the error estimates an appropriate auxiliary function, called reconstruction, is constructed and the error between the exact and numerical solution is split into a part between error and reconstruction as well as an error of approximation type for the stationary problem. This approach is in some sense dual to using the Ritz projection (for parabolic problems) in the a priori analysis. More precisely, the reconstruction exhibits two fundamental properties: (i) its difference to the numerical solution can be estimated via known elliptic a posteriori results, and (ii) its solution of an error equation (structurally the same as the Stokes equation) with a right-hand side that can be bounded by a posteriori estimators in an optimal way. This technique has been used in many applications; let us mention Akrivis et al. (2006), Karakatsani & Makridakis (2007), Akrivis et al. (2009), Demlow et al. (2009), Georgoulis et al. (2011) and Bänsch et al. (2012). However, to the best of our knowledge there is no proof of optimality for the estimators for the corresponding fully discrete schemes for the Stokes equations. In the semidiscrete case (continuous in space) optimality theory is easily reduced to the regularity of the continuous solution and data, cf. Akrivis & Chatzipantelidis (2010, Chapter 3) for parabolic problems. For the fully discrete case, some additional technical effort has to be invested, for instance the semidiscrete solution (continuous in time, discrete in space) has to be considered as an intermediate step. Higher-order time derivatives of the semidiscrete solution have to be bounded and consequently we provide corresponding estimates, which we call discrete regularity, in Appendix B. However, due to technical difficulties we show the optimality result only for a constant mesh. For the ease of presentation we follow the study by Akrivis & Chatzipantelidis (2010) and restrict our presented analysis to the case of constant time steps. Variable time steps and an adaptive strategy for time step control are treated in Section 7. An adaptive strategy for-time dependent problems is still a challenging topic. A key problem is the required spatial mesh modifications with time. Their influences in the error estimates are often underestimated. Let us mention the classical article by Dupont (1982) showing that modifying the mesh too carelessly may lead to a wrong solution. In the context of Stokes equations the issue is even more complex. Using standard operators like interpolation or L2 projection for transferring the velocity from one mesh to the next may lead to devastating effects at least for the pressure, cf. Besier & Wollner (2011) and Brenner et al. (2014). From the theoretical point of view a solenoidal projection like Helmholtz or Stokes projection should be used, which is, however, from the computational point of view too expensive. Hence, we use the L2 projection for transferring the velocity from one mesh to the next and quantify the resulting error by our changing mesh estimators. The rest of this article is organized as follows: in Section 2 we introduce notation, recall some results for the Stokes projection and state corresponding a priori estimates. The main results, the a posteriori estimates and their optimality, are stated in Section 3. The corresponding proofs are given in Sections 4 and 5. The performance of the proposed error estimators are discussed in Section 6 by computational experiments. An extension for variable time steps will be presented in Section 7. Finally, for completeness a priori estimates are proved in Appendix A and the theory of discrete regularity is presented in Appendix B. 2. Notation and preliminaries Let $$\varOmega \subset \mathbb{R}^{d}$$ (d = 2, 3) be an open, connected and bounded domain with boundary $$\varGamma$$ := ∂Ω. We consider the incompressible time-dependent Stokes equations on a finite time interval ]0, T[: Find the velocity u and the pressure p fulfilling \begin{align} \partial_{t} u-\nu\varDelta u+\nabla p=f,\qquad \operatorname{div} u=0\;\textrm{in }\varOmega\times]0,T[, \qquad u_{|\varGamma}=0,\qquad u(t=0)=u_{0}\;\textrm{in }\varOmega. \end{align} (2.1) For the sake of simplicity we set ν = 1; u0 is the initial, sufficiently smooth velocity. In the following we assume that the unique solution (u, p) and the right-hand side f to the above system is sufficiently smooth in time and space (compare also the discussion in Appendix B regarding regularity for the continuous system). As usual the standard Sobolev spaces Hm(Ω) (m = 0, ±1, …) are used, whose norms are denoted by ∥⋅∥m. The norm and inner product of L2(Ω) = H0(Ω) is denoted by ∥⋅∥ and ⟨⋅, ⋅⟩, respectively. Moreover, define $${L^{2}_{0}}(\varOmega ):=\{q\in L^{2}(\varOmega ) \,|\, \int _{\varOmega } q(x)\,\mathrm{d}x=0\}$$. To account for the homogeneous Dirichlet boundary condition we introduce $${H_{0}^{1}}(\varOmega ):= \{v\in H^{1}(\varOmega )\: :\: v_{|\varGamma }=0 \}$$. Thanks to the Poincaré inequality, for $$v\in{H_{0}^{1}}(\varOmega )^{d}$$, the norms ∥∇v∥ and ∥v∥1 are equivalent, and thus in what follows ∥∇v∥ is taken as norm in $${H_{0}^{1}}(\varOmega )^{d}$$. Set $$X:={H_{0}^{1}}(\varOmega )^{d}$$ and $$Y:= {L^{2}_{0}}(\varOmega )$$, satisfying the continuous inf–sup condition: $$ \exists\ c>0,\quad\inf_{q\in Y}\sup_{v\in X}\frac{\langle q,\operatorname{div} v\rangle}{\|q\|\|\nabla v\|}\ \geq\ c.$$ Furthermore, define the spaces of solenoidal functions in X and L2(Ω)d, respectively: \begin{align*} J &:= \big\{v\in X\ :\ \operatorname{div} v=0\big\},\\ H &:= \left\{v \in L^{2}(\varOmega)^{d}\; :\operatorname{div} v=0,\ v\cdot n_{|\varGamma}=0\right\}\!. \end{align*} 2.1 Finite element spaces To discretize in time let 0 = t0 < t1 < ⋯ < tN = T be a partition of $$[0,T]$$, $$I_{n}:=(t_{n-1},t_{n}]$$ a time sub-interval and τn := tn − tn−1 the time step. In the following we assume constant time steps. An extension for variable time steps will be presented in Section 7. For the space discretization we use a standard finite element method and introduce a family $$\{\mathcal{T}_{n}\}_{n=0}^{N}$$ of conforming shape regular triangulations of the domain Ω. Each triangulation $$\mathcal{T}_{n}$$ corresponds to a time interval In. This allows arbitrary mesh changes. Let hn(x) be the mesh-size function for each given triangulation $$\mathcal{T}_{n}$$ defined by $$ h_{n}(x):=\operatorname{diam}(K),\quad K\in\mathcal{T}_{n},\ x\in K $$ and denote by $$h_{n}:=\max _{x\in \varOmega _{n}}h_{n}(x)$$. For each triangulation $$\mathcal{T}_{n}$$ we choose a pair of finite element spaces $$X_{n}:=X_{h_{n}}\subseteq X$$ and $$Y_{n}:=Y_{h_{n}}\subseteq Y$$ of X and Y. Assume for all n that Xn × Yn is of order $$k\in \mathbb{N}$$, i.e. for each v ∈ X ∩ Hk+1(Ω)d and q ∈ Y ∩ Hk(Ω) there exists approximations $$\mathcal{I}_{n} v\in X_{n}$$ and $$\mathcal{J}_{n} q\in Y_{n}$$ such that \begin{align} \|v-\mathcal{I}_{n} v\| + h_{n} \left\|\nabla(v-\mathcal{I}_{n} v)\right\|\leq C h_{n}^{k+1}\|v\|_{k+1}, \qquad \|q-\mathcal{J}_{n} q\|_{{L^{2}_{0}}(\varOmega)}\leq C {h_{n}^{k}}\|q\|_{k}. \end{align} (2.2) These are standard conditions satisfied for instance by Taylor–Hood elements $$\mathcal{P}_{k}-\mathcal{P}_{k-1}$$, k ≥ 2. To handle mesh changes let us denote by $$\varPi _{n}:=\varPi _{X_{n}}\ :X\to X_{n}$$ the L2 projection onto Xn. We introduce the linear operator −Δn : X → Xn, such that for all u ∈ X, vn ∈ Xn, −Δnu ∈ Xn with ⟨−Δnu, vn⟩ = ⟨∇u, ∇vn⟩, and the linear operator Bn : Xn → Yn and its transpose $$B_{n}^{\top }:Y_{n}\to X_{n}$$, so that for every couple (vn, qn) ∈ Xn × Yn the identities ⟨Bnvn, qn⟩ = −⟨div vn, qn⟩ and $$\langle v_{n},B_{n}^{\top } q_{n} \rangle =- \langle \operatorname{div} v_{n},q_{n} \rangle $$ hold. In the sequel we assume that the mixed approximation satisfies the discrete inf–sup condition \begin{align} \exists\ c>0,\quad\inf_{q_{n}\in Y_{n}}\sup_{v_{n}\in X_{n}}\frac{\langle B_{n} v_{n},q_{n} \rangle}{\|\nabla v_{n} \|\,\|q_{n} \|}\geq c \end{align} (2.3) with a constant c not depending on n. Remark 2.1 Our analysis is also valid for nonconforming finite elements spaces. In this case the definitions of the discrete operators has to be extended to \begin{align*} -\bigtriangleup_{n}:X\to X_{n}+X,\quad B_{n}: X_{n}+X\to Y_{n},\quad B_{n}^{\top} :Y_{n}\to X_{n}+X,\quad \varPi_{n}:X\to X_{n}+X, \end{align*} and we must require the equivalence of the norms ∥∇u∥ and ∥u∥1 for u ∈ Xn. This is fulfilled for instance for the Crouzeix–Raviart element of lowest order, cf. Crouzeix & Raviart (1973). The space of discretely divergence-free vector fields is denoted by $$ J_{n} := \left\{v_{n} \in X_{n}\: :\: \langle \operatorname{div} v_{n},q_{n}\rangle=0\;\;\forall\ q_{n}\in Y_{n}\right\} =\textrm{ker}\, B_{n} $$ and for u ∈ Hk+1(Ω)d ∩ X and p ∈ Hk(Ω) ∩ Y the discrete Stokes projection$$\mathcal{S}^{u}_{n}$$ is given by $$\mathcal{S}_{n}(u,p)=(\mathcal{S}^{u}_{n}(u,p),\mathcal{S}^{p}_{n}(u,p))\in X_{n}\times Y_{n}$$, solution of the problem \begin{align} \left\langle \nabla \mathcal{S}^{u}_{n}(u,p),\nabla v_{n}\right\rangle -\left\langle \mathcal{S}^{p}_{n}(u,p),\operatorname{div} v_{n}\right\rangle &= \langle \nabla u,\nabla v_{n}\rangle+\langle \nabla p,v_{n}\rangle & &\forall\:v_{n}\in X_{n},\nonumber\\ \left\langle \operatorname{div} \mathcal{S}^{u}_{n}(u,p),q_{n}\right\rangle &=0 & &\forall\:q_{n}\in Y_{n}. \end{align} (2.4) Note that in the case of a discrete pair $$(V_{n},Q_{n})\in X_{n}\times Y_{n}$$ one has $$(\mathcal{S}^{u}_{n} (V_{n},Q_{n}),\mathcal{S}^{p}_{n}(V_{n},Q_{n})) = (V_{n},Q_{n})$$. In what follows C will denote a generic constant (possibly changing values at different places) that is independent of h and τ, but may depend on the solution and the domain. Furthermore, we write $$a\lesssim b$$ for two functions or quantities a, b, whenever there is a generic constant C, such that a ≤ C b. In the study by Heywood & Rannacher (1988) the following estimates are proved: Lemma 2.2 Let $$m_{0}\in \mathbb{N}$$ and $$\partial _{t}^{(m)}u\in H^{k+1}(\varOmega )^{d}\cap X$$ and $$\partial _{t}^{(m)}p\in H^{k}(\varOmega )\cap Y$$ for 0 ≤ m ≤ m0. Then \begin{align*} \left\|u-\mathcal{S}^{u}_{n}(u,p)\right\|+h_{n}\left\|\nabla(u-\mathcal{S}^{u}_{n}(u,p))\right\| &\lesssim h_{n}^{k+1}\left(\|u\|_{k+1}+\|\,p\|_{k}\right)\!,\\ \left\|\partial_{t}^{(m)}\left(u-\mathcal{S}^{u}_{n}(u,p)\right)\right\|+h_{n}\left\|\partial_{t}^{(m)}\nabla\left(u-\mathcal{S}^{u}_{n}(u,p)\right)\right\| &\lesssim h_{n}^{k+1}\left(\left\|\partial_{t}^{(m)}u\right\|_{k+1}+\left\|\partial_{t}^{(m)}p\right\|_{k}\right)\!. \end{align*} Additionally we need an estimate for the associated pressure. Lemma 2.3 Let u ∈ Hk+1(Ω)d ∩ X and p ∈ Hk(Ω) ∩ Y. Then the following pressure estimate holds: $$ \left\|\mathcal{S}^{p}_{n}(u,p)-p\right\|\:\leq \: {h_{n}^{k}}\left(\|u\|_{k+1}+\|\,p\|_{k}\right)\!.$$ Proof. Use the inf–sup stability (2.3) and the interpolation operator $$\mathcal{J}_{h}$$ to obtain \begin{align*} \left\|\mathcal{S}^{p}_{n}(u,p)-p\right\| &\leq \left\|\mathcal{S}^{p}_{n}(u,p)-\mathcal{J}_{n}p\right\| + \left\|(I-\mathcal{J}_{n})p\right\|\\ &\lesssim \sup_{\substack{v_{n}\in X_{n}\\\|\nabla v_{n}\|=1}}\left\langle \mathcal{S}^{p}_{n}(u,p)-\mathcal{J}_{n}p,\operatorname{div} v_{n}\right\rangle + \left\|(I-\mathcal{J}_{n})p\right\|\\ &= \sup_{\substack{v_{n}\in X_{n}\\\|\nabla v_{n}\|=1}}\left\langle \mathcal{S}^{p}_{n}(u,p)-p,\operatorname{div} v_{n}\right\rangle+ \sup_{\substack{v_{n}\in X_{n}\\\|\nabla v_{n}\|=1}}\left\langle(I-\mathcal{J}_{n})p,\operatorname{div} v_{n}\right\rangle+\left\|(I-\mathcal{J}_{n})p\right\|\\ &\lesssim \left\|\nabla \left(u-\mathcal{S}^{u}_{n}(u,p)\right)\right\|+\left\|(I-\mathcal{J}_{n})p\right\|\!, \end{align*} where in the last step equation (2.4) was used. Then the lemma is concluded by equation (2.2) and Lemma 2.2. Let us define the Stokes operator \begin{align} \mathcal{A} u:=-\varPi_{H}\!\bigtriangleup\! u\quad\forall\ u\in D(\mathcal{A})=J\cap H^{2}(\varOmega)^{d}, \end{align} (2.5) where ΠH : L2(Ω)d → H is the Helmholtz projection, see for instance Sohr (2001, III.2). For a given g ∈ L2(Ω)d, $$w=\mathcal{A}^{-1}g$$ is solution of the Stokes system \begin{align} -\varDelta w+\nabla r=g,\;\operatorname{div} w=0\quad\textrm{in }\varOmega,\qquad w=0\quad\textrm{on }\partial\varOmega. \end{align} (2.6) In what follows, we assume that the Stokes problem is H2 × H1-regular. More precisely: Assumption 2.4 Let (w, r) be the solution of equation (2.6). Then (w, r) ∈ H2(Ω)d ∩ X × H1(Ω) ∩ Y and \begin{align} \|w\|_{2}+\|\nabla r\|\lesssim\|g\|. \end{align} (2.7) 2.2 Time discretization For the time discretization we introduce the finite difference operators $$\bar \partial ^{k}$$. For a sequence $$(v^{n})_{n=0}^{N}$$ with vn ∈ Xn we set recursively $$ \bar\partial v^{n}:=\frac{v^{n}-v^{n-1}}{\tau},\qquad \bar\partial^{2} v^{n}:=\bar\partial(\bar\partial v^{n}), \qquad \bar\partial^{k} v^{n}=\bar\partial(\bar\partial^{k-1}v^{n}),$$ and the BDF2 time discretization operator by \begin{align} \bar\partial^{B} v^{n}:= \frac{\tau}{2}\bar\partial^{2} v^{n} +\bar\partial v^{n} =\frac{3v^{n} - 4v^{n-1} + v^{n-2}}{2\tau}. \end{align} (2.8) An extension of $$\bar \partial ^{B}$$ to arbitrary time steps will be presented in Section 7. Note that in general $$\bar \partial v^{n}\notin X_{n}$$. Hence, define $$ {\bar\partial_{n}^{k}} v^{n}:=\varPi_{n}\bar\partial^{k} v^{n}\ \in\ X_{n},\qquad{\bar\partial_{n}^{B}} v^{n} :=\varPi_{n}\bar\partial^{B} v^{n}\ \in\ X_{n}.$$ For any Banach space Z, $$L^{p}((t_{1},t_{2});Z)$$ is the space of Lebesgue measurable Z-valued functions u : (t1, t2) → Z with $$ \|u\|_{L^{p}((t_{1},t_{2});Z)} := \left(\int_{t_{1}}^{t_{2}}\left\|u(t)\right\|^{p}_{Z}\mathrm{d}t\right)^{1/p},\qquad \|u\|_{L^{\infty}((t_{1},t_{2});Z)} := \textrm{ess} \sup_{t_{1}<t<t_{2}} \left\|u(t)\right\|_{Z}\!, $$$$1\leq p<\infty $$, and $$\|\cdot \|_{L^{p}(Z)}:=\|\cdot \|_{L^{p}((0,T);Z)}$$. Furthermore, for a sequence $$v^{i_{1}},\ldots ,v^{i_{m}}\in Z$$, $$i_{1},\ldots ,i_{m}\in \mathbb{N}$$, i1 ≤⋯ ≤ im, let $$ \|u\|_{\ell^{p}(i_{1},i_{m};Z)} := \left(\sum_{n=i_{1}}^{i_{m}}\tau\|u^{n}\|^{p}_{Z}\right)^{1/p}, \qquad \|u\|_{\ell^{\infty}(i_{1},i_{m};Z)} := \max_{i_{1}\leq n\leq i_{m}} \|u^{n}\|_{Z}, $$$$1\leq p<\infty $$, and set $$\|u\|_{\ell ^{p}(Z)}:=\|u\|_{\ell ^{p}(1,N;Z)}$$. Throughout this paper we denote space discrete quantities by capitals and continuous quantities by small letters. 2.3 The scheme: BDF2 for the fully discrete Stokes equation In order to discretize equation (2.1) define a sequence of approximate velocities Un ∈ Xn and pressures Pn ∈ Yn fulfilling: let X0 = X1 = X2 and U0 ∈ X0, U1 ∈ X1 be given; for n ≥ 2 determine $$\big (U^{n},P^{n}\big )\in X_{n}\times Y_{n}$$ from \begin{align} \boxed{{\bar\partial_{n}^{B}} U^{n}-\varDelta_{n} U^{n} + B_{n}^{\top} P^{n}= F^{n},\qquad B_{n} U^{n}=0,} \end{align} (2.9) where Fn := Πnf(tn). To initialize the scheme, values for U0, P0 and U1 are needed. To this end determine p0 := p(0) from u0 by taking the divergence of the first equation of equation (2.1) for t = 0 resulting in a Poisson problem for p0 with, say, Neumann boundary values obtained by testing the first equation of equation (2.1) (evaluated on the boundary) with the normal vector. Set $$(U^{0},P^{0}):=\mathcal{S}_{0}(u_{0},p_{0})=(\mathcal{S}_{0}^{u}(u_{0},p_{0}),\mathcal{S}_{0}^{p} (u_{0},p_{0} ))$$ and calculate $$(U^{1},P^{1})\in X_{1}\times Y_{1}$$ by the trapezoidal scheme: \begin{align} \bar\partial_{1} U^{1}-\varDelta_{1} U^{1/2}+B_{1}^{\top} P^{1/2}=F^{1/2},\qquad B_{1} U^{1} =0, \end{align} (2.10) where $$U^{1/2}:=\frac{1}{2}(U^{1}+U^{0})$$. The above discretization yields the following a priori estimates for constant meshes: Theorem 2.5 (A priori estimates for the time-dependent Stokes equations) Suppose the exact solution (u, p) of equation (2.1) is sufficiently regular. Let Xn ≡ Xh ⊆ X for all n ≥ 0 with mesh size h and (Un, Pn) be the solution of equation (2.9) initialized by (2.10). Then \begin{align} \|U-u\|^{2}_{\ell^{\infty}(L^{2})} \lesssim h^{2(k+1)}(A^{0}+A^{1}) \;+\; \tau^{4} \left\|{\partial_{t}^{3}} u\right\|^{2}_{L^{1}(H^{-1})}\!, \end{align} (2.11) \begin{align} {\hskip-22pt}\left\|\nabla(U-u)\right\|^{2}_{\ell^{2}(L^{2})} \lesssim h^{2k}(A^{0}+h^{2} A^{1}) \;+\; \tau^{4} \left\|{\partial_{t}^{3}} u\right\|^{2}_{L^{1}(H^{-1})}\!, \end{align} (2.12) \begin{align} {\hskip-100pt}\left\|\nabla(U-u)\right\|^{2}_{\ell^{\infty}(L^{2})}+\|P-p\|^{2}_{\ell^{2}(L^{2})} \lesssim h^{2k}(A^{0}+h^{2} A^{1}) \;+\; \tau^{4} \left\|{\partial_{t}^{3}} u\right\|^{2}_{L^{1}(L^{2})}\!, \end{align} (2.13) where $$A^{i} := \|{\partial ^{i}_{t}} u\|^{2}_{L^{\infty }(H^{k+1})}+\|{\partial ^{i}_{t}} p\|^{2}_{L^{\infty }(H^{k})}$$, i = 0, 1. The proof is outlined in Appendix A. 3. A posteriori estimates for the fully discrete scheme In this section we introduce a posteriori error estimators and state corresponding a posteriori estimates in Theorem 3.4. The proof of the theorem is outlined in Section 4. Furthermore, in Theorem 3.6 we state that our time estimators are optimal, i.e. that they have the same asymptotic behavior as the a priori estimates. The theorem is proven in Section 5 and utilizes discrete regularity estimates, which are provided in Appendix B. In order to derive optimal a posteriori estimates we need an approximation U(t) to u(t) ∀ t ∈ ]0, T[ to the right order. To this end we introduce a piecewise linear function, interpolating a sequence {Un}n=0, …, N at the time nodes, by \begin{align} U(t):=U^{n}+(t-t_{n})\bar\partial U^{n},\quad t\in I_{n},\; n=1,\ldots,N. \end{align} (3.1) Even though the interpolant U(t), defined in equation (3.1), is a second-order approximation, its residual is of first order only, cf. the study by Akrivis & Chatzipantelidis (2010) for parabolic problems. We follow the study by Akrivis & Chatzipantelidis (2010) and modify the interpolant U(t) using the piecewise quadratic three-point reconstruction$$\hat{U}(t)$$, interpolating a sequence {Un}n=0, …, N in tn−2, tn−1, tn: \begin{align} \hat{U}(t) := U(t)+\frac{1}{2}(t-t_{n})(t-t_{n-1})\bar{\partial}^{2} U^{n},\quad t\in I_{n},\; n=2,\ldots,N. \end{align} (3.2) Its time derivative is given by \begin{align} \partial_{t}\hat{U}(t) =\bar\partial^{B} U^{n}+(t-t_{n})\bar\partial^{2} U^{n},\quad t\in I_{n},\; n=2,\ldots,N. \end{align} (3.3) This three-point reconstruction was also proposed in the study by Lozinski et al. (2009) for the Crank–Nicolson scheme. Our main tool in deriving estimates is an application of the elliptic reconstruction technique introduced by Makridakis & Nochetto (2003) for parabolic problems and extended to the Stokes equations in the study by Karakatsani & Makridakis (2007). To this end define the Stokes Reconstruction$$\mathcal{R}_{u}^{n}$$ of a discrete function $$(U^{n},P^{n})\in X_{n}\times Y_{n}$$ by $$ \mathcal{R}_{u}^{n}:=\mathcal{A}^{-1}\left(-\varDelta_{n} U^{n}+B_{n}^{\top} P^{n}-F^{n}+f(t_{n})\right)\!,$$ i.e. there is a unique $$\mathcal{R}_{p}^{n}\in Y$$ such that $$(\mathcal{R}_{u}^{n},\mathcal{R}_{p}^{n})\in X\times Y$$ is the continuous solution of \begin{align} -\varDelta \mathcal{R}_{u}^{n}+\nabla \mathcal{R}_{p}^{n}=-\varDelta_{n} U^{n}+B_{n}^{\top} P^{n}-F^{n}+f(t_{n}),\qquad\operatorname{div} \mathcal{R}_{u}^{n}=0,\qquad{\mathcal{R}_{u}^{n}}{|_{\varGamma}}=0. \end{align} (3.4) The next lemma, which is straightforward, though crucial for our analysis, shows that the finite element approximation of $$(\mathcal{R}_{u}^{n},\mathcal{R}_{p}^{n})$$ in equation (3.4) is nothing else but $$(U^{n},P^{n})$$ itself. Lemma 3.1 Let $$ (V^{n},R^{n})\in X_{n}\times Y_{n}$$ be the finite element solution of equation (3.4), namely $$ -\varDelta_{n} V^{n}+B_{n}^{\top} R^{n}=-\varDelta \mathcal{R}_{u}^{n} + \nabla \mathcal{R}_{p}^{n} = -\varDelta_{n} U^{n}+B_{n}^{\top} P^{n}-F^{n}+f(t_{n}),\qquad B_{n} V^{n}=0. $$ Then, $$ V^{n} = U^{n}\qquad\textrm{and}\qquad R^{n} = P^{n}.$$ Proof. See Karakatsani & Makridakis (2007, Lemma 1.3). The above lemma implies that the difference of a given finite element function and its Stokes reconstruction can be estimated by well-known estimates for the stationary Stokes system. We thus make the assumption on the availability of such a posteriori error estimates. Assumption 3.2 Let $$(\mathcal{R}_{u}^{n},\mathcal{R}_{p}^{n})\in X\times Y$$ and $$(U^{n},P^{n})\in X_{n}\times Y_{n}$$ be as in equation (3.4). We assume that there exist a posteriori estimator functions $${\eta ^{n}_{u}} = {\eta ^{n}_{u}}(U^{n};Z)$$, $${\eta ^{n}_{p}}={\eta ^{n}_{p}}(P^{n};Y)$$, such that \begin{align} \begin{aligned} \left\|U^{n}-\mathcal{R}_{u}^{n}\right\|_{Z}^{2} &\leq{\eta^{n}_{u}}(U^{n};Z),\qquad Z\in\left\{J^{\ast},L^{2},{H_{0}^{1}}\right\}\!,\\ \left\|P^{n}-\mathcal{R}_{p}^{n}\right\|^{2} &\leq{\eta^{n}_{p}}(P^{n};Y). \end{aligned} \end{align} (3.5 ) Additionally there exists an estimator function $$\eta ^{0}(Z)$$ of the initial approximation error, such that $$ \left\|u_{0}-\mathcal{S}^{u}_{0}(u_{0},p_{0})\right\|^{2}_{Z} \leq \eta^{0}(Z).$$ Note that for a large class of finite element spaces Xn, Yn such estimators are available, see for example Verführt (2013). In the study by Bank & Welfert (1990) a comparison of several a posteriori error estimates for the stationary Stokes problem is given. Let us now summarize the notation. First we denote space discrete quantities by capitals and continuous quantities by small letters. Un ∈ Xn denotes the solution of equation (2.9) at discrete time tn, whereas U(t) and $$\hat{U}(t)$$ are interpolants of Un and Un−1 in the corresponding time interval In. Equally we apply this notation for the Stokes reconstruction $$\mathcal{R}_{u}^{n}$$, which is defined in equation (3.4) at discrete time tn and $$\mathcal{R}_{u}(t)$$ or $$\hat{\mathcal{R}}_{u}(t)$$ are the interpolation, respectively three-point reconstruction, in the time interval In. Now we are able to define the error estimators. Definition 3.3 (Error estimators). We assume X2 = X1 = X0 and set to simplify the presentation $$U^{-1}:=U^{1}-2\tau _{0}(\varDelta _{0} U^{0}-B_{0}^{\top } P^{0}+F^{0})\in X_{0}$$ and $$U^{-2}:=-6\tau _{0}(\varDelta _{0} U^{0}-B_{0}^{} P^{0}+F^{0})-3U^{0}+4U^{1}\in X_{0}$$, cf. Lemma 4.5. Let us introduce the time error estimators \begin{align*} \mathcal{E}_{1} &:= \sum_{n=2}^{N}\tau\left\|{\bar\partial_{n}^{2}} U^{n}\right\|^{2}, & \mathcal{E}_{2} &:= \sum_{n=2}^{N}\frac{1}{\tau}\left\|{\bar\partial_{n}^{2}} U^{n}-\bar\partial^{2}_{n-1}U^{n-1}\right\|^{2}_{J^{\ast}}, & \mathcal{E}_{3} &:= \max_{2\leq n\leq N}\left\|{\bar\partial_{n}^{2}} U^{n}\right\|^{2},\\ \mathcal{E}_{6} &:= \sum_{n=2}^{N}\frac{1}{\tau}\left\|{\bar\partial_{n}^{2}} U^{n}-\bar\partial^{2}_{n-1}U^{n-1}\right\|^{2}, & \mathcal{E}_{7} &:= \max_{2\leq n\leq N}\left\|{\nabla\bar\partial_{n}^{2}} U^{n}\right\|^{2}, & & \end{align*} >the Stokes reconstruction error estimators $$ \mathcal{E}_{4} := \sum_{n=2}^{N}\tau\left\|\bar\partial^{2}f(t_{n})-{\bar\partial^{2}_{n}}\bar\partial^{B} U^{n}\right\|^{2}_{J^{\ast}},\qquad \mathcal{E}_{5} := \sum_{n=2}^{N}\tau\left\|\bar\partial^{2}f(t_{n})-{\bar\partial^{2}_{n}}\bar\partial^{B} U^{n}\right\|^{2}, $$ the space error estimators \begin{align*} \varLambda_{1} &:= \sum_{n=0}^{N}\tau\ {\eta_{u}^{n}}(U^{n};L^{2}), & \varLambda_{2} &:= \max_{0\leq n\leq N}{\eta_{u}^{n}}(U^{n};L^{2}),\\ \varLambda_{3} &:= \sum_{n=1}^{N}\tau\ {\eta_{u}^{n}}\left(\bar\partial_{n} U^{n};J^{\ast}\right)\!, & \varLambda_{4} &:= \sum_{n=1}^{N}\tau\ {\eta_{u}^{n}}\left(U^{n};{H^{1}_{0}}\right)\!,\\ \varLambda_{5} &:= \max_{0\leq n\leq N}{\eta_{u}^{n}}\left(U^{n};{H_{0}^{1}}\right)\!, & \varLambda_{6} &:= \sum_{n=1}^{N}\tau\ {\eta_{u}^{n}}\left(\bar\partial_{n} U^{n};L^{2}\right)\!,\\ \varLambda_{7} &:= \sum_{n=0}^{N}\tau\ {\eta_{p}^{n}}(P^{n};Y), & \end{align*} the changing mesh estimators \begin{align*} \varXi_{1} &:= \sum_{n=2}^{N}\tau \xi_{B}^{n,n}, & \varXi_{4} &:= \sum_{n=2}^{N}\frac{{h_{n}^{2}}}{\tau}\left(\xi_{B}^{n,n}+\xi_{B}^{n-1,n} +\xi_{B}^{n-2,n} \right)\!, & \varXi_{6} &:= \sum_{n=2}^{N}\frac{1}{\tau}\xi_{B}^{n,n}, \\ \varXi_{2} &:= \sum_{n=2}^{N}\frac{{h_{n}^{2}}}{\tau}\xi_{B}^{n,n}, & \varXi_{5} &:= \sum_{n=2}^{N}\frac{1}{\tau}\left(\xi_{B}^{n,n}+\xi_{B}^{n-1,n} +\xi_{B}^{n-2,n} \right)\!, & \varXi_{7} &:= \tau^{4}\max_{2\leq n\leq N}\left\|\nabla\left(\bar\partial^{2}-{\bar\partial_{n}^{2}}\right)U^{n}\right\|^{2}\!, \\ \varXi_{3} &:= \max_{2\leq n\leq N}\xi_{B}^{n,n}, & & & \end{align*} where the local BDF2-changing mesh operator $$\xi _{B}^{a,b}$$ is defined by \begin{align} \xi_{B}^{a,b}:= \left(\|U^{a}\|^{2}-\|\varPi_{b} U^{a}\|^{2}\right) + \left(\|U^{a-1}\|^{2}-\|\varPi_{b} U^{a-1}\|^{2}\right) + \left(\|U^{a-2}\|^{2}-\|\varPi_{b} U^{a-2}\|^{2}\right)\!, \end{align} (3.6) and the data approximation error estimators \begin{align*} \mathcal{F}_{1} :=&\, \sum_{n=1}^{N}\int_{t_{n-1}}^{t_{n}}\!\left\|\,f(t_{n})+(t-t_{n})\bar\partial f(t_{n})-f(t)\right\|^{2}_{J^{\ast}}\mathrm{d}t,\quad \mathcal{F}_{2} :=\sum_{n=1}^{N}\int_{t_{n-1}}^{t_{n}}\!\left\|\,f(t_{n})+(t-t_{n})\bar\partial f(t_{n})-f(t)\right\|^{2}\!,\\ {\textit{I}}_{1} :=&\, \tau^{3}\|\varPsi_{1}\|^{2}_{J^{\ast}},\qquad\qquad \qquad\qquad\qquad\qquad\qquad\quad\ {\textit{I}}_{2} := \tau^{3}\|\varPsi_{1}\|^{2}, \end{align*} where $$\varPsi _{1}:= \bar \partial ^{2} U^{2}-\varDelta _{1} \bar \partial U^{1}+ B_{1}^{\top }\bar \partial P^{1}-\bar \partial F^{1}$$. In the following theorem a posteriori estimates in various norms are stated. Theorem 3.4 (A posteriori error estimates for the time-dependent Stokes system). Let $$(U,P)$$ be the solution of equation (2.9) initialized by equation (2.10) and $$(u,p)$$ the exact solution of equation (2.1). Then \begin{align} {\hskip-56pt}\|U-u\|^{2}_{L^{2}(L^{2})} \lesssim \eta^{0}(J^{\ast}) + \tau^{4}(\mathcal{E}_{1} + \mathcal{E}_{2}) + \varLambda_{1} +\varXi_{1}+\varXi_{2} + \mathcal{F}_{1} + {\textit{I}}_{1}, \end{align} (3.7) \begin{align} {\hskip32pt}\|U-u\|^{2}_{L^{\infty}(L^{2})} \lesssim \eta^{0}(L^{2}) + \tau^{4}(\mathcal{E}_{2}+\mathcal{E}_{3}+\mathcal{E}_{4}) + \varLambda_{2} + \varLambda_{3} + \varXi_{2}+\varXi_{3}+\varXi_{4} + \mathcal{F}_{1}+ {\textit{I}}_{1}, \end{align} (3.8) \begin{align} {\hskip-45pt}\|\nabla(U-u)\|^{2}_{L^{2}(L^{2})} \lesssim \eta^{0}(L^{2}) + \tau^{4}(\mathcal{E}_{2}+\mathcal{E}_{4}) + \varLambda_{3} + \varLambda_{4} + \varXi_{2}+\varXi_{4} + \mathcal{F}_{1}+ {\textit{I}}_{1}, \end{align} (3.9) \begin{align} \|\nabla(U-u)(t)\|^{2}_{L^{\infty}(L^{2})} \lesssim \eta^{0}\left({H_{0}^{1}}\right) \!+\! \tau^{4}(\mathcal{E}_{5}\!+\!\mathcal{E}_{6}\!+\!\mathcal{E}_{7}) \!+\! \varLambda_{5}\!+\!\varLambda_{6} \!+\! \varXi_{5}\!+\!\varXi_{6}\!+\!\varXi_{7} \!+\! \mathcal{F}_{2}\!+\! {\textit{I}}_{2}, \!\!\!\qquad\end{align} (3.10) \begin{align} {\hskip-22pt}\|P-p\|^{2}_{L^{2}(L^{2})} \lesssim \eta^{0}\left({H_{0}^{1}}\right) + \tau^{4}(\mathcal{E}_{5}+\mathcal{E}_{6}) + \varLambda_{6} + \varLambda_{7} + \varXi_{5}+\varXi_{6}+ \mathcal{F}_{2}+ {\textit{I}}_{2}, \end{align} (3.11) where the estimators are defined in Definition 3.3 and $$\eta ^{0}(Z)$$ fulfills Assumption 3.2. Remark 3.5 (Negative norms). The time error estimator $$\mathcal{E}_{2}$$, Stokes reconstruction estimator $$\mathcal{E}_{4}$$ and data approximation error estimator $$\mathcal{F}_{1}$$ are defined with respect to a negative norm, as usual for error estimates of BDF2 time discretizations, cf. Akrivis & Chatzipantelidis (2010). From a computational point of view this is completely prohibitive, since its evaluation may be computationally more expensive than solving the discrete scheme. However, there are possibilities for estimating it. The simplest way is to bound $$\|\cdot \|_{J^{\ast }}$$ by ∥⋅∥, which can be pessimistic in practice. Another way is to approximate the J* norm in an adequate way. In our case the estimator $$\mathcal{E}_{4}$$ (for the sake of convenience assume constant meshes) can be handled further in the following way. We approximate the norm J* by the discrete norm $$J^{\ast }_{h}$$: $$ \left\|\bar\partial^{2} f(t_{n})-\bar\partial^{2}\bar\partial^{B}U^{n}\right\|_{J^{\ast}} \approx \left\|\bar\partial^{2} f(t_{n})-\bar\partial^{2}\bar\partial^{B}U^{n}\right\|_{J_{h}^{\ast}}\!.$$ Now we use the triangle inequality and insert the scheme to get \begin{align*} \left\|\bar\partial^{2} f(t_{n})-\bar\partial^{2}\bar\partial^{B}U^{n}\right\|_{J_{h}^{\ast}} &\leq \left\|\bar\partial^{2}\left(\,f(t_{n})-F^{n}\right)\right\|_{J_{h}^{\ast}}+\left\|\bar\partial^{2}(F^{n}-\bar\partial^{B} U^{n})\right\|_{J_{h}^{\ast}}\\ &= \left\|\bar\partial^{2}\left(\,f(t_{n})-F^{n}\right)\right\|_{J_{h}^{\ast}}+\left\|\bar\partial^{2}(-\varDelta_{h} U^{n})\right\|_{J_{h}^{\ast}}\\ &\leq \left\|\bar\partial^{2}\left(\,f(t_{n})-F^{n}\right)\right\|_{J_{h}^{\ast}}+\|\nabla\bar\partial^{2} U^{n}\|. \end{align*} For the handling of the data approximation errors in negative norms let us mention the discussion in the study by Ern et al. (2016). Our next theorem states the optimality of the time error estimators $$\mathcal{E}_{i}$$. We understand optimality in the sense that the time error estimators have the same asymptotic behavior as corresponding a priori estimates. In view of Theorem 3.4 the time error estimators are of second order, which corresponds to the a priori estimates in Theorem 2.5. However, this requires that the estimators are independent of the time step and mesh size, i.e. there exists a constant C independent of τ and h, such that $$\mathcal{E}_{i}\ \leq \ C$$. Theorem 3.6 (Stability of the time estimators). Suppose the exact solution (u, p) and data f, u0 are sufficiently regular. Let $$(X_{h},Y_{h})\subseteq X\times Y$$ be a mixed finite element approximation on a quasi-uniform triangulation $$\mathcal{T}_{h}$$ of order k ≥ 2 and let (Un, Pn) be solution of equation (2.9) with $$(X_{n},Y_{n})= (X_{h},Y_{h} )$$ ∀ n = 0, …, N. Then the time error estimators $$\mathcal{E}_{1}$$–$$\mathcal{E}_{7}$$ defined in (3.3) are stable, i.e. $$ \mathcal{E}_{1}+\mathcal{E}_{2}+\mathcal{E}_{3}+\mathcal{E}_{4} + \mathcal{E}_{5} + \mathcal{E}_{6} + \mathcal{E}_{7} \leq C(u,p,f),$$ where C is independent of the time step τ and the mesh size h. The proof is outlined in Section 5. Due to its technical difficulties we have to restrict our approximations in equation (2.9) to a constant, quasi-uniform mesh. This assumption, of course, does not go together with spatial adaptivity. However, this restriction is only necessary for the theoretical result of stability, not for the proof of the a posteriori estimates in Theorem 3.4. 4. Proofs of the a posteriori estimates In this section the proof of the a posteriori error estimates in Theorem 3.4 is presented. Let us start with some auxiliary statements. In the first step we deduce the error equation using the three-point reconstruction $$\hat{U}(t)$$: Lemma 4.1 (Error equation). Let n ≥ 2 and t ∈ In. Then we have the following identity in H−1: \begin{align} \partial_{t}(\hat{U}-u)(t)-\varDelta (U-u)(t) + \nabla (P-p)(t) =-\varDelta( U-\mathcal{R}_{u})(t) + \nabla(P-\mathcal{R}_{p})(t)+\Upsilon(t), \end{align} (4.1) where for t ∈ In \begin{align} \Upsilon(t) := f(t_{n})-f(t)+ (t-t_{n})\left(\bar\partial^{2} U^{n} -\varDelta\bar\partial \mathcal{R}_{u}^{n} + \nabla\bar\partial \mathcal{R}_{p}^{n}\right)+\left(\bar\partial^{B}-{\bar\partial^{B}_{n}}\right)U^{n}. \end{align} (4.2) Proof. Using the definitions of $$\hat{U}(t)$$ in equation (3.2) and the Stokes reconstruction inequation (3.4) one has for n ≥ 2, \begin{align*} \partial_{t}\hat{U}(t)-\varDelta U(t)+\nabla P(t)=&\ \bar\partial^{B} U^{n} + (t-t_{n})\bar\partial^{2} U^{n} -\varDelta(U-\mathcal{R}_{u})(t) +\nabla (P-\mathcal{R}_{p})(t) -\varDelta \mathcal{R}_{u}^{n} + \nabla \mathcal{R}_{p}^{n}\\ &+ (t-t_{n})\left(-\varDelta\bar\partial \mathcal{R}_{u}^{n} + \nabla\bar\partial \mathcal{R}_{p}^{n}\right)\\ =&\ {\bar\partial_{n}^{B}} U^{n} -\varDelta_{n} U^{n} + B_{n}^{\top} P^{n} -F^{n}+f\big(t_{n}\big) + (t-t_{n})\left(\bar\partial^{2} U^{n} -\varDelta\bar\partial \mathcal{R}_{u}^{n} + \nabla\bar\partial \mathcal{R}_{p}^{n}\right)\\ &-\varDelta(U-\mathcal{R}_{u})(t) +\nabla (P-\mathcal{R}_{p})(t)+\left(\bar\partial^{B}-{\bar\partial^{B}_{n}}\right)U^{n}\\ =&\ f(t_{n})+ (t-t_{n})\left(\bar\partial^{2} U^{n} -\varDelta\bar\partial \mathcal{R}_{u}^{n} + \nabla\bar\partial \mathcal{R}_{p}^{n}\right)-\varDelta(U-\mathcal{R}_{u})(t) \\ &+\nabla (P-\mathcal{R}_{p})(t)+\left(\bar\partial^{B}-{\bar\partial^{B}_{n}}\right)U^{n}.\\ \end{align*} Now subtract equation (2.1) to finish the proof. 4.1 Estimating initial errors Next we present upper bounds for the initial errors. Lemma 4.2 (Initial a posteriori error estimates). Let $$(u,p)$$ be the exact solution of equation (2.1), X2 = X1 = X0 and $$ (U^{0},P^{0}):=(\mathcal{S}^{u}_{n}(u_{0},p_{0}),\mathcal{S}^{p}_{n}(u_{0},p_{0}))$$ the solution of equation (2.4). Let $$(U^{1},P^{1})$$ be given by equation (2.10) and $$(U^{2},P^{2})$$ by equation (2.9). Then \begin{align} &\|\hat{U}-u\|^{2}_{L^{\infty}((0,t_{1});J^{\ast})}+\|U-u\|^{2}_{L^{2}((0,t_{1});L^{2})}\nonumber\\ &\quad\lesssim \eta^{0}(J^{\ast})+\int_{0}^{t_{1}}\left(\left\|(\hat{U}-U)(t)\right\|^{2} +\left\|(U-\mathcal{R}_{u})(t)\right\|^{2}+ \left\|\Upsilon(t)\right\|^{2}_{J^{\ast}}\right)\mathrm{d}t, \end{align} (4.3) \begin{align} &\|U-u\|^{2}_{L^{\infty}((0,t_{1});L^{2})}\nonumber\\ &\quad\lesssim \eta^{0}(L^{2})+ \max_{t\in[0,t_{1}]}\left\|(U-\hat{U})(t)\right\|^{2}+\max_{t\in[0,t_{1}]}\left\|(\hat{\mathcal{R}}_{u}-\hat{U})(t)\right\|^{2}\nonumber \\ &\qquad+\int_{0}^{t_{1}}\left(\left\|\partial_{t}(\hat{\mathcal{R}}_{u}-\hat{U})(t)\right\|^{2}_{J^{\ast}} +\left\|\nabla(\mathcal{R}_{u}-\hat{\mathcal{R}}_{u})(t)\right\|^{2}+\left\|\Upsilon(t)\right\|^{2}_{J^{\ast}}\right)\mathrm{d}t, \end{align} (4.4) \begin{align} &\left\|\nabla(U-u)\right\|^{2}_{L^{2}((0,t_{1});L^{2})}\nonumber\\ &\quad\lesssim \eta^{0}(L^{2})+\int_{0}^{t_{1}}\left\|\nabla(U-\mathcal{R}_{u})(t)\right\|^{2}\mathrm{d}t\nonumber\\ &\qquad+\int_{0}^{t_{1}}\left(\left\|\partial_{t}(\hat{\mathcal{R}}_{u}-\hat{U})(t)\right\|^{2}_{J^{\ast}} +\left\|\nabla(\mathcal{R}_{u}-\hat{\mathcal{R}}_{u})(t)\right\|^{2}+\left\|\varUpsilon(t)\right\|^{2}_{J^{\ast}}\right)\mathrm{d}t. \end{align} (4.5) \begin{align} &\left\|\nabla(U-u)\right\|^{2}_{L^{\infty}((0,t_{1});L^{2})}\nonumber\\ &\quad\lesssim \eta^{0}\left({H^{1}_{0}}\right) + \max_{t\in[0,t_{1}]}\left\|\nabla(U-\hat{U})(t)\right\|^{2}+\max_{t\in[0,t_{1}]}\left\|\nabla(\hat{\mathcal{R}}_{u}-\hat{U})(t)\right\|^{2}\nonumber\\ &\qquad+\int_{0}^{t_{1}}\left(\left\|\partial_{t}(\hat{\mathcal{R}}_{u}-\hat{U})(t)\right\|^{2} +\left\|\varDelta(\mathcal{R}_{u}-\hat{\mathcal{R}}_{u})(t)\right\|^{2}+\left\|\varUpsilon(t)\right\|^{2}\right)\mathrm{d}t, \end{align} (4.6) \begin{align} &\|P-p\|^{2}_{L^{2}((0,t_{1});L^{2})}\nonumber\\ &\lesssim \eta^{0}\left({H^{1}_{0}}\right)+\int_{0}^{t_{1}}\left\|(P-\mathcal{R}_{p})(t)\right\|^{2}\mathrm{d}t\nonumber\\ &\qquad+\int_{0}^{t_{1}}\left(\left\|\partial_{t}(\hat{\mathcal{R}}_{u}-\hat{U})(t)\right\|^{2} +\left\|\varDelta(\mathcal{R}_{u}-\hat{\mathcal{R}}_{u})(t)\right\|^{2}+\left\|\varUpsilon(t)\right\|^{2}\right)\mathrm{d}t, \end{align} (4.7) where η0 is defined in Assumption 3.2 and \begin{align} \varUpsilon := (t-t_{1/2})\left(\bar\partial^{2}U^{2} - \varDelta_{1}\bar\partial U^{1}+ B_{1}^{\top} \bar\partial P^{1} -\bar\partial F^{1}\right)+ f(t_{1})+(t-t_{1})\bar\partial f(t_{1})-f(t),\qquad t\in I_{1}. \end{align} (4.8) Proof. For t ∈ I1 = (0, t1] define the three-point reconstruction \begin{align} \hat{U}(t) = U(t)+\frac{1}{2}(t-t_{0})(t-t_{1})\bar\partial^{2} U^{2},\qquad \partial_{t}\hat{U}(t)=\bar\partial U^{1}+(t-t_{1/2})\bar\partial^{2} U^{2}, \end{align} (4.9) where the identity $$t-t_{1/2}=\frac{\tau }{2}+(t-t_{1})$$ was used. The interpolant $$\hat{U}(t)$$ fulfills (in H−1) \begin{align*} &\partial_{t} \hat{U}(t)-\varDelta U(t)+\nabla P(t) \\ &\quad=\bar\partial U^{1}+(t-t_{1/2})\bar\partial^{2} U^{2}-\varDelta(U-\mathcal{R}_{u})(t)+\nabla (P-\mathcal{R}_{p})(t) \\ &\qquad-\varDelta \mathcal{R}_{u}^{1}+\nabla \mathcal{R}_{p}^{1} + (t-t_{1})\left(-\varDelta\bar\partial \mathcal{R}_{u}^{1}+\nabla \bar\partial \mathcal{R}_{p}^{1}\right)\\ &\quad=\varDelta_{1} U^{1/2}-B_{1}^{\top} P^{1/2}+F^{1/2}+(t-t_{1/2})\bar\partial^{2} U^{2}-\varDelta(U-\mathcal{R}_{u})(t)+\nabla (P-\mathcal{R}_{p})(t) \\ &\qquad-\varDelta_{1} U^{1}+B_{1}^{\top} P^{1}-F^{1}+f(t_{1}) + (t-t_{1})\bar\partial\left(-\varDelta_{1} U^{1}+B_{1}^{\top} P^{1}-F^{1}+f(t_{1})\right)\\ &\quad=(t-t_{1/2})\left(\bar\partial^{2} U^{2}-\varDelta_{1}\bar\partial U^{1}+ B_{1}^{\top}\bar\partial P^{1}-\bar\partial F^{1}\right)+f(t_{1})+(t-t_{1})\bar\partial f(t_{1})\\ &\qquad-\varDelta(U-\mathcal{R}_{u})(t)+\nabla (P-\mathcal{R}_{p})(t). \end{align*} Note we have used the fact that X2 = X1 = X0. Subtract equation (2.1) from the above identity to get the error equation for t ∈ I1 \begin{align} \partial_{t}(\hat{U}-u)(t)-\varDelta (U-u)(t)+\nabla (P-p)(t) =\varUpsilon(t)-\varDelta(U-\mathcal{R}_{u})(t)+\nabla (P-\mathcal{R}_{p})(t), \end{align} (4.10) where $$\varUpsilon $$ is defined inequation (4.8). To prove equation (4.3) test equation (4.10) by $$w(t):=\mathcal{A}^{-1}(\hat{U}-u)(t)$$, i.e. (w(t), q(t)) is solution of $$ -\varDelta w(t)+\nabla q(t)=(\hat{U}-u)(t),\quad \textrm{div}\, w(t)=0,\quad w(t)|_{\varGamma}=0. $$ For the first term on the left-hand side, one gets $$ \left\langle\partial_{t}(\hat{U}-u)(t),w(t)\right\rangle = \frac{1}{2}\frac{d}{\mathrm{d}t}\left\|\nabla w(t)\right\|^{2}\!. $$ The second term can be treated with the help of the identity \begin{align} 2\langle a-b,c-b\rangle=\|a-b\|^{2}+\|c-b\|^{2}-\|a-c\|^{2} \end{align} (4.11) and the fact that the interpolant of the Stokes reconstruction $$\mathcal{R}_{u}(t)$$ is divergence free: \begin{align*} \left\langle\nabla(U-u)(t),\nabla w(t)\right\rangle &= \left\langle(U-u)(t),-\varDelta w(t)\right\rangle=\left\langle(U-u)(t),(\hat{U}-u)(t)\right\rangle-\left\langle(U-u)(t),\nabla q(t)\right\rangle\\ &= \frac{1}{2}\left\|(U\!-u)(t)\right\|^{2}\!+\frac{1}{2}\left\|(\hat{U}\!-u)(t)\right\|^{2}\!-\frac{1}{2}\left\|(\hat{U}\!-U)(t)\right\|^{2}\!-\left\langle(U\!-\mathcal{R}_{u})(t),\nabla q(t)\right\rangle\\ &\geq \frac{1}{2}\left\|(U-u)(t)\right\|^{2}+\frac{1}{2}\left\|(\hat{U}-u)(t)\right\|^{2}-\frac{1}{2}\left\|(\hat{U}-U)(t)\right\|^{2}\\ &\quad- C\left\|(U-\mathcal{R}_{u})(t)\right\|\,\left\|(\hat{U}-u)(t)\right\|\!. \end{align*} For the last inequality the regularity estimate (2.7) for ∇q(t) has been used. For the terms on the right-hand side of equation (4.10), we obtain \begin{align*} \left\langle-\varDelta (U-\mathcal{R}_{u})(t)+ \varUpsilon(t),w(t)\right\rangle &= \left\langle (U-\mathcal{R}_{u})(t),-\varDelta w(t)\right\rangle+\left\langle\varUpsilon(t),w(t)\right\rangle\\ &= \left\langle(U-\mathcal{R}_{u})(t),-\nabla q(t)+(\hat{U}-u)(t)\right\rangle+\left\langle\varUpsilon(t),w(t)\right\rangle\\ &\lesssim \left\|(U-\mathcal{R}_{u})(t)\right\|^{2} + \left\|(\hat{U}-u)(t)\right\| + \left\|\varUpsilon(t)\right\|_{J^{\ast}}\left\|\nabla w(t)\right\|\!. \end{align*} Now, integrating from t = 0 to t1, yields \begin{align} &\max_{t\in[0,t_{1}]}\left\|(\hat{U}-u)(t)\right\|^{2}_{J^{\ast}}+\int_{0}^{t_{1}}\left(\left\|(U-u)(t)\right\|^{2}+\left\|(\hat{U}-u)(t)\right\|^{2}\right)\mathrm{d}t\nonumber\\ &\quad\lesssim \left\|(\hat{U}-u)(0)\right\|^{2}_{J^{\ast}}+\int_{0}^{t_{1}}\left(\left\|(\hat{U}-U)(t)\right\|^{2}+ \left\|(U-\mathcal{R}_{u})(t)\right\|^{2}+ \left\|\varUpsilon(t)\right\|^{2}_{J^{\ast}}\right)\mathrm{d}t, \end{align} (4.12) with $$\|(\hat{U}-u)(0)\|^{2}_{J^{\ast }} = \|U^{0}-u_{0}\|^{2}_{J^{\ast }}=\|\mathcal{S}^{u}_{n}(u_{0},p_{0})-u_{0}\|^{2}_{J^{\ast }}\leq \eta ^{0}(J^{\ast })$$. Rearrange equation (4.10) to get for t ∈ I1 \begin{align} \partial_{t}(\hat{\mathcal{R}}_{u} -u)(t)-\varDelta\left(\mathcal{R}_{u}-u\right)\!(t) +\nabla (\mathcal{R}_{p}-p)(t) =\partial_{t}(\hat{\mathcal{R}}_{u}-\hat{U})(t)+\varUpsilon(t). \end{align} (4.13) Test equation (4.13) with $$\left (\mathcal{R}_{u}-u\right )\!(t)\in J$$, use identity (4.11) and integrate from t = 0 to t1 to obtain \begin{align} &\max_{t\in[0,t_{1}]}\left\|(\hat{\mathcal{R}}_{u}-u)(t)\right\|^{2}+\int_{0}^{t_{1}}\left(\left\|\nabla(\hat{\mathcal{R}}_{u}-u)(t)\right\|^{2}+\left\|\nabla(\mathcal{R}_{u}-u)(t)\right\|^{2}\right)\mathrm{d}t\nonumber\\ &\quad\lesssim \left\|(\hat{\mathcal{R}}_{u}-u)(0)\right\|^{2}+\int_{0}^{t_{1}}\left(\left\|\partial_{t}(\hat{\mathcal{R}}_{u}-\hat{U})(t)\right\|^{2}_{J^{\ast}} +\left\|\nabla(\mathcal{R}_{u}-\hat{\mathcal{R}}_{u})(t)\right\|^{2}+\left\|\varUpsilon(t)\right\|^{2}_{J^{\ast}}\right)\mathrm{d}t. \end{align} (4.14) Finally, the triangle inequality yields \begin{align*} \max_{t\in[0,t_{1}]}\left\|(U-u)(t)\right\|^{2} \leq&\, \max_{t\in[0,t_{1}]}\left\|(U-\hat{U})(t)\right\|^{2} +\max_{t\in[0,t_{1}]}\left\|(\hat{\mathcal{R}}_{u}-\hat{U})(t)\right\|^{2}+\max_{t\in[0,t_{1}]}\left\|(\hat{\mathcal{R}}_{u}-u)(t)\right\|^{2}\\ \lesssim&\, \eta^{0}(L^{2})+ \max_{t\in[0,t_{1}]}\left\|(U-\hat{U})(t)\right\|^{2}+\max_{t\in[0,t_{1}]}\left\|(\hat{\mathcal{R}}_{u}-\hat{U})(t)\right\|^{2}\\ &+\int_{0}^{t_{1}}\left(\left\|\partial_{t}(\hat{\mathcal{R}}_{u}-\hat{U})(t)\right\|^{2}_{J^{\ast}} +\left\|\nabla(\mathcal{R}_{u}-\hat{\mathcal{R}}_{u})(t)\right\|^{2}+\left\|\varUpsilon(t)\right\|^{2}_{J^{\ast}}\right)\mathrm{d}t. \end{align*} To prove equation (4.5) the error is split into $$ \int_{0}^{t_{1}}\left\|\nabla(U-u)(t)\right\|^{2}\mathrm{d}t \leq \int_{0}^{t_{1}}\left\|\nabla(U-\mathcal{R}_{u})(t)\right\|^{2} \mathrm{d}t + \int_{0}^{t_{1}}\left\|\nabla(\mathcal{R}_{u}-u)(t)\right\|^{2}$$ and then equation (4.14) is used. The proofs of equations (4.6) and (4.7) are omitted, since they can be achieved in exactly the same way as the proofs of equations (3.10) and (3.11) in Theorem 3.4 using the error equation (4.13). 4.2 Error estimator In the following lemma we study how to handle mesh changes within the a posteriori framework. One fundamental technical difficulty arises from the fact that because of mesh changes Un ∈ Xn≠Xn−1 ∋ Un−1 in general, and thus terms like $$\bar \partial U^{n}$$ are not well defined as a finite element function. Lemma 4.3 (Changing meshes). For an arbitrary function Ws ∈ Xs, s = 1, …, N and for arbitrary $$n=1,\dots , N$$ as well m = 1, …, k + 1, we have \begin{align} {\hskip-138pt}\left\|(I-\varPi_{n})W^{s}\right\|^{2} = \|W^{s}\|^{2}-\|\varPi_{n} W^{s}\|^{2}, \end{align} (4.15) \begin{align} {\hskip-113pt}\left\|(I-\varPi_{n})W^{s}\right\|^{2}_{-m} \lesssim \ h^{2m}_{n}\left(\|W^{s}\|^{2}-\|\varPi_{n} W^{s}\|^{2}\right)\!, \end{align} (4.16) \begin{align} \|\bar\partial W^{s}\|^{2}_{-m} \lesssim \|\bar\partial_{s} W^{s}\|^{2}_{-m} +C\frac{h_{n}^{2m}}{\tau^{2}} \left(\|W^{s-1}\|^{2}-\|\varPi_{n} W^{s-1}\|^{2}\right)\!, \end{align} (4.17) \begin{equation} {\hskip-186pt}\left\|(I-\varPi_{n})\bar\partial^{B} W^{s}\right\|_{-m}^{2} \lesssim \frac{h^{2m}_{n}}{\tau^{2}}\xi_{B}^{s,n}(W), \end{equation} (4.18) \begin{align} {\hskip-89pt}\|\bar\partial^{2} W^{s}\|^{2} \leq \left\|{\bar\partial_{s}^{2}} W^{s}\right\|^{2} + C\frac{1}{\tau^{4}}\xi_{B}^{s,s}(W), \end{align} (4.19) where the local BDF2-changing mesh operator $$\xi _{B}^{s,n}(W)$$ is defined in equation (3.6). Proof. By definition of the L2 projection $$ \left\langle(I-\varPi_{n})W^{s},v_{n}\right\rangle=0\qquad \forall\ v_{n}\in X_{n}$$ and the identity \begin{align} 2\langle a-b, a\rangle = \|a\|^{2}-\|b\|^{2}+\|a-b\|^{2} \end{align} (4.20) it follows $$ 0=2\left\langle(I-\varPi_{n})W^{s},\varPi_{n} W^{s}\right\rangle = -\|\varPi_{n}W^{s}\|^{2}+\|W^{s}\|^{2}-\left\|(I-\varPi_{n})W^{s}\right\|^{2}\!, $$ which implies equation (4.15). To prove equation (4.16) observe that $$ \left\|(I\!-\varPi_{n})W^{s}\right\|_{-m}=\!\sup_{v\in{H_{0}^{1}}\cap H^{m}}\!\frac{\left\langle(I-\varPi_{n})W^{s},v\right\rangle}{\|v\|_{H^{m}}} =\!\sup_{v\in{H_{0}^{1}}\cap H^{m}}\!\frac{\left\langle(I-\varPi_{n})W^{s},v-\mathcal{I}_{n}v\right\rangle}{\|v\|_{H^{m}}} \!\leq\! C\ {h^{m}_{n}} \left\|(I-\varPi_{n})W^{s}\right\|\!. $$ Now using equation (4.15) yields the result. To derive equation (4.17) use the estimate $$ \|\bar\partial W^{s}\|^{2}_{-m} \lesssim \|\bar\partial_{s} W^{s}\|^{2}_{-m} + \left\|(I-\varPi_{s})\bar\partial W^{s}\right\|^{2}_{-m}= \|\bar\partial_{s} W^{s}\|^{2}_{-m} + \frac{1}{\tau^{2}}\left\|(I-\varPi_{s}) W^{s-1}\right\|^{2}_{-m} $$ and apply equation (4.16) to the last term on the right-hand side. Equation (4.18) follows from the definition of the $$\bar \partial ^{B}$$ operator and equation (4.16): $$ \left\|(I-\varPi_{n})\bar\partial^{B} W^{s}\right\|^{2}_{-m} \lesssim \frac{1}{\tau^{2}}\left( \left\|(I-\varPi_{n})W^{s}\right\|^{2}_{-m} + \left\|(I-\varPi_{n})W^{s-1}\right\|^{2}_{-m} + \left\|(I-\varPi_{n})W^{s-2}\right\|^{2}_{-m} \right)\!. $$ To prove equation (4.19) rewrite $$\bar \partial ^{2} W^{s}$$ as $$ \bar\partial^{2} W^{s} = \frac{2}{\tau}(\bar\partial^{B} W^{s}-\bar\partial W^{s}).$$ Then one obtains by using equations (4.15) and (4.18) \begin{align*} \|\bar\partial^{2} W^{s}\|^{2} &= \left\|{\bar\partial^{2}_{s}} W^{s}\right\|^{2} + \left\|(I-\varPi_{s})\bar\partial^{2} W^{s}\right\|^{2}\\ &\leq \left\|{\bar\partial^{2}_{s}} W^{s}\right\|^{2} + C\frac{1}{\tau^{2}}\left(\left\|(I-\varPi_{s})\bar\partial^{B} W^{s}\right\|^{2} + \frac{1}{\tau^{2}}\left\|(I-\varPi_{s}) W^{s-1}\right\|^{2}\right)\\ &\leq \left\|{\bar\partial^{2}_{s}} W^{s}\right\|^{2} + C\frac{1}{\tau^{4}}\xi_{B}^{s,s}(W). \end{align*} Note that the difference ∥(I − Πs)Ws−1∥2 is already contained in the definition of $$\xi _{B}^{s,s}$$. Below we derive the spatial and reconstruction estimators. We start with bounds of the error between the discrete solution and the reconstruction. Lemma 4.4 Let $$\left (U^{n}, P^{n}\right )$$ be the discrete solution of equation (2.1) and equation (2.10) and $$(\mathcal{R}_{u}^{n},\mathcal{R}_{p}^{n})$$ be the reconstruction defined in equation (3.2). Then we have the following estimates for the interpolants \begin{align}{\hskip-11pt} {\int_{0}^{T}}\left\|(U-\mathcal{R}_{u})(t)\right\|^{2} \lesssim \ \varLambda_{1}, \end{align} (4.21) \begin{align} {\hskip-22pt}\max_{t\in[0,T]}\left\|(\hat{U}-\hat{\mathcal{R}}_{u})(t)\right\|^{2} \lesssim \ \varLambda_{2}, \end{align} (4.22) \begin{align} {\int_{0}^{T}}\left\|\partial_{t}(\hat{U}-\hat{\mathcal{R}}_{u})(t)\right\|^{2}_{J^{\ast}}\lesssim \ \varLambda_{3}+\varXi_{2}, \end{align} (4.23) \begin{align} {\hskip-21pt}{\int_{0}^{T}}\left\|\nabla(U-\mathcal{R}_{u})(t)\right\|^{2} \lesssim \ \varLambda_{4}, \end{align} (4.24) \begin{align} {\hskip-30pt}\max_{t\in[0,T]}\left\|\nabla(\hat{U}-\hat{\mathcal{R}}_{u})(t)\right\|^{2} \lesssim \ \varLambda_{5}, \end{align} (4.25) \begin{align}{\hskip3pt} {\int_{0}^{T}}\left\|\partial_{t}(\hat{U}-\hat{\mathcal{R}}_{u})(t)\right\|^{2} \lesssim \ \varLambda_{6}+\varXi_{6}, \end{align} (4.26) \begin{align} {\hskip-13pt}{\int_{0}^{T}}\left\|(P-\mathcal{R}_{p})(t)\right\|^{2} \lesssim \ \varLambda_{7}, \end{align} (4.27) where the estimators are defined in Definition 3.3. Proof. Due to the definitions of the interpolant in equation (3.1) and the reconstruction in equation (3.2) and equation (4.9), it holds \begin{align*} \left\|(U-\mathcal{R}_{u})(t)\right\| \ &\lesssim \ \left\|U^{n}-\mathcal{R}_{u}^{n}\right\| + \left\|U^{n-1}-\mathcal{R}_{u}^{n-1}\right\|\!, & &t\in I_{n}, \ n\geq 1,\\ \left\|(\hat{U}- \hat{\mathcal{R}}_{u})(t)\right\| \ &\lesssim \ \left\|U^{n}-\mathcal{R}_{u}^{n}\right\| + \left\|U^{n-1}-\mathcal{R}_{u}^{n-1}\right\|+ \left\|U^{n-2}-\mathcal{R}_{u}^{n-2}\right\|\!, & &t\in I_{n}, \ n\geq 2,\\ \left\|(\hat{U}- \hat{\mathcal{R}}_{u})(t)\right\| \ &\lesssim \ \left\|U^{2}-\mathcal{R}_{u}^{2}\right\| + \left\|U^{1}-\mathcal{R}_{u}^{1}\right\|+ \left\|U^{0}-\mathcal{R}_{u}^{0}\right\|\!, & &t\in I_{1}. \end{align*} Hence equations (4.21), (4.22), (4.24), (4.25) and (4.27) follow directly by applying the above inequalities and integrating over time. To prove equation (4.23) and equation (4.26), the definition of the three-point reconstruction yields \begin{align*} {\int_{0}^{T}}\left\|\partial_{t}(\hat{U}-\hat{\mathcal{R}}_{u})(t)\right\|^{2}_{*} \lesssim&\, \sum_{n=1}^{N} \tau \left\|\bar\partial\left(U^{n}-\mathcal{R}_{u}^{n}\right)\right\|^{2}_{*}\\ \lesssim&\, \sum_{n=1}^{N} \tau \left\|\bar\partial_{n}\big(U^{n}-\bar\partial \mathcal{R}_{u}^{n}\big)\right\|^{2}_{*} + \sum_{n=2}^{N}\tau \left\|(\bar\partial-\bar\partial_{n})U^{n}\right\|^{2}_{*} ,\quad *\in\{L^{2},J^{\ast}\}, \end{align*} respectively. This proves equations (4.23) and (4.26), where the last term on the right-hand side was absorbed in the estimators Ξ2 and Ξ6, respectively. In the next lemma, upper bounds for the Stokes reconstruction are given. Lemma 4.5 Let Un be the discrete solution of equation (2.1) and equation (2.10) and $$\mathcal{R}_{u}^{n}$$ the corresponding reconstruction defined in equation (3.2). Further let X2 = X1 = X0 and set $$U^{-1}:= U^{1}-2\tau (\varDelta _{0} U^{0}-B_{0}^{\top } P^{0}+F^{0} )\in X_{0}$$ and $$U^{-2}:=-6\tau (\varDelta _{0} U^{0}-B_{0}^{\top } P^{0}+F^{0} )-3U^{0}+4U^{1}\in X_{0}$$ to simplify the presentation. Then \begin{align} {\int_{0}^{T}}\left\|\nabla(\mathcal{R}_{u}-\hat{\mathcal{R}}_{u})(t)\right\|^{2}\mathrm{d}t \lesssim \ \tau^{4}\mathcal{E}_{4} + \varXi_{4}, \end{align} (4.28) \begin{align} {\int_{0}^{T}}\left\|\varDelta(\mathcal{R}_{u}-\hat{\mathcal{R}}_{u})(t)\right\|^{2}\mathrm{d}t \lesssim \ \tau^{4}\mathcal{E}_{5} + \varXi_{5}, \end{align} (4.29) where the estimators are defined in Definition 3.3. Proof. By definition of the reconstruction in equation (3.2) one has for n ≥ 2 \begin{align*} -\varDelta\bar\partial^{2} \mathcal{R}_{u}^{n}+\nabla\bar\partial^{2} \mathcal{R}_{p}^{n} =&\, \frac{1}{\tau^{2}}\left( \left(-\varDelta_{n} U^{n} +B_{n}^{\top} P^{n}-F^{n}\right) -2\left(-\varDelta_{n-1}U^{n-1}+B_{n-1}^{\top} P^{n-1}-F^{n-1}\right)\right.\\ &\quad\left.+ \left(-\varDelta_{n-2}U^{n-2}+B_{n-2}^{\top} P^{n-2}-F^{n-2}\right)\right) + \bar\partial^{2}f^{n}. \end{align*} By applying equation (2.9), we get \begin{align*} -\varDelta\bar\partial^{2} \mathcal{R}_{u}^{n}+\nabla\bar\partial^{2} \mathcal{R}_{p}^{n} &= \frac{1}{\tau^{2}}\left(-{\bar\partial^{B}_{n}} U^{n}+2\bar\partial^{B}_{n-1}U^{n-1}-\bar\partial^{B}_{n-2}U^{n-2}\right) + \bar\partial^{2} f^{n}\\ &= -{\bar\partial^{2}_{n}}\bar\partial^{B} U^{n} + \bar\partial^{2} f^{n} +\frac{1}{\tau^{2}}\left(2\left(\bar\partial^{B}_{n-1}-{\bar\partial^{B}_{n}}\right)U^{n-1}-\left(\bar\partial^{B}_{n-2}-{\bar\partial^{B}_{n}}\right)U^{n-2}\right)\!. \end{align*} Note that in the case n = 2, 3 it holds $${\bar \partial ^{B}_{1}} U^{1}=\varDelta _{1} U^{1}+F^{1}$$ and $${\bar \partial ^{B}_{0}} U^{0} = \varDelta _{0} U^{0} +F^{0}$$ due to the definition of U−1 and U−2. Now we use the definition of the three-point reconstruction in equation (3.2) and equation (4.9) to get \begin{align*} {\int_{0}^{T}}\left\|\nabla(\mathcal{R}_{u}-\hat{\mathcal{R}}_{u})(t)\right\|^{2}\mathrm{d}t \lesssim&\, \sum_{n=2}^{N}\tau^{5}\left\|\nabla\bar\partial^{2} \mathcal{R}_{u}^{n}\right\|^{2}\\ \lesssim&\, \sum_{n=2}^{N}\tau^{5}\left\|\bar\partial^{2}f^{n}-{\bar\partial^{2}_{n}}\bar\partial^{B} U^{n} \right\|^{2}_{J^{\ast}}\\ &+\sum_{n=2}^{N}\tau\left(\underbrace{\left\|\left(\bar\partial^{B}_{n-1}-{\bar\partial^{B}_{n}}\right)U^{n-1}\right\|_{J^{\ast}}^{2}+\left\|\left(\bar\partial^{B}_{n-2}-{\bar\partial^{B}_{n}}\right)U^{n-2}\right\|^{2}_{J^{\ast}}}_{=(*)}\right)\!. \end{align*} The last term on the right-hand side is handled by using equation (4.18): \begin{align*} (*) &\lesssim \left\|(I\!-\!\varPi_{n-1})\bar\partial^{B}U^{n-1}\right\|^{2}_{J^{\ast}}\!+\!\left\|(I\!-\!\varPi_{n})\bar\partial^{B}U^{n-1}\right\|^{2}_{J^{\ast}} + \left\|(I\!-\!\varPi_{n-2})\bar\partial^{B} U^{n-2}\right\|^{2}_{J^{\ast}} \!+\!\left\|(I\!-\!\varPi_{n})\bar\partial^{B} U^{n-2}\right\|_{J^{\ast}}^{2}\\ &\lesssim \frac{h_{n-1}^{2}}{\tau^{2}}\xi_{B}^{n-1,n-1}(U) + \frac{{h_{n}^{2}}}{\tau^{2}}\xi_{B}^{n-1,n}(U) +\frac{h_{n-2}^{2}}{\tau^{2}}\xi_{B}^{n-2,n-2}(U)+\frac{{h_{n}^{2}}}{\tau^{2}}\xi_{B}^{n-2,n}(U). \end{align*} Thus, we obtain $$ {\int_{0}^{T}}\left\|\nabla(\mathcal{R}_{u}-\hat{\mathcal{R}}_{u})(t)\right\|^{2}\mathrm{d}t \lesssim \mathcal{E}_{4} + \varXi_{4}.$$ In the same way equation (4.29) is shown by using Assumption 2.4. Now we bound the residual $$\varUpsilon $$ of the error equation (4.1) and the difference between the linear interpolant (3.1) and three-point reconstruction (3.2). Lemma 4.6 Let Un be the discrete solution of equation (2.1) and equation (2.10), respectively. Then we have \begin{align} {\hskip-78pt}{\int_{0}^{T}}\left\|(\hat{U}-U)(t)\right\|^{2} \lesssim \ \tau^{4}\mathcal{E}_{1} + \varXi_{1}, \end{align} (4.30) \begin{align} {\int_{0}^{T}}\left\|\varUpsilon(t)\right\|_{J^{\ast}}^{2} \lesssim \ \tau^{4}\mathcal{E}_{2} + \varXi_{2} + \mathcal{F}_{1}(t) + {\textit{I}}_{1}, \end{align} (4.31) \begin{align} {\hskip-86pt}\max_{t\in[0,T]}\left\|(\hat{U}-U)(t)\right\|^{2} \lesssim \ \tau^{4}\mathcal{E}_{3} + \varXi_{3}, \end{align} (4.32) \begin{align} {\hskip-5pt}{\int_{0}^{T}}\left\|\varUpsilon(t)\right\|^{2}\mathrm{d}t \lesssim \ \tau^{4}\mathcal{E}_{6} + \varXi_{6} + \mathcal{F}_{2}(t)+ {\textit{I}}_{2}, \end{align} (4.33) \begin{align} {\hskip-93pt}\max_{t\in[0,T]}\left\|\nabla(\hat{U}-U)(t)\right\|^{2} \lesssim \ \tau^{4}\mathcal{E}_{7} + \varXi_{7}, \end{align} (4.34) where the estimators are defined in Definition 3.3. Proof. Equations (4.30), (4.32) and (4.34) are easily shown by the definition of the reconstruction, cf. equations (3.2) and (4.9), and equation (4.19): $$ \left\|(\hat{U}-U)(t)\right\|^{2}\ \lesssim\ \tau^{4}\|\bar\partial^{2} U^{n}\|^{2}\ \lesssim \ \tau^{4}\left\|{\bar\partial^{2}_{n}} U^{n}\right\|^{2} + \xi_{B}^{n,n}(U).$$ For the proof of equations (4.31) and (4.33) we reformulate $$\varUpsilon{(t)}$$ . In the time interval In, n ≥ 2, we have \begin{align*} \varUpsilon(t) &= f(t_{n})-f(t)+(t-t_{n})\left(\bar\partial^{2}U^{n} - \varDelta\bar\partial \mathcal{R}_{u}^{n}+\nabla\bar\partial \mathcal{R}_{p}^{n}\right)+\left(\bar\partial^{B}-{\bar\partial^{B}_{n}}\right)U^{n}\\ &=f(t_{n})-f(t)+\left(\bar\partial^{B}-{\bar\partial^{B}_{n}}\right)U^{n}+(t-t_{n})\underbrace{\left(\bar\partial^{2} U^{n} +\frac{1}{\tau}\left(-\varDelta \mathcal{R}_{u}^{n}+\nabla \mathcal{R}_{p}^{n} +\varDelta \mathcal{R}_{u}^{n-1}-\nabla \mathcal{R}_{p}^{n-1}\right)\right)}_{=(*)}\!. \end{align*} Let us simplify the last bracket further by using equation (2.9). \begin{align*} (*) &= \bar\partial^{2} U^{n} + \frac{1}{\tau}\left(-\varDelta_{n} U^{n} + B_{n}^{\top} P^{n}-F^{n}+f(t_{n}) - \left(-\varDelta_{n-1} U^{n-1} + B_{n-1}^{\top} P^{n-1}-F^{n-1}+f(t_{n-1})\right) \right)\\ &= \bar\partial f(t_{n}) + \frac{1}{\tau}\left(\bar\partial U^{n} - \bar\partial U^{n-1}-{\bar\partial^{B}_{n}} U^{n} + \bar\partial^{B}_{n-1} U^{n-1}\right)\\ &= \bar\partial f(t_{n}) - \left({\bar\partial^{2}_{n}} U^{n}-\bar\partial^{2}_{n-1} U^{n-1}\right) + \frac{1}{\tau}(\bar\partial-\bar\partial_{n}) U^{n} - \frac{1}{\tau}(\bar\partial-\bar\partial_{n-1})U^{n-1}. \end{align*} Note that we have used the definition of U−1 in Lemma 4.5. Then the triangle inequality yields \begin{align*} \|\varUpsilon(t)\|_{*} \ \lesssim& \ \left\|\,f(t_{n})+(t-t_{n})\bar\partial f(t_{n})-f(t)\right\|_{*} + \left\|(\bar\partial-{\bar\partial^{B}_{n}})U^{n}\right\|_{*} \\ &+ \tau\left\|{\bar\partial^{2}_{n}} U^{n}-\bar\partial^{2}_{n-1}U^{n-1}\right\|_{*} + \left\|(\bar\partial-\bar\partial_{n})U^{n}\right\|_{*}+\left\|(\bar\partial-\bar\partial_{n-1})U^{n-1}\right\|_{*},\quad *\in\{L^{2},J^{\ast}\}. \end{align*} For t ∈ I1, we have cf. equation (4.8), $$ \left\|\varUpsilon(t)\right\|_{*} \ \lesssim \ \left\|\,f(t_{1})+(t-t_{1})\bar\partial f(t_{1})-f(t)\right\|_{*} + \tau\|\varPsi_{1}\|_{*},\quad *\in\{L^{2},J^{\ast}\}.$$ Together we obtain \begin{align*} {\int_{0}^{T}}\left\|\varUpsilon(t)\right\|^{2}_{*} \ \lesssim&\ \tau^{3}\|\varPsi_{1}\|^{2}_{*} + {\int_{0}^{T}}\left\|\,f(t_{n})+(t-t_{n})\bar\partial f(t_{n})-f(t)\right\|_{*}\mathrm{d}t \\ &+ \sum_{n=2}^{N}\tau^{3}\left\|{\bar\partial^{2}_{n}} U^{n}-\bar\partial^{2}_{n-1}U^{n-1}\right\|_{*}^{2} +\sum_{n=2}^{N} \tau \left\|\left(\bar\partial-{\bar\partial^{B}_{n}}\right)U^{n}\right\|_{*}^{2}+\sum_{n=1}^{N}\tau\left\|(\bar\partial-\bar\partial_{n})U^{n}\right\|_{*}^{2}. \end{align*} Now apply equation (4.18) to the fourth term on the right-hand side to prove equations (4.31) and (4.33). Note that the last term on the right-hand side can be absorbed by the fourth term. 4.3 Proof of Theorem 3.4 Proof. (Theorem 3.4) First we prove equation (3.7). To this end we test equation (4.1) with y ∈ J to get $$ \left\langle\partial_{t}(\hat{U}-u)(t),y\right\rangle+\left\langle\nabla(U-u)(t),\nabla y\right\rangle =\left\langle-\varDelta (U-\mathcal{R}_{u})(t),y\right\rangle+\left\langle \varUpsilon(t),y\right\rangle\!,\qquad t\in I_{n},\ n\geq2. $$ Next, set $$y=w(t):=\mathcal{A}^{-1}(\hat{U}-u)(t)$$, i.e. (w(t), q(t)) is solution of $$ -\varDelta w(t)+\nabla q(t)=(\hat{U}-u)(t),\quad \textrm{div}\, w(t)=0,\quad w(t)|_{\varGamma}=0. $$ For the first term on the left-hand side, one gets $$ \langle\partial_{t}(\hat{U}-u)(t),w(t)\rangle = \frac{1}{2}\frac{\mathrm{d}}{\mathrm{d}t}\|\nabla w(t)\|^{2}. $$ The second term is handled by the use of regularity estimate (2.7) for ∇q(t), the identity equation (4.11) and the fact that the interpolant of the Stokes reconstruction $$\mathcal{R}_{u}(t)$$ is divergence free: \begin{align*} \left\langle\nabla(U-u)(t),\nabla w(t)\right\rangle &= \left\langle(U-u)(t),-\varDelta w(t)\right\rangle=\left\langle(U-u)(t),(\hat{U}-u)(t)\right\rangle-\left\langle(U-u)(t),\nabla q(t)\right\rangle\\ &= \frac{1}{2}\left\|(U-u)(t)\right\|^{2}\!+\!\frac{1}{2}\left\|(\hat{U}-u)(t)\right\|^{2}\!-\!\frac{1}{2}\left\|(\hat{U}-U)(t)\right\|^{2}\!-\!\left\langle(U-\mathcal{R}_{u})(t),\nabla q(t)\right\rangle\\ &\geq \frac{1}{2}\left\|(U-u)(t)\right\|^{2}+\frac{1}{2}\left\|(\hat{U}-u)(t)\right\|^{2}-\frac{1}{2}\left\|(\hat{U}-U)(t)\right\|^{2}\\ &\quad-\left\|(U-\mathcal{R}_{u})(t)\right\|\left\|(\hat{U}-u)(t)\right\|\!. \end{align*} The terms on the right-hand side are treated as follows \begin{align*} \left\langle-\varDelta (U-\mathcal{R}_{u})(t)+ \varUpsilon(t),w(t)\right\rangle &= \left\langle (U-\mathcal{R}_{u})(t),-\varDelta w(t)\right\rangle+\left\langle\varUpsilon(t),w(t)\right\rangle\\ &= \left\langle(U-\mathcal{R}_{u})(t),-\nabla q(t)+(\hat{U}-u)(t)\right\rangle+\left\langle\varUpsilon(t),w(t)\right\rangle\\ &\lesssim \left\|(U-\mathcal{R}_{u})(t)\right\|^{2} + \frac{1}{8}\left\|(\hat{U}-u)(t)\right\| + \left\|\varUpsilon(t)\right\|_{J^{\ast}}\left\|\nabla w(t)\right\|\!. \end{align*} Collecting these results, integrating from t = t1 to T, using Gronwall’s Lemma and Young’s inequality one arrives at \begin{align*} &\max_{t\in[t_{1},T]}\left\|\nabla w(t)\right\|^{2}+\int_{t_{1}}^{T}\left(\left\|(U-u)(t)\right\|^{2}+\left\|(\hat{U}-u)(t)\right\|^{2}\right)\mathrm{d}t\\ &\quad\lesssim \left\|\nabla w(t_{1})\right\|^{2}+\int_{t_{1}}^{T}\left(\left\|(\hat{U}-U)(t)\right\|^{2} + \left\|(U-\mathcal{R}_{u})(t)\right\|^{2} + \left\|\varUpsilon(t)\right\|_{J^{\ast}}^{2}\right)\mathrm{d}t. \end{align*} This combined with the initial estimate (4.3) yields $$ {\int_{0}^{T}}\left\|(U-u)(t)\right\|^{2}\mathrm{d}t \ \lesssim \ \eta^{0}(J^{\ast})+{\int_{0}^{T}}\left(\left\|(\hat{U}-U)(t)\right\|^{2} + \left\|(U-\mathcal{R}_{u})(t)\right\|^{2} + \left\|\varUpsilon(t)\right\|_{J^{\ast}}^{2}\right)\mathrm{d}t.$$ Now use equations (4.21), (4.30) and (4.31) to prove equation (3.7). To show equations (3.8) and (3.9), rearrange terms in equation (4.1) to get the following error equation for t ∈ In, n ≥ 2, \begin{align} \partial_{t}(\hat{\mathcal{R}}_{u}-u)(t)-\varDelta(\mathcal{R}_{u}-u)(t)+\nabla(\mathcal{R}_{p}-p)(t) = \partial_{t}(\hat{\mathcal{R}}_{u}-\hat{U})(t)+\varUpsilon(t). \end{align} (4.35) Test equation (4.35) by $$(\hat{\mathcal{R}}_{u}-u)(t)\in J$$, use identity (4.11) and Young’s inequality to get \begin{align*} &\frac{\mathrm{d}}{\mathrm{d}t}\left\|(\hat{\mathcal{R}}_{u}-u)(t)\right\|^{2}+\left\|\nabla(\hat{\mathcal{R}}_{u}-u)(t)\right\|^{2}+\left\|\nabla(\mathcal{R}_{u}-u)(t)\right\|^{2}\\ &\quad\lesssim\left\|\partial_{t}(\hat{\mathcal{R}}_{u}-\hat{U})(t)\right\|^{2}_{J^{\ast}}+\left\|\nabla(\mathcal{R}_{u}-\hat{\mathcal{R}}_{u})(t)\right\|^{2}+\|\varUpsilon\|^{2}_{J^{\ast}}. \end{align*} Integrating from t = t1 to T yields \begin{align} &\max_{t\in[t_{1},T]}\left\|(\hat{\mathcal{R}}_{u}-u)(t)\right\|^{2}+\int_{t_{1}}^{T}\left(\left\|\nabla(\hat{\mathcal{R}}_{u}-u)(t)\right\|^{2}+\left\|\nabla(\mathcal{R}_{u}-u)(t)\right\|^{2}\right)\mathrm{d}t\nonumber \\ &\quad\lesssim \left\|(\hat{\mathcal{R}}_{u}-u)(t_{1})\right\|^{2}+\int_{t_{1}}^{T}\left(\left\|\partial_{t}(\hat{\mathcal{R}}_{u}-\hat{U})(t)\right\|^{2}_{J^{\ast}}+\left\|\nabla(\mathcal{R}_{u}-\hat{\mathcal{R}}_{u})(t)\right\|^{2} +\left\|\varUpsilon(t)\right\|^{2}_{J^{\ast}}\right)\mathrm{d}t. \end{align} (4.36) Now combine this estimate with equation (4.4) and apply the triangle inequality to get \begin{align*} \max_{t\in[0,T]}\left\|(U-u)(t)\right\|^{2} \lesssim& \max_{t\in[0,t_{1}]}\left\|(U-u)(t)\right\|^{2}+ \max_{t\in[t_{1},T]}\left\|(U-\hat{U})(t)\right\|^{2}\\ &+ \max_{t\in[t_{1},T]}\left\|(\hat{\mathcal{R}}_{u}-\hat{U})(t)\right\|^{2}+ \max_{t\in[t_{1},T]}\left\|(\hat{\mathcal{R}}_{u}-u)(t)\right\|^{2}\\ \lesssim& \eta^{0}(L^{2})+ \max_{t\in[0,T]}\left\|(U-\hat{U})(t)\right\|^{2}+ \max_{t\in[0,T]}\left\|(\hat{\mathcal{R}}_{u}-\hat{U})(t)\right\|^{2}\\ &+{\int_{0}^{T}}\left(\left\|\partial_{t}(\hat{\mathcal{R}}_{u}-\hat{U})(t)\right\|^{2}_{J^{\ast}}+\left\|\nabla(\mathcal{R}_{u}-\hat{\mathcal{R}}_{u})(t)\right\|^{2}+\left\|\varUpsilon(t)\right\|^{2}_{J^{\ast}}\right)\mathrm{d}t. \end{align*} With equations (4.22), (4.23), (4.28), (4.31) and (4.32), the proof of equation (3.8) is finished. In a similar way equation (3.9) is bounded. Use the triangle inequality and the initial estimate (4.5) to get \begin{align*} {\int_{0}^{T}}\left\|\nabla(U-u)(t)\right\|^{2} \lesssim&\, \int_{0}^{t_{1}}\left\|\nabla(U-u)(t)\right\|^{2}+\int_{t_{1}}^{T}\left\|\nabla(U-\mathcal{R}_{u})(t)\right\|^{2}+\int_{t_{1}}^{T}\left\|\nabla(\mathcal{R}_{u}-u)(t)\right\|^{2}\\ \lesssim&\, \eta^{0}(L^{2})+ {\int_{0}^{T}}\left\|\nabla(U-\mathcal{R}_{u})(t)\right\|^{2}\\ &+{\int_{0}^{T}}\left(\left\|\partial_{t}(\hat{\mathcal{R}}_{u}-\hat{U})(t)\right\|^{2}_{J^{\ast}}+\left\|\nabla(\mathcal{R}_{u}-\hat{\mathcal{R}}_{u})(t)\right\|^{2}+\left\|\varUpsilon(t)\right\|^{2}_{J^{\ast}}\right)\mathrm{d}t, \end{align*} and then use equations (4.23), (4.24), (4.28) and (4.31). To prove equations (3.10) and (3.11), test equation (4.35) by $$\partial _{t}(\hat{\mathcal{R}}_{u}-u)(t)\in J$$ to obtain for t ∈ In, n ≥ 2, $$ \left\|\partial_{t}(\hat{\mathcal{R}}_{u}-u)(t)\right\|^{2}+\left\langle-\varDelta(\mathcal{R}_{u}-u)(t),\partial_{t}(\hat{\mathcal{R}}_{u}-u)(t)\right\rangle \lesssim \left\|\partial_{t}(\hat{\mathcal{R}}_{u}-\hat{U})(t)\right\|^{2}+\left\|\varUpsilon(t)\right\|^{2}\!, $$ where the inner product is treated as follows: $$ \left\langle-\varDelta(\mathcal{R}_{u}-u)(t),\partial_{t}(\hat{\mathcal{R}}_{u}-u)(t)\right\rangle = \left\langle\nabla(\hat{\mathcal{R}}_{u}-u)(t),\nabla\partial_{t}(\hat{\mathcal{R}}_{u}-u)(t)\right\rangle + \left\langle-\varDelta(\mathcal{R}_{u}-\hat{\mathcal{R}}_{u}),\partial_{t}(\hat{\mathcal{R}}_{u}-u)\right\rangle\!.$$ Then, integrate from t = t1 to T to get \begin{align} &\int_{t_{1}}^{T}\left\|\partial_{t}(\hat{\mathcal{R}}_{u}-u)(t)\right\|^{2}+\max_{t\in[t_{1},T]}\left\|\nabla(\hat{\mathcal{R}}_{u}-u)\right\|^{2}\nonumber \\ &\quad\lesssim \left\|\nabla(\hat{\mathcal{R}}_{u}-u)(t_{1})\right\|^{2} +\int_{t_{1}}^{T}\left(\left\|\partial_{t}(\hat{\mathcal{R}}_{u}-\hat{U})(t)\right\|^{2} + \left\|\varDelta(\mathcal{R}_{u}-\hat{\mathcal{R}}_{u})(t)\right\|^{2}+\left\|\varUpsilon(t)\right\|^{2}\right)\mathrm{d}t. \end{align} (4.37) This estimate combined with equation (4.4) yields by using the triangle inequality \begin{align*} \max_{t\in[0,T]}\left\|\nabla(U-u)(t)\right\|^{2} \lesssim& \max_{t\in[0,t_{1}]}\left\|\nabla(U-u)(t)\right\|^{2}+ \max_{t\in[t_{1},T]}\left\|\nabla(U-\hat{U})(t)\right\|^{2}\\ &+ \max_{t\in[t_{1},T]}\left\|\nabla(\hat{\mathcal{R}}_{u}-\hat{U})(t)\right\|^{2}+ \max_{t\in[t_{1},T]}\left\|\nabla(\hat{\mathcal{R}}_{u}-u)(t)\right\|^{2}\\ \lesssim&\, \eta^{0}\left({H_{0}^{1}}\right)+ \max_{t\in[0,T]}\left\|\nabla(U-\hat{U})(t)\right\|^{2}+ \max_{t\in[0,T]}\left\|\nabla(\hat{\mathcal{R}}_{u}-\hat{U})(t)\right\|^{2}\\ &+{\int_{0}^{T}}\left(\left\|\partial_{t}(\hat{\mathcal{R}}_{u}-\hat{U})(t)\right\|^{2}+\left\|\varDelta(\mathcal{R}_{u}-\hat{\mathcal{R}}_{u})(t)\right\|^{2}+\left\|\varUpsilon(t)\right\|^{2}\right)\mathrm{d}t. \end{align*} Then equation (3.10) is shown by equations (4.25), (4.26), (4.29), (4.33) and (4.34). It remains to prove equation (3.11). The inf–sup condition and equation (4.37) yield for t ∈ In \begin{align*} \left\|(\mathcal{R}_{p}-p)(t)\right\|&\leq \sup_{v\in X,\|\nabla v\|=1}\left\langle \nabla(\mathcal{R}_{p}-p)(t),v\right\rangle\\ &=\sup_{v\in X,\|\nabla v\|=1}\left\langle-\partial_{t}(\hat{\mathcal{R}}_{u}-u)(t)+\varDelta(\mathcal{R}_{u}-u)(t)+\partial_{t}(\hat{\mathcal{R}}_{u}-\hat{U})(t)+\varUpsilon(t),v\right\rangle\\ &\lesssim \left\|\partial_{t}(\hat{\mathcal{R}}_{u}-u)(t)\right\|_{-1}+\left\|\nabla(\mathcal{R}_{u}-u)(t)\right\|+\left\|\partial_{t}(\hat{\mathcal{R}}_{u}-\hat{U})(t)\right\|_{-1}+\left\|\varUpsilon(t)\right\|_{-1}\\ &\lesssim \left\|\partial_{t}(\hat{\mathcal{R}}_{u}-u)(t)\right\|+\left\|\nabla(\mathcal{R}_{u}-u)(t)\right\|+\left\|\partial_{t}(\hat{\mathcal{R}}_{u}-\hat{U})(t)\right\|_{-1}+\left\|\varUpsilon(t)\right\|\!. \end{align*} Upon squaring this equation, integrating from t = t1 to T and using equations (4.36) and (4.37), one gets \begin{align} \int_{t_{1}}^{T}\!\left\|(\mathcal{R}_{p}-p)(t)\right\|^{2} \!\lesssim\! \left\|\nabla(\hat{\mathcal{R}}_{u}-u)(t_{1})\right\|^{2} \!+\!\int_{t_{1}}^{T}\!\left(\left\|\partial_{t}(\hat{\mathcal{R}}_{u}-\hat{U})(t)\right\|^{2} \!+\! \left\|\varDelta(\mathcal{R}_{u}-\hat{\mathcal{R}}_{u})(t)\right\|^{2}\!+\!\left\|\varUpsilon(t)\right\|^{2}\right)\!\mathrm{d}t. \end{align} (4.38) Next, one gets by the triangle inequality, equation (4.7) and equation (4.38) \begin{align*} {\int_{0}^{T}}\left\|(P-p)(t)\right\|^{2} &\lesssim \int_{0}^{t_{1}}\left\|(P-p)(t)\right\|^{2}+\int_{t_{1}}^{T}\left\|(P-\mathcal{R}_{p})(t)\right\|^{2}+\int_{t_{1}}^{T}\left\|(\mathcal{R}_{p}-p)(t)\right\|^{2}\\ &\lesssim \ \eta^{0}\left({H_{0}^{1}}\right)+{\int_{0}^{T}}\left\|(P-\mathcal{R}_{p})(t)\right\|^{2} \\ &\quad+{\int_{0}^{T}}\left(\left\|\partial_{t}(\hat{\mathcal{R}}_{u}-\hat{U})(t)\right\|^{2} + \left\|\varDelta(\mathcal{R}_{u}-\hat{\mathcal{R}}_{u})(t)\right\|^{2}+\left\|\varUpsilon(t)\right\|^{2}\right)\mathrm{d}t. \end{align*} Then using equations (4.26), (4.27), (4.29) and (4.33) to complete the proof. 5. Proof of optimality, Theorem 3.6 In this section we show that the time error estimators $$\mathcal{E}_{1}$$–$$\mathcal{E}_{7}$$ in Theorem 3.4 are bounded independently of the time step τ and the mesh size h, and in turn the a posteriori estimates yield the same asymptotic convergence rates in time as the a priori estimates. Due to the complexity we assume constant meshes. The proof is subdivided into four steps. First we bound the estimators by difference quotients of the initial values in Lemma 5.1. Hence, for proving optimality of the estimators, corresponding difference quotients of the initial values have to be bounded. To the best of our knowledge such a result is available only for the semidiscrete case, i.e. continuous in space, see for instance the study by Akrivis & Chatzipantelidis (2010) and Bänsch & Brenner (2016). Let u be the exact solution and w be the corresponding semidiscrete solution. In the study by Akrivis & Chatzipantelidis (2010), error estimates for semidiscrete parabolic problems using BDF2 for time discretization are considered, and the resulting error estimator is stable if the third-order difference quotient $$\bar \partial ^{3} w^{3}$$ of initial values can be bounded. This is easily done by inserting the exact solution u and estimating $$ \|\bar\partial^{3} w^{3}\|\leq \left\|\bar\partial^{3} \left(w^{3}-u(t_{3})\right)\right\| + \left\|\bar\partial^{3} u(t_{3})\right\|\leq \frac{1}{\tau^{3}}\sum_{i=0}^{3}\left\|w^{i}-u(t_{i})\right\|+\left\|\bar\partial^{3} u(t_{3})\right\|\!.$$ Since the initial errors ∥wi − u(ti)∥ are of order $$\mathcal{O}(\tau ^{3})$$, the third-order difference quotient $$\bar \partial ^{3} w^{3}$$ is bounded independently of τ. However, in our case of the fully discrete solution Un of equation (2.9) the initial errors Ui − u(ti), i = 0, 1, 2, 3, fulfill $$ \left\|U^{i} - u(t_{i})\right\| \leq \ C(\tau^{3} + h^{k+1})$$ and applying the above technique directly would result in a condition on the minimal size of the time step depending on h. Therefore, we introduce as a second step in Lemma 5.2 the corresponding time-dependent semidiscrete problem, and estimate the error between a fully and semidiscrete solution. As a result we are able to bound the initial errors by corresponding time derivatives of this semidiscrete solution. In the third step the discrete regularity results of Appendix B are used in Lemma 5.3. This allows us to bound the semidiscrete values by interpolation estimates, and regularity of the exact solution and given data. Finally, in the fourth step, we combine these results to prove Theorem 3.6. Let us first state two stability estimates. Lemma 5.1 (Stability). Assume Xn = Xh ∀ n = 0, …, N. Then the scheme defined in equation (2.9) fulfills the following stability results for all i ≥ 0: \begin{align} {\hskip-40pt}\max_{2+i\leq n\leq N}\|\bar\partial^{i} U^{n}\|^{2} + \sum_{n=2+i}^{N}\tau\|\nabla\bar\partial^{i} U^{n}\|^{2} \lesssim \sum_{n=2+i}^{N}\tau \|\bar\partial^{i} F^{n}\|^{2}_{J_{h}^{\ast}}+\|\bar\partial^{i} U^{1+i}\|^{2} + \|\bar\partial^{i} U^{i}\|^{2}, \end{align} (5.1) \begin{align} \max_{2+i\leq n\leq N}\|\nabla\bar\partial^{i} U^{n}\|^{2} + \sum_{n=2+i}^{N}\tau\|\bar\partial^{i+1} U^{n}\|^{2} \lesssim \sum_{n=2+i}^{N}\tau \|\bar\partial^{i} F^{n}\|^{2}+\|\nabla \bar\partial^{i} U^{1+i}\|^{2} + \tau \|\bar\partial^{i+1} U^{i+1}\|^{2}. \end{align} (5.2) Proof. Apply the time difference operator $$\bar \partial ^{i}$$ to equation (2.9) to obtain for n ≥ i + 2, i ≥ 0, \begin{align} \bar\partial^{i}\bar\partial^{B} U^{n} - \varDelta_{h}\bar\partial^{i} U^{n} + B_{h}^{\top} \bar\partial^{i} P^{n} = \bar\partial^{i} F^{n}. \end{align} (5.3) To prove equation (5.1), test equation (5.3) with $$4\tau \bar \partial ^{i} U^{n}$$. The first term on the left-hand side results in \begin{align*} 4\tau\langle \bar\partial^{i}\bar\partial^{B} U^{n},\bar\partial^{i} U^{n}\rangle =&\, 2\langle3\bar\partial^{i} U^{n}-4\bar\partial^{i} U^{n-1}+ \bar\partial^{i} U^{n-2},\bar\partial^{i} U^{n}\rangle\\ =&\,\|\bar\partial^{i} U^{n}\|^{2}-\|\bar\partial^{i} U^{n-1}\|^{2} + \tau^{4}\|\bar\partial^{i+2}U^{n}\|^{2} + \|2\bar\partial^{i} U^{n}-\bar\partial^{i} U^{n-1}\|^{2}\\ &-\|2\bar\partial^{i} U^{n-1}-\bar\partial^{i} U^{n-2}\|^{2}, \end{align*} where we have used identity (A.8) in the last step. Use Young’s inequality on the right-hand side to get \begin{align*} &\|\bar\partial^{i} U^{n}\|^{2}-\|\bar\partial^{i} U^{n-1}\|^{2} + \tau^{4}\|\bar\partial^{i+2}U^{n}\|^{2} + \|2\bar\partial^{i} U^{n}-\bar\partial^{i} U^{n-1}\|^{2}-\|2\bar\partial^{i} U^{n-1}-\bar\partial^{i} U^{n-2}\|^{2}\\ &\quad+ 4\tau\|\nabla\bar\partial^{i} U^{n}\|^{2} \leq C\tau\|\bar\partial^{i} F^{n}\|^{2}_{J^{\ast}_{h}} + \tau\|\nabla\bar\partial^{i} U^{n}\|^{2}. \end{align*} Summing up from n = 2 + i to N yields equation (5.1). For proving equation (5.2) we test equation (5.3) with $$4\tau \bar \partial ^{i+1}U^{n}$$. By using identity equation (4.20), we obtain for the left-hand side \begin{align*} 4\tau\langle \bar\partial^{i}\bar\partial^{B} U^{n},\bar\partial^{i+1} U^{n}\rangle =&\, 4\tau\left\langle \frac{\tau}{2}\bar\partial^{i+2}U^{n} + \bar\partial^{i+1}U^{n},\bar\partial^{i+1} U^{n}\right\rangle\\ =&\,\tau\left(\|\bar\partial^{i+1}U^{n}\|^{2}-\|\bar\partial^{i+1}U^{n-1}\|^{2}+\tau^{2}\|\bar\partial^{i+2} U^{n}\|^{2} + 4\|\bar\partial^{i+1} U^{n}\|^{2}\right)\!. \end{align*} Use again equation (4.11) for the second term on the left-hand side and Young’s inequality for the right-hand side of equation (5.3) to obtain the error inequality $$ \tau\left(\|\bar\partial^{i+1}U^{n}\|^{2}-\|\bar\partial^{i+1}U^{n-1}\|^{2}\right)+\tau\|\bar\partial^{i+1} U^{n}\|^{2} + \|\nabla\bar\partial^{i}U^{n}\|^{2}-\|\nabla\bar\partial^{i} U^{n-1}\|^{2} \lesssim \tau\|\bar\partial^{i} F^{n}\|^{2}. $$ Summing up from n = 2 + i to N yields equation (5.2). In the next step we present a priori estimates of the corresponding semidiscrete solution of equations (2.9) and (2.10). Lemma 5.2 Let (W(t), Q(t)) ∈ Xh × Yh be solution of the semidiscrete scheme \begin{align} \partial_{t} W(t)-\varDelta_{h} W(t) + B_{h}^{\top} Q(t) = \varPi_{h}\, f(t),\qquad B_{h}W=0,\qquad W(0)= U^{0}. \end{align} (5.4) Then for i = 1, 2, 3 one can estimate $$ \left\|U^{i}-W(t_{i})\right\|+\tau^{1/2}\left\|\nabla\left(U^{i}-W(t_{i})\right)\right\| \! \lesssim\! \tau^{5/2}\!\left(\left\|\partial_{t}^{(3)}W(t)\right\|_{L^{2}((0,3\tau);L^{2}(\varOmega)^{d})} +\!\left\|\partial_{t}^{(2)}f(t)\right\|_{L^{2}((0,\tau);L^{2}(\varOmega)^{d})}\right)\!.$$ Proof. Set Θi := Ui − W(ti) and $$\varPi _{J_{h}}:X\to J_{h}$$ the L2 projection on Jh. The error equation in $$J_{h}^{\ast }$$ corresponding to equation (2.10) reads $$ \varTheta^{1}-\frac{\tau}{2}\varDelta_{h} \varTheta^{1} = \tau R^{1},$$ where $$R^{1}:=(\partial _{t}-\bar \partial )W(t_{1})-\varDelta _{h}(W(t_{1})-W(t_{1/2}))+\varPi _{J_{h}}(f(t_{1})-f(t_{1/2}))$$ fulfilling $$ \|R^{1}\| \ \lesssim \ \tau^{3/2} \left(\left\|\partial_{t}^{(3)}W(t)\right\|_{L^{2}((0,\tau);L^{2}(\varOmega)^{d})} +\left\|\partial_{t}^{(2)}f(t)\right\|_{L^{2}((0,\tau);L^{2}(\varOmega)^{d})}\right)\!. $$ Test this equation by Θ1 to get $$ \|\varTheta^{1}\|^{2} + \tau\left\|\nabla{e_{1}^{1}}\right\|^{2} \lesssim \tau^{2} \|R^{1}\|^{2}.$$ To bound the error Θ2, set $$R^{2}:=(\partial _{t}-\bar \partial ^{B})W(t_{2})$$ fulfilling $$\|R^{2}\| \ \lesssim \ \tau ^{3/2} \|\partial _{t}^{(3)}W(t)\|_{L^{2}((\tau ,2\tau );L^{2}(\varOmega )^{d})}$$. This gives $$ 3\varTheta^{2}-2\tau\varDelta_{h} \varTheta^{2} = 4\varTheta^{1} - 2\tau R^{2}.$$ The same procedure as for Θ1 then yields the estimate for Θ2 and also for Θ3. The above lemma bounds the initial errors Ui − W(ti) in terms of the third-order time derivative of the semidiscrete velocity W. In the next lemma the results from Appendix B are used to further estimate this term. In what follows we have to make use of the inverse estimate \begin{align} \|\nabla V\|\lesssim \frac{1}{h}\|V\|\qquad\forall\ V\in X_{h}. \end{align} (5.5) This is why a condition of quasi-uniform meshes is required. In view of adaptive meshes we would like to drop this condition and use a local variant of equation (5.5). However, the proof of the next lemma needs a global inverse estimate due to the global definition of the operator $$\mathcal{A}_{h}$$ and the Stokes projection $$\mathcal{S}^{u}_{n}$$. Lemma 5.3 Assume that the exact solution u of equation (2.1) is sufficiently regular. Let W be as in Lemma 5.2. Then \begin{align*} \left\|{\partial_{t}^{3}}W\right\|_{L^{2}((0,3\tau);L^{2}(\varOmega)^{d})} \lesssim&\, \left\|{\partial_{t}^{3}}u-\mathcal{S}^{u}_{n}\left({\partial_{t}^{3}}u,{\partial_{t}^{3}}p\right)\right\|_{L^{2}((0,3\tau);L^{2}(\varOmega)^{d})} + \left\|{\partial_{t}^{3}}u\right\|_{L^{2}((0,3\tau);L^{2}(\varOmega)^{d})}\\ &+\frac{1}{h^{3}}\left\|u_{0}-\mathcal{S}^{u}_{n}\left(u_{0},p_{0}\right)\right\| + \frac{1}{h}\left\|\partial_{t}u_{0}-\mathcal{S}^{u}_{n}(\partial_{t} u_{0},\partial_{t} p_{0})\right\|\!. \end{align*} Proof. To simplify the notation we set $$ \mathcal{S}^{u}_{n}(u,p)(t):=\mathcal{S}^{u}_{n}\left(u(t),p(t)\right)\qquad\textrm{and}\qquad{\partial_{t}^{l}}\mathcal{S}^{u}_{n}(u,p)(t) := \mathcal{S}^{u}_{n}\left({\partial_{t}^{l}}u(t),{\partial_{t}^{l}}p(t)\right)\!.$$ The triangle inequality yields $$ \left\|{\partial_{t}^{3}}W\right\|\leq \left\|{\partial_{t}^{3}}(W-\mathcal{S}^{u}_{n}(u,p))\right\|+\left\|{\partial_{t}^{3}}(u-\mathcal{S}^{u}_{n}(u,p))\right\|+\left\|{\partial_{t}^{3}}u\right\|\!.$$ It remains to bound the term $$\|{\partial _{t}^{3}}(W-\mathcal{S}^{u}_{n}(u,p))\|_{L^{2}((0,3\tau );L^{2}(\varOmega )^{d})}$$. To this end we use the results on discrete regularity provided in Appendix B. Subtract the continuous system (2.1) from the semidiscrete system (5.4) and combine the resulting equation with the Stokes projection (2.4) to get $$ \partial_{t}\left(W-\mathcal{S}^{u}_{n}(u,p)\right)(t) + \mathcal{A}_{h}\left(W-\mathcal{S}^{u}_{n}(u,p)\right)(t)= \varPi_{J_{h}}\partial_{t}\left(u(t)-\mathcal{S}^{u}_{n}(u,p)\right)(t),$$ where $$\varPi _{J_{h}}$$ is the L2 projection onto Jh and $$\mathcal{A}_{h}$$ the discrete Stokes operator defined in equation (B.3). For the initial error one has $$ W(0)-\mathcal{S}^{u}_{n}(u_{0},p_{0}) = U^{0} - \mathcal{S}^{u}_{n}(u_{0},p_{0}) = \mathcal{S}^{u}_{n}(u_{0},p_{0})-\mathcal{S}^{u}_{n}(u_{0},p_{0})=0.$$ We use equation (B.7) of Theorem B.2 in the case k = 2 applied to $$ Z(t)=W(t)-\mathcal{S}^{u}_{n}(u,p)(t),\; G=\varPi_{J_{h}}\partial_{t}\left(u-\mathcal{S}^{u}_{n}(u,p)\right)(t)\;\textrm{ and }Z(0)=0.$$ Hence we get \begin{align*} \left\|{\partial_{t}^{3}}\left(W-\mathcal{S}^{u}_{n}(u,p)\right)\right\|_{L^{2}((0,3\tau);L^{2}(\varOmega)^{d})} \lesssim&\, \left\|{\partial_{t}^{3}}\left(u-\mathcal{S}^{u}_{n}( u,p)\right)\right\|_{L^{2}((0,3\tau);L^{2}(\varOmega)^{d}) }\\ &+ \left\|\nabla \varPi_{J_{h}}\left(u_{0}-\mathcal{S}^{u}_{n}(u_{0},p_{0})\right)\right\| + \left\|\nabla \varPi_{J_{h}}\partial_{t}\left(u_{0}-\mathcal{S}^{u}_{n}(u_{0},p_{0})\right)\right\|\\ &+ \left\|\nabla \mathcal{A}_{h} \varPi_{J_{h}}\left(u_{0}-\mathcal{S}^{u}_{n}(u_{0},p_{0})\right)\right\|\!. \end{align*} The second and third term on the right-hand side can be bounded with the help of the inverse estimate (5.5) and the stability of the L2 projection: \begin{align*} \left\|\nabla \varPi_{J_{h}}\left(u_{0}-\mathcal{S}^{u}_{n}(u_{0},p_{0})\right)\right\| &\lesssim \frac{1}{h}\left\| \varPi_{J_{h}}\left(u_{0}-\mathcal{S}^{u}_{n}(u_{0},p_{0})\right)\right\| \leq \frac{1}{h} \left\|\left(u_{0}-\mathcal{S}^{u}_{n}(u_{0},p_{0})\right)\right\|\!,\\ \left\|\nabla \varPi_{J_{h}}\partial_{t}\left(u_{0}-\mathcal{S}^{u}_{n}(u_{0},p_{0})\right)\right\| &\lesssim \frac{1}{h} \left\|\partial_{t}\left(u_{0}-\mathcal{S}^{u}_{n}(u_{0},p_{0})\right)\right\|\!. \end{align*} For the last term on the right-hand side we apply the inverse estimate (5.5) once again and use the definition of the discrete Stokes operator, cf. (B.12): \begin{align*} \left\|\nabla \mathcal{A}_{h} \varPi_{J_{h}}\left(u_{0}-\mathcal{S}^{u}_{n}(u_{0},p_{0})\right)\right\|^{2} &\lesssim \frac{1}{h^{2}}\left\|\mathcal{A}_{h} \varPi_{J_{h}}\left(u_{0}-\mathcal{S}^{u}_{n}(u_{0},p_{0})\right)\right\|^{2} \\ &=\frac{1}{h^{2}}\left\langle \nabla \varPi_{J_{h}}\left(u_{0}-\mathcal{S}^{u}_{n}(u_{0},p_{0})\right),\nabla \mathcal{A}_{h}\varPi_{J_{h}}\left(u_{0}-\mathcal{S}^{u}_{n}(u_{0},p_{0})\right)\right\rangle \\ &\leq \frac{1}{h^{3}}\left\|\left(u_{0}-\mathcal{S}^{u}_{n}(u_{0},p_{0})\right)\right\|\left\|\nabla \mathcal{A}_{h}\varPi_{J_{h}}\left(u_{0}-\mathcal{S}^{u}_{n}(u_{0},p_{0})\right)\right\|\!. \end{align*} Dividing the last equation by $$\|\nabla \mathcal{A}_{h} \varPi _{J_{h}}(u_{0}-\mathcal{S}^{u}_{n}(u_{0},p_{0}))\|$$ finally together with the above estimates concludes the lemma. Now we have all the ingredients to prove Theorem 3.6. Proof. (Theorem 3.6) Use equation (5.1) for i = 2 to bound the estimators $$\mathcal{E}_{1}$$ and $$\mathcal{E}_{3}$$ $$ \mathcal{E}_{1}\lesssim \mathcal{E}_{3} \lesssim \sum_{n=4}^{N}\tau\|\bar\partial^{2} F^{n}\|^{2}_{J_{h}^{\ast}} + \|\bar\partial^{2} U^{2}\|^{2} + \|\bar\partial^{2} U^{3}\|^{2}$$ and use equation (5.2) for i = 2 to bound the estimators $$\mathcal{E}_{2}$$ and $$\mathcal{E}_{6}$$–$$\mathcal{E}_{7}$$: $$ \mathcal{E}_{2} + \mathcal{E}_{7} \leq \mathcal{E}_{6} + \mathcal{E}_{7} \lesssim \sum_{n=4}^{N}\tau\|\bar\partial^{2} F^{n}\|^{2} + \|\nabla\bar\partial^{2} U^{2}\|^{2} + \|\nabla\bar\partial^{2} U^{3}\|^{2} +\tau\|\bar\partial^{3}U^{2}\|+\tau\|\bar\partial^{3}U^{3}\|.$$ The estimators $$\mathcal{E}_{4}$$ and $$\mathcal{E}_{5}$$ can easily be handled by the triangle inequality: $$ \mathcal{E}_{4}\leq \mathcal{E}_{5} \lesssim \sum_{n=2}^{N}\tau\left\|\bar\partial^{2} f(t_{n})\right\|^{2} + \tau \|\bar\partial^{3} U^{1}\|^{2} +\mathcal{E}_{6}.$$ It is clear that the term $$\sum _{n=4}^{N}\tau \|\bar \partial ^{2} F^{n}\|^{2}$$ is bounded by regularity of f. It remains to bound the quantities $$\|\bar \partial ^{2} U^{2}\|^{2}$$, $$\|\bar \partial ^{2} U^{3}\|^{2}$$, $$\|\nabla \bar \partial ^{2} U^{2}\|^{2}$$, $$\|\nabla \bar \partial ^{2} U^{3}\|^{2}$$ and $$\tau \|\bar \partial ^{3} U^{3}\|$$. Note that the terms $$\tau \|\bar \partial ^{3} U^{1}\|$$ and $$\tau \|\bar \partial ^{3} U^{2}\|$$ can be absorbed into the data approximation error estimators $$ {\textit{I}}_{1}$$ and $$ {\textit{I}}_{2}$$ and in lower-order difference quotients. Let W(ti) be the semidiscrete solution of equation (5.4). Then for instance the term $$\tau \|\bar \partial ^{3} U^{3}\|^{2}$$ can be split by the triangle inequality: $$ \sqrt{\tau}\|\bar\partial^{3} U^{3}\| \leq \sqrt{\tau}\left\|\bar\partial^{3} \big(U^{3}-W(t_{3})\big)\right\| + \sqrt{\tau}\left\|\bar\partial^{3} W(t_{3})\right\|\!.$$ The first term on the right-hand side can be estimated with the help of Lemma 5.2: \begin{align*} \sqrt{\tau}\left\|\bar\partial^{3} \left(U^{3}-W(t_{3})\right)\right\| &\lesssim \tau^{-5/2}\left(\left\|U^{1}-W(t_{1})\right\|+\left\|U^{2}-W(t_{2})\right\|+\left\|U^{3}-W(t_{3})\right\|\right)\\ &\lesssim \left\|\partial_{t}^{(3)}W(t)\right\|_{L^{2}((0,3\tau);L^{2}(\varOmega)^{d})}+\left\|\partial_{t}^{(2)}f(t)\right\|_{L^{2}((0,\tau);L^{2}(\varOmega)^{d})}\!. \end{align*} It is easily shown that the second term fulfills $$ \sqrt{\tau }\|\bar \partial ^{3} W(t_{3})\|\lesssim \|\partial _{t}^{(3)}W(t)\|_{L^{2}((0,3\tau );L^{2}(\varOmega )^{d})}$$. Collect these results and use Lemma 5.3 to get \begin{align*} \sqrt{\tau}\|\bar\partial^{3} U^{3}\| \lesssim&\, \left\|\partial_{t}^{(3)}u-\mathcal{S}^{u}_{n}\left(\partial_{t}^{(3)}u,\partial_{t}^{(3)}p\right)\right\|_{L^{2}((0,3\tau);L^{2}(\varOmega)^{d})} + \left\|\partial_{t}^{(3)}u\right\|_{L^{2}((0,3\tau);L^{2}(\varOmega)^{d})}\\ &+\frac{1}{h^{3}}\left\|u_{0}-\mathcal{S}^{u}_{n}(u_{0},p_{0})\right\| + \frac{1}{h}\left\|\partial_{t}u_{0} - \mathcal{S}^{u}_{n}(\partial_{t} u_{0},\partial_{t} p_{0})\right\| +\left\|\partial_{t}^{(2)}f(t)\right\|_{L^{2}((0,\tau);L^{2}(\varOmega)^{d})}\!. \end{align*} Finally use Lemma 2.2 to bound the approximation errors $$\|u(\cdot )-\mathcal{S}^{u}_{n}(u(\cdot ),p(\cdot ))\|$$ to conclude the proof of the error bound of $$\sqrt{\tau }\|\bar \partial ^{3} U^{3}\|$$. Note that at this point we need at least order k = 2 elements for the velocity space Xh to bound the term $$\frac{1}{h^{3}}\|u_{0}-\mathcal{S}^{u}_{n}(u_{0},p_{0})\|$$. The other terms $$\|\bar \partial ^{2} U^{2}\|^{2}$$, $$\|\bar \partial ^{2} U^{3}\|^{2}$$, $$\|\nabla \bar \partial ^{2} U^{2}\|^{2}$$ and $$\|\nabla \bar \partial ^{2} U^{3}\|^{2}$$ can be treated in a similar way. 6. Computational results In this section computational examples are presented to numerically verify our analysis. The numerical simulations are calculated by NAVIER (Bänsch, 1998) and FENICS (Alnæs et al., 2015). 6.1 Asymptotic behavior of the temporal estimators In this first example we show the asymptotic behavior of the time and Stokes reconstruction error estimators as stated in Theorem 3.4. We chose Ω =]0, 1[2 and \begin{align*} u(t, x) &= \left(\pi\sin(t)\sin(2\pi x_{2})\sin^{2}(\pi x_{1}),\: -\pi\sin(t)\sin(2\pi x_{1})\sin^{2}(\pi x_{2})\right)^{\top}\! , \\ p(t, x) &= \sin(t)\cos(\pi x_{1})\sin(\pi x_{2}), \end{align*} with a right-hand side f, such that the above pair (u, p) is a solution to equation (2.1). The time interval was [0, 1]. As we are only interested here in the time estimators, we use a constant finite element $$\mathcal{P}_{2}-\mathcal{P}_{1}$$ Taylor–Hood discretization to compute the numerical solutions and a fine grid consisting of 16641 vertices. Consequently, the error due to space discretization was very small. For nonchanging meshes the estimators of Definition 3.3 read \begin{align*} \mathcal{E}_{1} &:= \sum_{n=2}^{N}\tau\|\bar\partial^{2} U^{n}\|^{2}, & \mathcal{E}_{4} &:= \sum_{n=2}^{N}\tau\left\|\bar\partial^{2}\left(f(t_{n})-\bar\partial^{B} U^{n}\right)\right\|^{2}_{J^{\ast}}, & \mathcal{E}_{7} &:= \max_{2\leq n\leq N}\|\nabla\bar\partial^{2} U^{n}\|^{2},\\ \mathcal{E}_{2} &:= \sum_{n=3}^{N}\tau\|\bar\partial^{3} U^{n}\|^{2}_{J^{\ast}}, & \mathcal{E}_{5} &:= \sum_{n=2}^{N}\tau\left\|\bar\partial^{2}\left(\,f(t_{n})-\bar\partial^{B} U^{n}\right)\right\|^{2}\, & &\\ \mathcal{E}_{3} &:= \max_{2\leq n\leq N}\|\bar\partial^{2} U^{n}\|^{2}, & \mathcal{E}_{6} &:= \sum_{n=3}^{N}\tau\|\bar\partial^{3} U^{n}\|^{2}, & & \end{align*} and we set \begin{align*} {\varTheta^{2}_{1}} &:= \tau^{4}(\mathcal{E}_{1}+\mathcal{E}_{2}), & {\varTheta^{2}_{2}} &:= \tau^{4}(\mathcal{E}_{2}+\mathcal{E}_{3}+\mathcal{E}_{4}), & {\varTheta^{2}_{3}} &:= \tau^{4}(\mathcal{E}_{2}+\mathcal{E}_{4}),\\{\varTheta^{2}_{4}} &:= \tau^{4}(\mathcal{E}_{5}+\mathcal{E}_{6}+\mathcal{E}_{7}), & {\varTheta^{2}_{5}} &:= \tau^{4}(\mathcal{E}_{5}+\mathcal{E}_{6}). & & \end{align*} With this notation Theorem 3.4 reads \begin{align*} \|U-u\|^{2}_{L^{2}(L^{2})} &\lesssim{\varTheta^{2}_{1}}+ \eta^{0}(J^{\ast}) + \varLambda_{1} + \mathcal{F}_{1} + {\textit{I}}_{1},\\ \|U-u\|^{2}_{L^{\infty}(L^{2})} &\lesssim{\varTheta^{2}_{2}} + \eta^{0}(L^{2}) + \varLambda_{2} + \varLambda_{3} + \mathcal{F}_{1}+ {\textit{I}}_{1},\\ \left\|\nabla(U-u)\right\|^{2}_{L^{2}(L^{2})} &\lesssim{\varTheta^{2}_{3}} + \eta^{0}(L^{2}) + \varLambda_{3} + \varLambda_{4} + \mathcal{F}_{1}+ {\textit{I}}_{1},\\ \left\|\nabla(U-u)(t)\right\|^{2}_{L^{\infty}(L^{2})} &\lesssim{\varTheta^{2}_{4}} + \eta^{0}\left({H_{0}^{1}}\right) + \varLambda_{5}+\varLambda_{6} + \mathcal{F}_{2}+ {\textit{I}}_{2},\\ \|P-p\|^{2}_{L^{2}(L^{2})} &\lesssim{\varTheta^{2}_{5}} +\eta^{0}\left({H_{0}^{1}}\right) + \varLambda_{6} + \varLambda_{7} +\mathcal{F}_{2}+ {\textit{I}}_{2}. \end{align*} For simplicity only the Θ- terms are presented, since the other terms are less interesting in the present context. Figure 1 shows the asymptotic orders of the estimators for velocity and pressure. As can be seen, the order of estimators and errors are in perfect agreement as long as the time error is dominant. For smaller time step sizes the curves for the errors eventually start to stagnate, since then the spatial discretization error becomes dominant. Fig. 1. View largeDownload slide BDF2: errors and (scaled) estimators vs. time step size. Triangles have slope 2.0. Fig. 1. View largeDownload slide BDF2: errors and (scaled) estimators vs. time step size. Triangles have slope 2.0. 6.2 Robustness of the error estimators In this second computation we want to test the estimator of Definition 3.3 for robustness in a more complex example whose solution is nonseparable with respect to the time and space variable. We chose the time interval [0, 1], Ω =]0, 1[2 and \begin{align*} u(t, x) &= \Big(x_{2}\sin\!\left(2\pi\ t\left({x_{1}^{2}}+ {x_{2}^{2}}\right)\right),\: -x_{1}\sin\!\left(2\pi\ t\left({x_{1}^{2}} +{x_{2}^{2}}\right)\right)\Big)^{\top}, \end{align*} \begin{align*} p(t, x) &= x_{1}x_{2}\sin\!\left(2\pi\ t\left({x_{1}^{2}}+{x_{2}^{2}}\right)\right)\!,\end{align*} with a right-hand side f, such that (u, p) is a solution to equation (2.1). For the spatial estimators we follow Verführt (2013) and set \begin{align*} {\eta_{u}^{n}}\left(U^{n};{H_{0}^{1}}\right)&:=\sum_{T\in\mathcal{T}_{n}}\left({h_{T}^{2}}\left\|\varDelta_{n} U^{n} + F^{n}- B_{n}^{\top} P^{n}\right\|^{2}_{T} + \|\!\operatorname{div} U^{n}\|^{2}\right) +\sum_{e\in\mathcal{E}_{\mathcal{T}}}h_{e}\left\|\left[n_{e}\cdot(\nabla U^{n}-P^{n}I)\right]\right\|^{2}_{e},\\{\eta_{u}^{n}}(U^{n};L^{2}) &:=\sum_{T\in\mathcal{T}_{n}}\left({h_{T}^{4}}\left\|\varDelta_{n} U^{n} + F^{n}- B_{n}^{\top} P^{n}\right\|^{2}_{T} + {h_{T}^{2}}\|\!\operatorname{div} U^{n}\|^{2}\right) +\sum_{e\in\mathcal{E}_{\mathcal{T}}}{h_{e}^{3}}\left\|\left[n_{e}\!\cdot\!(\nabla U^{n}-P^{n}I)\right]\right\|^{2}_{e}. \end{align*} To shorten the presentation we present only the spatial estimators$$\varLambda_{1} $$, $$\varLambda_{2} $$, $$\varLambda_{4} $$ and $$\varLambda_{5} $$. Table 1 shows the values of the estimators on a constant finite element $$\mathcal{P}_{2}-\mathcal{P}_{1}$$ Taylor–Hood discretization with 16641 vertices (h = 0.011). As claimed in Theorem 3.6 the error estimators $$\mathcal{E}_{1}$$–$$\mathcal{E}_{7}$$ are independent of the time step τ. Also the space estimators do not depend on the time step size. Table 1 Temporal and space estimators for different time step sizes τ on a constant mesh with h = 0.011 τ $$\mathcal{E}_{1}^{1/2}$$ $$\mathcal{E}_{2}^{1/2}$$ $$\mathcal{E}_{3}^{1/2}$$ $$\mathcal{E}_{4}^{1/2}$$ $$\mathcal{E}_{5}^{1/2}$$ $$\mathcal{E}_{6}^{1/2}$$ $$\mathcal{E}_{7}^{1/2}$$ $$\varLambda _{1}^{1/2}$$ $$\varLambda _{2}^{1/2}$$ $$\varLambda _{4}^{1/2}$$ $$\varLambda _{5}^{1/2}$$ 0.2 6.49 0.15 8.88 1.68 263 32 54 0.000251 0.000288 0.0227 0.0261 0.1 7.47 0.17 9.60 2.18 374 35 58 0.000244 0.000305 0.0221 0.0276 0.05 7.28 0.16 9.83 2.01 331 34 59 0.000224 0.000313 0.0203 0.0283 0.025 7.35 0.16 9.88 2.05 340 33 59 0.000223 0.000316 0.0202 0.0286 0.0125 7.44 0.16 9.90 2.10 353 33 59 0.000225 0.000316 0.0204 0.0286 0.00625 7.42 0.16 9.90 2.10 351 33 59 0.000224 0.000317 0.0203 0.0287 τ $$\mathcal{E}_{1}^{1/2}$$ $$\mathcal{E}_{2}^{1/2}$$ $$\mathcal{E}_{3}^{1/2}$$ $$\mathcal{E}_{4}^{1/2}$$ $$\mathcal{E}_{5}^{1/2}$$ $$\mathcal{E}_{6}^{1/2}$$ $$\mathcal{E}_{7}^{1/2}$$ $$\varLambda _{1}^{1/2}$$ $$\varLambda _{2}^{1/2}$$ $$\varLambda _{4}^{1/2}$$ $$\varLambda _{5}^{1/2}$$ 0.2 6.49 0.15 8.88 1.68 263 32 54 0.000251 0.000288 0.0227 0.0261 0.1 7.47 0.17 9.60 2.18 374 35 58 0.000244 0.000305 0.0221 0.0276 0.05 7.28 0.16 9.83 2.01 331 34 59 0.000224 0.000313 0.0203 0.0283 0.025 7.35 0.16 9.88 2.05 340 33 59 0.000223 0.000316 0.0202 0.0286 0.0125 7.44 0.16 9.90 2.10 353 33 59 0.000225 0.000316 0.0204 0.0286 0.00625 7.42 0.16 9.90 2.10 351 33 59 0.000224 0.000317 0.0203 0.0287 Table 1 Temporal and space estimators for different time step sizes τ on a constant mesh with h = 0.011 τ $$\mathcal{E}_{1}^{1/2}$$ $$\mathcal{E}_{2}^{1/2}$$ $$\mathcal{E}_{3}^{1/2}$$ $$\mathcal{E}_{4}^{1/2}$$ $$\mathcal{E}_{5}^{1/2}$$ $$\mathcal{E}_{6}^{1/2}$$ $$\mathcal{E}_{7}^{1/2}$$ $$\varLambda _{1}^{1/2}$$ $$\varLambda _{2}^{1/2}$$ $$\varLambda _{4}^{1/2}$$ $$\varLambda _{5}^{1/2}$$ 0.2 6.49 0.15 8.88 1.68 263 32 54 0.000251 0.000288 0.0227 0.0261 0.1 7.47 0.17 9.60 2.18 374 35 58 0.000244 0.000305 0.0221 0.0276 0.05 7.28 0.16 9.83 2.01 331 34 59 0.000224 0.000313 0.0203 0.0283 0.025 7.35 0.16 9.88 2.05 340 33 59 0.000223 0.000316 0.0202 0.0286 0.0125 7.44 0.16 9.90 2.10 353 33 59 0.000225 0.000316 0.0204 0.0286 0.00625 7.42 0.16 9.90 2.10 351 33 59 0.000224 0.000317 0.0203 0.0287 τ $$\mathcal{E}_{1}^{1/2}$$ $$\mathcal{E}_{2}^{1/2}$$ $$\mathcal{E}_{3}^{1/2}$$ $$\mathcal{E}_{4}^{1/2}$$ $$\mathcal{E}_{5}^{1/2}$$ $$\mathcal{E}_{6}^{1/2}$$ $$\mathcal{E}_{7}^{1/2}$$ $$\varLambda _{1}^{1/2}$$ $$\varLambda _{2}^{1/2}$$ $$\varLambda _{4}^{1/2}$$ $$\varLambda _{5}^{1/2}$$ 0.2 6.49 0.15 8.88 1.68 263 32 54 0.000251 0.000288 0.0227 0.0261 0.1 7.47 0.17 9.60 2.18 374 35 58 0.000244 0.000305 0.0221 0.0276 0.05 7.28 0.16 9.83 2.01 331 34 59 0.000224 0.000313 0.0203 0.0283 0.025 7.35 0.16 9.88 2.05 340 33 59 0.000223 0.000316 0.0202 0.0286 0.0125 7.44 0.16 9.90 2.10 353 33 59 0.000225 0.000316 0.0204 0.0286 0.00625 7.42 0.16 9.90 2.10 351 33 59 0.000224 0.000317 0.0203 0.0287 Table 2 Temporal and space estimators for different mesh sizes h and fixed τ = 0.02 h $$\mathcal{E}_{1}^{1/2}$$ $$\mathcal{E}_{2}^{1/2}$$ $$\mathcal{E}_{3}^{1/2}$$ $$\mathcal{E}_{4}^{1/2}$$ $$\mathcal{E}_{5}^{1/2}$$ $$\mathcal{E}_{6}^{1/2}$$ $$\mathcal{E}_{7}^{1/2}$$ $$\varLambda _{1}^{1/2}$$ $$\varLambda _{2}^{1/2}$$ $$\varLambda _{4}^{1/2}$$ $$\varLambda _{5}^{1/2}$$ 0.3535 7.39 0.16 9.90 2.03 337 34 59 0.85383 1.90111 2.415 5.377 0.1767 7.37 0.16 9.89 2.06 342 33 59 0.12283 0.26697 0.694 1.510 0.0883 7.37 0.16 9.89 2.06 342 33 59 0.01962 0.03554 0.222 0.402 0.0441 7.37 0.16 9.89 2.06 342 33 59 0.00393 0.00541 0.089 0.122 0.0220 7.37 0.16 9.89 2.06 342 33 59 0.00091 0.00126 0.041 0.057 0.0110 7.37 0.16 9.89 2.06 342 33 59 0.00022 0.00031 0.020 0.028 h $$\mathcal{E}_{1}^{1/2}$$ $$\mathcal{E}_{2}^{1/2}$$ $$\mathcal{E}_{3}^{1/2}$$ $$\mathcal{E}_{4}^{1/2}$$ $$\mathcal{E}_{5}^{1/2}$$ $$\mathcal{E}_{6}^{1/2}$$ $$\mathcal{E}_{7}^{1/2}$$ $$\varLambda _{1}^{1/2}$$ $$\varLambda _{2}^{1/2}$$ $$\varLambda _{4}^{1/2}$$ $$\varLambda _{5}^{1/2}$$ 0.3535 7.39 0.16 9.90 2.03 337 34 59 0.85383 1.90111 2.415 5.377 0.1767 7.37 0.16 9.89 2.06 342 33 59 0.12283 0.26697 0.694 1.510 0.0883 7.37 0.16 9.89 2.06 342 33 59 0.01962 0.03554 0.222 0.402 0.0441 7.37 0.16 9.89 2.06 342 33 59 0.00393 0.00541 0.089 0.122 0.0220 7.37 0.16 9.89 2.06 342 33 59 0.00091 0.00126 0.041 0.057 0.0110 7.37 0.16 9.89 2.06 342 33 59 0.00022 0.00031 0.020 0.028 Table 2 Temporal and space estimators for different mesh sizes h and fixed τ = 0.02 h $$\mathcal{E}_{1}^{1/2}$$ $$\mathcal{E}_{2}^{1/2}$$ $$\mathcal{E}_{3}^{1/2}$$ $$\mathcal{E}_{4}^{1/2}$$ $$\mathcal{E}_{5}^{1/2}$$ $$\mathcal{E}_{6}^{1/2}$$ $$\mathcal{E}_{7}^{1/2}$$ $$\varLambda _{1}^{1/2}$$ $$\varLambda _{2}^{1/2}$$ $$\varLambda _{4}^{1/2}$$ $$\varLambda _{5}^{1/2}$$ 0.3535 7.39 0.16 9.90 2.03 337 34 59 0.85383 1.90111 2.415 5.377 0.1767 7.37 0.16 9.89 2.06 342 33 59 0.12283 0.26697 0.694 1.510 0.0883 7.37 0.16 9.89 2.06 342 33 59 0.01962 0.03554 0.222 0.402 0.0441 7.37 0.16 9.89 2.06 342 33 59 0.00393 0.00541 0.089 0.122 0.0220 7.37 0.16 9.89 2.06 342 33 59 0.00091 0.00126 0.041 0.057 0.0110 7.37 0.16 9.89 2.06 342 33 59 0.00022 0.00031 0.020 0.028 h $$\mathcal{E}_{1}^{1/2}$$ $$\mathcal{E}_{2}^{1/2}$$ $$\mathcal{E}_{3}^{1/2}$$ $$\mathcal{E}_{4}^{1/2}$$ $$\mathcal{E}_{5}^{1/2}$$ $$\mathcal{E}_{6}^{1/2}$$ $$\mathcal{E}_{7}^{1/2}$$ $$\varLambda _{1}^{1/2}$$ $$\varLambda _{2}^{1/2}$$ $$\varLambda _{4}^{1/2}$$ $$\varLambda _{5}^{1/2}$$ 0.3535 7.39 0.16 9.90 2.03 337 34 59 0.85383 1.90111 2.415 5.377 0.1767 7.37 0.16 9.89 2.06 342 33 59 0.12283 0.26697 0.694 1.510 0.0883 7.37 0.16 9.89 2.06 342 33 59 0.01962 0.03554 0.222 0.402 0.0441 7.37 0.16 9.89 2.06 342 33 59 0.00393 0.00541 0.089 0.122 0.0220 7.37 0.16 9.89 2.06 342 33 59 0.00091 0.00126 0.041 0.057 0.0110 7.37 0.16 9.89 2.06 342 33 59 0.00022 0.00031 0.020 0.028 Table 2 presents the values of the estimators for refining meshes with halving mesh size h and constant time step τ = 0.02. The time estimators $$\mathcal{E}_{1}$$–$$\mathcal{E}_{7}$$ are independent of the mesh size h. This shows the robustness of the temporal estimators with respect to the spatial discretization. The space estimators behave as expected. We have quadratic convergence for the L2 estimators $$\varLambda _{1}^{1/2}$$ and $$\varLambda _{2}^{1/2}$$ and linear convergence for the H1 estimators $$\varLambda _{4}^{1/2}$$ and $$\varLambda _{5}^{1/2}$$. In our next computation we want to show the robustness of our time error estimator with respect to the end time T. To this end we consider a solution of equation (2.1), which is periodic in time: $$ u(t, x) = \Big(\sin(t)\sin(2\pi(x_{1}+ x_{2})),\: -\sin(t)\sin(2\pi(x_{1} +x_{2}))\Big)^{\top},\qquad p(t, x) = 0, $$ with a corresponding right-hand side f in the time interval [0, 4π] on Ω =]0, 1[2. We set τ = 0.05 and use the same spatial discretization as in the example before. In Fig. 2 we compare the local contributions $$\|\bar \partial ^{2} U^{n}\|$$ and $$\|\bar \partial ^{3} U^{n}\|$$ of the time estimators with the time derivatives $$\|{\partial _{t}^{2}} u(t_{n})\|$$ and $$\|{\partial _{t}^{3}} u(t_{n})\|$$ of the exact solution in view of a dependence on the final time T. As can be seen the difference quotients of the discrete solution are in perfect agreement with the continuous time derivatives, and we conclude the robustness of our estimators with respect to the end time T. Fig. 2. View largeDownload slide Time versus local contributions $$\|\bar \partial ^{2} U^{n}\|$$ (left), $$\|\bar \partial ^{3} U^{n}\|$$ (right) of the time estimator and $$\|{\partial _{t}^{2}} u(t_{n})\|$$ (left), $$\|{\partial _{t}^{3}} u(t_{n})\|$$ (right) of the exact solution for end time T = 4π. Fig. 2. View largeDownload slide Time versus local contributions $$\|\bar \partial ^{2} U^{n}\|$$ (left), $$\|\bar \partial ^{3} U^{n}\|$$ (right) of the time estimator and $$\|{\partial _{t}^{2}} u(t_{n})\|$$ (left), $$\|{\partial _{t}^{3}} u(t_{n})\|$$ (right) of the exact solution for end time T = 4π. Fig. 3. View largeDownload slide Example with mesh change. L2 pressure error versus time (initial mesh with 1024 elements, τ=1.25e-3). Fig. 3. View largeDownload slide Example with mesh change. L2 pressure error versus time (initial mesh with 1024 elements, τ=1.25e-3). 6.3 Mesh changes With this example we study the effect of mesh modification. The domain here is Ω = ] − 0.5, 0.5[2, T = 1.5 and the exact solution is \begin{align} \begin{aligned} u(x,y,t)& = \pi\sin(t+0.25)(\sin(2\pi y)\sin(\pi x)^{2},\sin(2\pi x)\sin(\pi y)^{2})^{T},\\ p(x,y,t)&= \sin(t+0.25)\cos(\pi x)\sin(\pi y). \end{aligned} \end{align} (6.1) In order to focus on the mesh change, the setting is as follows. The computations are started with different uniform meshes with mesh size h0 (corresponding to 256, 1024, 4096 and 16384 elements, respectively) and uniform time step sizes τ. The initial mesh is created by uniformly refining a given macro triangulation with the bisection method (Bänsch, 1991b). At the time instant tjump = 1.0 the mesh is derefined, resulting in a coarsened mesh with mesh size 2 h0. This scenario is as in the study by Besier & Wollner (2012). Since the pressure is the unknown most sensitive to mesh changes, we study the effect of mesh modification for this variable. In the study by Besier & Wollner (2012), it was shown that a derefinement of the mesh results in a jump of the pressure error of the order 1/τ. This behavior is confirmed by our experiments. In Fig. 3 the L2 error of the pressure is shown as a function of time. The huge jump in the error at tjump is clearly visible. More quantitative information is given in Tables 3 and 4. In Table 3 the jumps δEp in the pressure error are listed for various τ and h0. Here, δEp is defined as the ‘additional’ contribution to the pressure error by the mesh change: \begin{align} \delta E_{p} := \| P^{n_{0}}-p(t_{\textrm{jump}}) \| - \| P^{n_{0}-1}-p(t_{\textrm{jump}}-\tau) \|, \end{align} (6.2) Table 3 Example with mesh change: ‘additional’ pressure error δEp equation (6.2) for various meshes (nt = no. of initial elements) and time step sizes τ τ 2.e−2 1.e−2 5.e−3 2.5e−3 1.25e−3 nt= 256 8.3534e−02 1.5880e−01 3.0628e−01 5.9729e−01 1.1755e+00 nt=1024 5.6903e−03 1.0481e−02 2.0098e−02 3.9038e−02 7.6344e−02 nt=4096 5.9363e−04 8.1377e−04 1.3692e−03 2.5847e−03 5.0541e−03 nt=16384 1.2083e−04 1.2638e−04 1.4346e−04 1.9564e−04 3.3151e−04 τ 2.e−2 1.e−2 5.e−3 2.5e−3 1.25e−3 nt= 256 8.3534e−02 1.5880e−01 3.0628e−01 5.9729e−01 1.1755e+00 nt=1024 5.6903e−03 1.0481e−02 2.0098e−02 3.9038e−02 7.6344e−02 nt=4096 5.9363e−04 8.1377e−04 1.3692e−03 2.5847e−03 5.0541e−03 nt=16384 1.2083e−04 1.2638e−04 1.4346e−04 1.9564e−04 3.3151e−04 View Large Table 3 Example with mesh change: ‘additional’ pressure error δEp equation (6.2) for various meshes (nt = no. of initial elements) and time step sizes τ τ 2.e−2 1.e−2 5.e−3 2.5e−3 1.25e−3 nt= 256 8.3534e−02 1.5880e−01 3.0628e−01 5.9729e−01 1.1755e+00 nt=1024 5.6903e−03 1.0481e−02 2.0098e−02 3.9038e−02 7.6344e−02 nt=4096 5.9363e−04 8.1377e−04 1.3692e−03 2.5847e−03 5.0541e−03 nt=16384 1.2083e−04 1.2638e−04 1.4346e−04 1.9564e−04 3.3151e−04 τ 2.e−2 1.e−2 5.e−3 2.5e−3 1.25e−3 nt= 256 8.3534e−02 1.5880e−01 3.0628e−01 5.9729e−01 1.1755e+00 nt=1024 5.6903e−03 1.0481e−02 2.0098e−02 3.9038e−02 7.6344e−02 nt=4096 5.9363e−04 8.1377e−04 1.3692e−03 2.5847e−03 5.0541e−03 nt=16384 1.2083e−04 1.2638e−04 1.4346e−04 1.9564e−04 3.3151e−04 View Large Table 4 Example with mesh change: estimator $$\sqrt{\tilde{\xi }_{B}^{n}}$$ for various meshes (nt = no. of initial elements) and time step sizes τ τ 2.e−2 1.e−2 5.e−3 2.5e−3 1.25e−3 nt=256 9.2077e−01 1.8415e+00 3.6831e+00 7.3662e+00 1.4732e+01 nt=1024 1.3887e−01 2.7773e−01 5.5546e−01 1.1109e+00 2.2218e+00 nt=4096 1.8840e−02 3.7681e−02 7.5361e−02 1.5072e−01 3.0144e−01 nt=16384 2.4194e−03 4.8389e−03 9.6779e−03 1.9356e−02 3.8712e−02 τ 2.e−2 1.e−2 5.e−3 2.5e−3 1.25e−3 nt=256 9.2077e−01 1.8415e+00 3.6831e+00 7.3662e+00 1.4732e+01 nt=1024 1.3887e−01 2.7773e−01 5.5546e−01 1.1109e+00 2.2218e+00 nt=4096 1.8840e−02 3.7681e−02 7.5361e−02 1.5072e−01 3.0144e−01 nt=16384 2.4194e−03 4.8389e−03 9.6779e−03 1.9356e−02 3.8712e−02 View Large Table 4 Example with mesh change: estimator $$\sqrt{\tilde{\xi }_{B}^{n}}$$ for various meshes (nt = no. of initial elements) and time step sizes τ τ 2.e−2 1.e−2 5.e−3 2.5e−3 1.25e−3 nt=256 9.2077e−01 1.8415e+00 3.6831e+00 7.3662e+00 1.4732e+01 nt=1024 1.3887e−01 2.7773e−01 5.5546e−01 1.1109e+00 2.2218e+00 nt=4096 1.8840e−02 3.7681e−02 7.5361e−02 1.5072e−01 3.0144e−01 nt=16384 2.4194e−03 4.8389e−03 9.6779e−03 1.9356e−02 3.8712e−02 τ 2.e−2 1.e−2 5.e−3 2.5e−3 1.25e−3 nt=256 9.2077e−01 1.8415e+00 3.6831e+00 7.3662e+00 1.4732e+01 nt=1024 1.3887e−01 2.7773e−01 5.5546e−01 1.1109e+00 2.2218e+00 nt=4096 1.8840e−02 3.7681e−02 7.5361e−02 1.5072e−01 3.0144e−01 nt=16384 2.4194e−03 4.8389e−03 9.6779e−03 1.9356e−02 3.8712e−02 View Large where n0 corresponds to the time instant tjump = n0τ. Notice that our theory does not provide error control for δEp. This term is used in our experiments only as a tool to quantify the observed jump in the pressure. The comparison then to our coarsening estimator is indicative only in order to access whether or not our estimator can detect the jump in pressure. Since we perform only one mesh change, a simplified mesh change estimator $$\tilde{\xi }_{B}^{n}$$ is used: $$ \tilde{\xi}_{B}^{n} := \| \tau^{-1}(\varPi_{n} - I)U^{n-1}\|^{2}. $$ In Table 4, the coarsening estimator terms $$\sqrt{\tilde{\xi }_{B}^{n}}$$ are reported. Note that because of orthogonality the relation $$ \tilde{\xi}_{B}^{n} =\| \tau^{-1}U^{n-1}\|^{2} - \| \tau^{-1}\varPi^{n}U^{n-1}\|^{2}$$ holds, simplifying the computation of this term. Notice that in this example $$\tilde{\xi }_{B}^{n}=0$$ for n≠n0. The projection ΠnUn−1 can easily be computed without the need of (inexact) interpolation or quadrature, see Bänsch, 1991a for details. From Table 3 one deduces the following behavior of δEp. If the mesh is fine enough, the ‘extra contribution’ of the error is negligibly small. However, from a certain small time step size τ (depending on h0), the extra contribution to the error behaves like 1/τ (reading the rows of Table 3). In contrast, fixing a time step size and reading the columns, δEp is decreasing. A similar behavior can be observed for the coarsening estimator $$\tilde{\xi }_{B}^{n}$$. However, the 1/τ increase is valid for all h0. For fixed time step size, $$\sqrt{\tilde{\xi }_{B}^{n}}$$ decreases approximately like $${h_{0}^{3}}$$, which is the order one would expect for $$\mathcal{P}_{2}-\mathcal{P}_{1}$$ Taylor–Hood element. We conclude that, in this experiment, our estimator has the right qualitative behavior and captures the significant increase of the error introduced by the mesh change. 7. Variable time steps In this section we present a generalization of the analysis of this paper to the case of variable time steps. For the ease of presentation we restrict ourselves to a constant mesh and present only the estimate in the L2–L2 norm. The other a posteriori estimates can be obtained in an analogous manner. In contrast, it is an open problem whether the optimality proof in Section 5 carries over to the case of variable time step sizes. At least we expect an assumption on the ratio of successive time steps, see Emmrich (2005) and the literature cited there. 7.1 A posteriori estimates For a partition 0 = t0 < t1 < ⋯ < tN = T define the time step τn := tn − tn−1 and the time step ratio $$\omega _{n}:=\frac{\tau _{n}}{\tau _{n-1}}$$. The finite difference operator $$\bar \partial $$ for variable time steps is defined recursively $$ \bar\partial U^{n}:=\bar\partial^{1} U^{n} := \frac{U^{n}-U^{n-1}}{\tau_{n}},\qquad \bar\partial^{k} U^{n}:= \frac{\bar\partial^{k-1}U^{n}-\bar\partial^{k-1}U^{n-1}}{\tau_{n}},\;k=2,\ldots,n.$$ The BDF2 operator is then given by $$ \bar\partial^{B} U^{n}:=\frac{1}{\tau_{n}}\left(\frac{1+2\omega_{n}}{1+\omega_{n}}U^{n}-(1+\omega_{n})U^{n-1}+\frac{{\omega_{n}^{2}}}{1+\omega_{n}}U^{n-2}\right) =\tau_{n}\frac{\omega_{n}}{1+\omega_{n}}\bar\partial^{2}U^{n}+\bar\partial U^{n}. $$ Let Xh, Yh be a mixed finite element approximation of X and Y. The discrete instationary Stokes system, cf. equations (2.9) and (2.10), with variable time steps reads: set U0 := u0, P0 := p0 and calculate U1 from \begin{align} \bar\partial U^{1}-\varDelta_{h} U^{1/2}+ B_{h}^{\top} P^{1/2}=F^{1/2},\qquad B_{h} U^{1} = 0,\qquad U^{1}|_{\varGamma}=0. \end{align} (7.1) Then for n ≥ 2 compute Un from \begin{align} \bar\partial^{B}U^{n} -\varDelta_{h} U^{n}+B_{h}^{\top} P^{n} =F^{n},\qquad B_{h} U^{n}=0. \end{align} (7.2) In the next theorem we present the estimators for variable time steps. Theorem 7.1 (L2–L2 a posteriori error estimates for variable time steps) The discrete solutions (Un, Pn) of equations (7.1) and (7.2) satisfy the estimates $$ \| U-u\|_{L^{2}(L^{2})}^{2}\ \lesssim\ \check{\mathcal{E}}_{1}+\check{\mathcal{E}}_{2} + \varLambda_{1} + {\mathcal{F}}_{1} + {\textit{I}}_{1}, $$ where $$\varLambda_{1} $$, $$\mathcal{F}_{1}$$ and $$ {\textit{I}}_{1} $$ are given in Definition 3.3 and \begin{align} \check{\mathcal{E}}_{1} =\sum_{n=2}^{N} {\tau_{n}^{5}}\|\bar\partial^{2} U^{n}\|^{2}\qquad \textrm{and}\qquad \check{\mathcal{E}}_{2} = \sum_{n=2}^{N} \frac{{\tau_{n}^{3}}}{{\omega_{n}^{2}}}\left\|\frac{\omega_{n}}{\omega_{n}+1}\bar\partial^{2} U^{n} -\frac{\omega_{n-1}}{\omega_{n-1}+1}\bar\partial^{2} U^{n-1}\right\|^{2}_{J^{\ast}}\!. \end{align} (7.3) Proof. For n ≥ 2, t ∈ In, define the three-point reconstruction for variable time steps $$ \hat{U}(t):= U(t)+\frac{\omega_{n}}{\omega_{n}+1}(t-t_{n})(t-t_{n-1})\bar\partial U^{n}.$$ Its time derivative is $$ \partial_{t} \hat{U}(t)=\bar\partial^{B} U^{n} + \frac{2\omega_{n}}{\omega_{n}+1}(t-t_{n})\bar\partial^{2} U^{n}.$$ With this definition, deduce the error equation like for equation (4.1). Now proceed exactly like in the proof of equation (3.7) of Theorem 3.4. Note that in case of constant time steps the above theorem reduces to Theorem 3.4 (in the case of constant meshes). 7.2 Adaptive strategy Based on the above result for non constant time steps we devise a simple adaptive algorithm and present some computational results. For n ≥ 1 define the local error indicator \begin{align} {\vartheta_{n}}^{2} :=\frac{{\tau_{n}^{2}}}{{\omega_{n}^{2}}} \left\|\frac{\omega_{n}}{\omega_{n}+1}\bar\partial^{2} U^{n}-\frac{\omega_{n-1}}{\omega_{n-1}+1}\bar\partial^{2} U^{n-1}\right\|^{2}_{J^{\ast}} +{\tau_{n}^{4}}\|\bar\partial^{2} U^{n}\|^{2}. \end{align} (7.4) Note that for n ≤ 3 the first term is dropped. From Theorem 7.1 we conclude $$ \|U-u\|_{L^{2}(L^{2})}^{2} \lesssim\ \sum_{n=1}^{N} \tau_{n}{\vartheta_{n}}^{2} \; + \varLambda_{1} + \mathcal{F}_{1} + {I}_{1}. $$ The optimal strategy to choose the time step sizes with respect to the L2–norm in time would consist in equilibrating the local contributions τnϑn2. This, however, is not possible within a time-marching scheme, since it would require to a priori know the total number N of time steps. Instead, one contents oneself with equilibrating the local contribution ϑn, which is more suited for the $$L^{\infty }$$–norm (compare also the discussion in the study by Kreuzer et al., 2012). Thus we propose the following simple adaptive time-stepping strategy, which is rather common for ordinary differential equations, see Deuflhard & Weiser (2012): in each time step a tentative size of the next time step is chosen according to the extrapolation formula $$ \tau_{n+1}:= \tau_{n} {\textrm{TOL}/{\vartheta_{n}}}^{\frac{1}{\gamma+1}}, $$ where γ is the order of the method, γ = 2 in our case. If the new time step generates too big an error, this time step is rejected and recomputed with a reduced time step. Otherwise the time step is accepted. With some ‘safety’ parameters ρ1, ρ2, ρ2 the algorithm reads as follows. Algorithm 1 Let ρ1, ρ2, ρ3 > 1 and TOL > 0 be given. Initialize all unknowns. Let τ1 be given. do n ← n + 1 loop forever [Un, Pn] = solve (Un−1, Un−2) compute ϑnfrom Eq. (83) if ϑnρ1* TOL *     time step rejected τn ← τn/ρ2 else *     time step accepted break end if end loop forever τn+1 := min{τn(TOL/ϑn)1/3, ρ3τn} tn+1 := tn + τn+1 whiletn+1 < T Note that solve returns the unknowns for time tn by solving one time step of equation (2.9). The above adaptive method is tested with the same example as in Section 6, except for taking \begin{align*} u(t,x) & = g(t)\pi\,\left[\sin(2\pi x_{2})\sin(\pi x_{1})^{2},\;-\sin(2\pi x_{1})\sin(\pi x_{2})^{2}\right]^{\top}\!,\\ p(t,x) & = g(t)\cos(\pi x_{1})\sin(\pi x_{2}) \end{align*} with $$g(t) = \sin (t)\left (1-\exp \left (-5.e2(t-0.25)^{2}\right )\right )$$ exhibiting a ‘bump’ around t = 0.25, see Fig. 4. This behavior of g is well suited to study the performance of a time adaptive strategy (compare also Kreuzer et al., 2012). Fig. 4. View largeDownload slide Graph of function g. Fig. 4. View largeDownload slide Graph of function g. Fig. 5. View largeDownload slide Instantaneous L2(Ω) error versus time with uniform time step and adaptive strategy (left) and local error estimator θn versus time (right). Fig. 5. View largeDownload slide Instantaneous L2(Ω) error versus time with uniform time step and adaptive strategy (left) and local error estimator θn versus time (right). Fig. 6. View largeDownload slide Time step sizes τn versus time. Fig. 6. View largeDownload slide Time step sizes τn versus time. Figure 5 (left) shows the result for a computation with uniform time step size and with adaptive strategy. For a fair comparison the uniform time step is chosen such that the number of time steps is the same as for the adaptive run. As can be seen, the local error of the computation with uniform time step is rather big around t = 0.25, otherwise they are quite small, thus reflecting the character of the function g. The run with the adaptive strategy is done with TOL = 5.e − 3, ρ1 := 1.2, ρ2 = 1.43 and ρ3 = 1.8. As one can see the error is fairly well controlled. Figure 5 (right) shows the error estimator θn, equilibrated around the value 5.e − 3, and Fig. 6 reports the time step size as a function of time for the adaptive strategy. We conclude that in principle the derived error estimator is capable of effectively monitoring the time step size. References Ainsworth , M. & Oden , J. T. ( 2011 ) A Posteriori Error Estimation in Finite element Analysis , vol. 37. New York : John Wiley . Akrivis , G. & Chatzipantelidis , P. ( 2010 ) A posteriori error estimates for the two-step backward differentiation formula method for parabolic equations . SIAM J. Numer. Anal. , 48 , 109 – 132 . Google Scholar CrossRef Search ADS Akrivis , G. , Makridakis , Ch. & Nochetto , R. H. ( 2006 ) A posteriori error estimates for the Crank–Nicolson method for parabolic equations . Math. Comp. , 75 , 511 – 531 . Google Scholar CrossRef Search ADS Akrivis , G. , Makridakis , Ch. & Nochetto , R. H. ( 2009 ) Optimal order a posteriori error estimates for a class of Runge–Kutta and Galerkin methods . Numer. Math. , 114 , 133 – 160 . Google Scholar CrossRef Search ADS Alnæs , M. , Blechta , J. , Hake , J. , Johansson , A. , Kehlet , B. , Logg , A. , Richardson , C. , Ring , J. , Rognes , M. & Wells , G. ( 2015 ) The FEniCS project version 1.5 . Archive of Num. Software , 3 , 9 – 23 . Bank , R. E. & Welfert , B. D. ( 1990 ) A posteriori error estimates for the Stokes equations: a comparison . Comput. Methods Appl. Mech. Eng. , 82 , 323 – 340 . Google Scholar CrossRef Search ADS Bänsch , E. ( 1991a ) An adaptive finite-element strategy for the three-dimensional time-dependent Navier–Stokes equations . J. Comput. Appl. Math. , 36 , 3 – 28 . Google Scholar CrossRef Search ADS Bänsch , E. ( 1991b ) Local mesh refinement in 2 and 3 dimensions . IMPACT Comput. Sci. Engrg. , 3 , 181 – 191 . Google Scholar CrossRef Search ADS Bänsch , E. ( 1998 ) Simulation of instationary, incompressible flows . Acta Math. Univ. Comenianae , 67 , 101 – 114 . Bänsch , E. & Brenner , A. ( 2016 ) A posteriori error estimates for pressure-correction schemes . SIAM J. Numer. Anal. , 54 , 2323 – 2358 . Google Scholar CrossRef Search ADS Bänsch , E. , Karakatsani , F. & Makridakis , Ch. ( 2012 ) A posteriori error control for fully discrete Crank–Nicolson schemes . SIAM J. Numer. Anal. , 50 , 2845 – 2872 . Google Scholar CrossRef Search ADS Besier , M. & Wollner , W. ( 2012 ) On the pressure approximation in nonstationary incompressible flow simulations on dynamically varying spatial meshes . Int. J. Numer. Methods Fluids , 69 , 1045 – 1064 . Google Scholar CrossRef Search ADS Brenner , A. , Bänsch , E. & Bause , M. ( 2014 ) A priori error analysis for finite element approximations of the stokes problem on dynamic meshes . IMA J. Numer. Anal. , 34 , 123 – 146 . Google Scholar CrossRef Search ADS Crouzeix , M. & Raviart , P.-A. ( 1973 ) Conforming and nonconforming finite element methods for solving the stationary stokes equations i . Esaim. Math. Model. and Numer. Anal. , 7 , 33 – 75 . Demlow , A. , Lakkis , O. & Makridakis , Ch. ( 2009 ) A posteriori error estimates in the maximum norm for parabolic problems . SIAM J. Numer. Anal. , 47 , 2157 – 2176 . Google Scholar CrossRef Search ADS Deuflhard , P. & Weiser , M. ( 2012 ) Adaptive numerical solution of PDEs . Berlin : de Gruyter . Google Scholar CrossRef Search ADS Dupont , T. ( 1982 ) Mesh modification for evolution equations . Math. Comp. , 39 , 85 – 107 . Google Scholar CrossRef Search ADS Emmrich , E. ( 2005 ) Stability and error of the variable two-step BDF for semilinear parabolic problems . JAMS , 19 , 33 – 55 . Ern , A. , Smears , I. & Vohralík , M. ( 2016 ) Guaranteed, locally space-time efficient, and polynomial-degree robust a posteriori error estimates for high-order discretizations of parabolic problems . Technical report . arXiv:1610.01804 . Evans , L. C. ( 2010 ) Partial Differential Equations , 2nd edn. Graduate Studies in Mathematics , vol. 33 . Providence, RI : American Mathematical Society . Georgoulis , E. , Lakkis , O. & Virtanen , J. ( 2011 ) A posteriori error control for discontinuous Galerkin methods for parabolic problems . SIAM J. Numer. Anal. , 49 , 427 – 458 . Google Scholar CrossRef Search ADS Heywood , J. G. ( 1980 ) The Navier–Stokes equations: on the existence, regularity and decay of solutions . Indiana Univ. Math. J. , 29 , 639 – 681 . Google Scholar CrossRef Search ADS Heywood , J. G. & Rannacher , R. ( 1982 ) Finite-element approximation of the nonstationary Navier–Stokes problem. I. Regularity of solutions and second-order error estimates for spatial discretization. SIAM J. Numer. Anal. , 19 , 275 – 311 . Heywood , J. G. & Rannacher , R. ( 1988 ) Finite-element approximation of the nonstationary Navier–Stokes problem. III. Smoothing property and higher order error estimates for spatial discretization . SIAM J. Numer. Anal. , 25 , 489 – 512 . Google Scholar CrossRef Search ADS Karakatsani , F. & Makridakis , Ch. ( 2007 ) A posteriori estimates for approximations of time-dependent Stokes equations . IMA J. Numer. Anal. , 27 , 741 – 764 . Google Scholar CrossRef Search ADS Kreuzer , C. , Schmidt , A. , Möller , C. A. & Siebert , K. G. ( 2012 ) Design and convergence analysis for an adaptive discretization of the heat equation . IMA J. Numer. Anal. , 32 , 1375 – 1403 . Google Scholar CrossRef Search ADS Lakkis , O. & Makridakis , Ch. ( 2006 ) Elliptic reconstruction and a posteriori error estimates for fully discrete linear parabolic problems . Math. Comp. , 75 , 1627 – 1658 . Google Scholar CrossRef Search ADS Lozinski , A. , Picasso , M. & Prachittham , V. ( 2009 ) An anisotropic error estimator for the Crank–Nicolson method: application to a parabolic problem . SIAM J. Sci. Comput. , 31 , 1757 – 1783 . Google Scholar CrossRef Search ADS Makridakis , Ch. & Nochetto , R. H. ( 2003 ) Elliptic reconstruction and a posteriori error estimates for parabolic problems . SIAM J. Numer. Anal. , 41 , 1585 – 1594 . Google Scholar CrossRef Search ADS Sohr , H. ( 2001 ) The Navier–Stokes Equations . Basel : Birkhäuser . Temam , R. ( 1977 ) Navier–Stokes Equations , 3rd edn . Amsterdam : North-Holland . Verfürth , R. ( 1996 ) A Review of A Posteriori Error Estimation and Adaptive Mesh-refinement Techniques . Wiley-Teubner Series . Chichester-Stuttgart . Verfürth , R. ( 2003 ) A posteriori error estimates for finite element discretizations of the heat equation . Calcolo , 40 , 195 – 212 . Google Scholar CrossRef Search ADS Verführt , R. ( 2013 ) A Posteriori Error Estimation Techniques for Finite Element Methods . Oxford Science Publications . Google Scholar CrossRef Search ADS Appendix A. Proof of Theorem 2.5 In the first step a priori estimates of the initial errors are presented: Lemma A.1 Assume the exact solution of the Stokes system (2.1) is sufficiently regular. Determine p0 from u0 and equation (2.1). Set $$U^{0}=\mathcal{S}_{0}^{u}(u_{0},p_{0})$$ and $$P^{0}:=\mathcal{S}^{p}_{0}(u_{0},p_{0})$$ of Lemma 2.3. Calculate U1 ∈ Xh and P1 ∈ Yh by equation (2.10). Then the approximation estimate of the Stokes operator holds \begin{align*} \|U^{1}-\mathcal{S}_{1}^{u}(u(t_{1}),p(t_{1}))\|^{2}+\tau\|\nabla(U^{1}-\mathcal{S}_{1}^{u}(u(t_{1}),p(t_{1})))\|^{2}+\tau^{2}\|P^{1}-\mathcal{S}_{1}^{p}(u(t_{1}),p(t_{1}))\|^{2} \end{align*} \begin{align} \lesssim \tau^{2} h^{2(k+1)}A^{1} + \tau^{6}\|{\partial^{3}_{t}} u\|^{2}_{L^{\infty}((0,t_{1});L^{2})} \end{align} (A.1) and the fully discrete estimates \begin{align} \|U^{1}-u(t_{1})\|^{2} \lesssim h^{2(k+1)} A^{0} + \tau^{6}\|{\partial^{3}_{t}} u\|^{2}_{L^{\infty}((0,t_{1});L^{2})}, \end{align} (A.2) \begin{align} \|\nabla(U^{1}-u(t_{1}))\|^{2} \lesssim h^{2k} A^{0}+\tau h^{2(k+1)} A^{1} + \tau^{5}\|{\partial^{3}_{t}} u\|^{2}_{L^{\infty}((0,t_{1});L^{2})}, \end{align} (A.3) \begin{align} \|P^{1}-p(t_{1})\|^{2} \lesssim h^{2k} A^{0}+ h^{2(k+1)} A^{1} + \tau^{4}\|{\partial^{3}_{t}} u\|^{2}_{L^{\infty}((0,t_{1});L^{2})}, \end{align} (A.4) where $$ A^{i}:=\|{\partial^{i}_{t}} u\|^{2}_{L^{\infty}((0,t_{1});H^{k+1})}+ \|{\partial^{i}_{t}} p\|^{2}_{L^{\infty}((0,t_{1});H^{k})}.$$ Proof. Set $$\theta ^{i}:=U^{i}-\mathcal{S}_{i}^{u}(u_{i},p_{i})$$. Subtract the half of equation (2.1) at times t = 0 and t = t1 from equation (2.10), use the Stokes projection defined in Lemma 2.3 and note that θ0 = 0 and $$P^{0}-~\mathcal{S}_{0}^{p}(u_{0},p_{0})=0$$ to deduce the error equation \begin{align} \frac{1}{\tau}\theta^{1}-\frac{1}{2}\varDelta_{h}\theta^{1}+\frac{1}{2}B_{h}^{\top}(P^{1}-\mathcal{S}_{1}^{p}(u(t_{1}),p(t_{1}))) = \bar\partial(u(t_{1})-\mathcal{S}^{u}_{n}(u(t_{1}),p(t_{1})))+(\partial_{t} u(t_{1/2})-\bar\partial u(t_{1})). \end{align} (A.5) Test this equation by τθ1 ∈ Jh to obtain \begin{align} \|\theta^{1}\|^{2}+\tau\|\nabla\theta^{1}\|^{2} &\lesssim \tau^{2}\|\bar\partial(u(t_{1})-\mathcal{S}_{1}^{u} (u(t_{1}),p(t_{1})))\|^{2}+\tau^{2}\|(\partial_{t}u(t_{1/2})-\bar\partial u(t_{1}))\|^{2}\nonumber\\ &\lesssim \tau^{2}h^{2(k+1)}A^{1} + \tau^{6}\|{\partial^{3}_{t}} u\|^{2}_{L^{\infty}((0,t_{1});L^{2})}, \end{align} (A.6) where Lemma 2.2 and Taylor expansion has been used. This proves the estimate for the first and second term in equation (A.1). Equations (A.2) and (A.3) are then easily deduced by the triangle inequality and Lemma 2.2. To show the pressure estimate (A.4) let us first show the estimate for $$\|P^{1}-\mathcal{S}_{1}^{p}(u(t_{1}),p(t_{1}))\|$$ in equation (A.1). Use the inf–sup condition and insert the error equation (A.5) to get \begin{align*} \|P^{1}-\mathcal{S}_{1}^{p}(u(t_{1}),p(t_{1}))\| &\lesssim \sup_{v_{h}\in X_{h}}\frac{|\langle P^{1}-\mathcal{S}_{1}^{p}(u(t_{1}),p(t_{1})),\operatorname{div} v_{h}\rangle|^{2}}{\|\nabla v_{h}\|}\\ &\leq \frac{1}{\tau}\|\theta^{1}\|_{X_{h}^{\ast}}\!+\!\|\nabla\theta^{1}\|\!+\!\|\bar\partial(u(t_{1})-\mathcal{S}_{1}^{u}(u(t_{1}),p(t_{1})))\|_{X_{h}^{\ast}}+\!\|\partial_{t} u(t_{1/2})\!-\!\bar\partial u(t_{1})\|_{X^{\ast}}\\ &\leq h^{2(k+1)}A^{1} + \tau^{4}\|{\partial^{3}_{t}} u\|^{2}_{L^{\infty}((0,t_{1});L^{2})}, \end{align*} where we have bounded $$\|\theta ^{1}\|_{X_{h}^{\ast }}\leq \|\theta ^{1}\|$$ and used equation (A.6) in the last step. The pressure estimate (A.4) is then obtained by the triangle inequality and Lemma 2.3: $$ \|P^{1}-p(t_{1})\|\;\leq \|P^{1}-\mathcal{S}_{1}^{p}(u(t_{1}),p(t_{1}))\|+\|p(t_{1})-\mathcal{S}_{1}^{p}(u(t_{1}),p(t_{1}))\|.$$ Now we are able to prove Theorem 2.5: Proof. (Theorem 2.5) Set $$\theta ^{n}:= U^{n}-\mathcal{S}^{u}_{n}(u(t_{n}),p(t_{n}))$$. Subtract equation (2.1) from equation (2.9) and combine with the Stokes projector $$\mathcal{S}^{u}_{n}$$ to get the error equation \begin{align} \bar\partial^{B}\theta^{n}-\varDelta_{h}\theta^{n} + B_{h}^{\top}(P^{n}-\mathcal{S}^{p}_{n}(u(t_{n}),p(t_{n})))\; =\; \bar\partial^{B}(u(t_{n})-\mathcal{S}^{u}_{n}(u(t_{n}),p(t_{n})))+(\partial_{t}-\bar\partial^{B})u(t_{n}),\quad n\geq2. \end{align} (A.7) Test this equation by 4τθn ∈ Jh. By the identity \begin{align} 2\langle 3a^{n}-4a^{n-1}+a^{n-2},a^{n}\rangle = \|a^{n}\|^{2}-\|a^{n-1}\|^{2}+\tau^{4}\|\bar\partial^{2} a^{n}\|^{2}+\|2a^{n}-a^{n-1}\|^{2}-\|2a^{n-1}-a^{n-2}\|^{2} \end{align} (A.8) the first term can be transformed into \begin{align*} 4\tau\langle\bar\partial^{B}\theta^{n},\theta^{n}\rangle &= 2\langle 3\theta^{n}-4\theta^{n-1}+\theta^{n-2},\theta^{n}\rangle\\ &=\|\theta^{n}\|^{2}-\|\theta^{n-1}\|^{2}+\tau^{4}\|\bar\partial^{2} \theta^{n}\|^{2}+\|2\theta^{n}-\theta^{n-1}\|^{2}-\|2\theta^{n-1}-\theta^{n-2}\|^{2}. \end{align*} Since θn ∈ Jh the pressure terms drop out and one obtains the error inequality \begin{align*} \|\theta^{n}\|^{2}-\|\theta^{n-1}\|^{2}+\tau^{4}\|\bar\partial^{2} \theta^{n}\|^{2}+\|2\theta^{n}-\theta^{n-1}\|^{2}-\|2\theta^{n-1}-\theta^{n-2}\|^{2} + \tau\|\nabla\theta^{n}\|^{2}\\ \lesssim \tau\|\bar\partial^{B}(u(t_{n})-\mathcal{S}^{u}_{n}(u(t_{n}),p(t_{n})))\|^{2}_{J_{h}^{\ast}}+\tau\|(\partial_{t}-\bar\partial^{B})u(t_{n})\|^{2}_{J_{h}^{\ast}}. \end{align*} Summing up from n = 2 to N yields \begin{align*} \max_{1\leq n\leq N}\|\theta^{n}\|^{2}+\sum_{n=2}^{N}\tau\|\nabla\theta^{n}\|^{2} &\lesssim \sum_{n=2}^{N} \tau\|\bar\partial^{B}(u(t_{n})-\mathcal{S}^{u}_{n}(u(t_{n}),p(t_{n})))\|^{2}_{J_{h}^{\ast}}\\ &\quad+\sum_{n=2}^{N}\tau\|(\partial_{t}-\bar\partial^{B})u(t_{n})\|^{2}_{J_{h}^{\ast}}+\|\theta^{1}\|^{2}+\|\theta^{1}-\theta^{0}\|^{2}.\end{align*} For the first term on the right-hand side we have by using Lemma 2.2 \begin{align} \sum_{n=2}^{N}\tau\|\bar\partial^{B}(u(t_{n})-\mathcal{S}^{u}_{n}(u(t_{n}),p(t_{n})))\|^{2}_{J_{h}^{\ast}} \lesssim \sum_{n=1}^{N} \tau\|\bar\partial u(t_{n})-\mathcal{S}^{u}_{n}(\bar\partial u(t_{n}),\bar\partial p(t_{n}))\|^{2} \lesssim h^{2(k+1)} A^{1}. \end{align} (A.9) For the second term on the right-hand side use standard estimates for the BDF2 operator $$\bar \partial ^{B}$$ and get \begin{align} \max_{1\leq n\leq N}\|\theta^{n}\|^{2}+\sum_{n=2}^{N}\tau\|\nabla\theta^{n}\|^{2} \lesssim \|\theta^{1}\|^{2} +h^{2(k+1)} A^{1} + \tau^{4}\|{\partial_{t}^{3}} u\|^{2}_{L^{\infty}(H^{-1})}. \end{align} (A.10) The velocity estimate (2.11) is finally obtained using the triangle inequality: $$ \max_{1\leq n\leq N}\|U^{n}-u(t_{n})\|\lesssim\max_{1\leq n\leq N}\|\theta^{n}\|+\max_{1\leq n\leq N} \|u(t_{n})-\mathcal{S}^{u}_{n}(u(t_{n}),p(t_{n}))\|,$$ the initial estimate (A.1), equation (A.10) and Lemma 2.2. In the same way equation (2.12) is proved. To derive the pressure estimate (2.13) we first bound the term $$\|\bar \partial ^{B} U^{n}-\partial _{t} u\|$$. To this end test the error equation (A.7) by $$4\tau \bar \partial ^{B}\theta ^{n}$$ and use the identity (A.8) to get \begin{align*} \tau\|\bar\partial^{B}\theta^{n}\|^{2}+\|\nabla\theta^{n}\|^{2}-\|\nabla\theta^{n-1}\|^{2}+\tau^{4}\|\nabla\bar\partial\theta^{n}\|^{2}+ \|\nabla(2\theta^{n}-\theta^{n-1})\|^{2}-\|\nabla(2\theta^{n-1}-\theta^{n-2})\|^{2}\\ \lesssim\tau\|\bar\partial^{B}(u(t_{n})-\mathcal{S}^{u}_{n}(u(t_{n}),p(t_{n})))\|^{2}+\tau\|(\partial_{t}-\bar\partial^{B})u(t_{n})\|^{2}. \end{align*} Sum up from n = 2 to N, observe that θ0 = 0 to obtain \begin{align} \sum_{n=2}^{N}\tau\|\bar\partial^{B}\theta^{n}\|^{2}+\max_{1\leq n\leq N}\|\nabla\theta^{n}\|^{2} &\lesssim\|\nabla\theta^{1}\|^{2}+\sum_{n=2}^{N}\tau\|\bar\partial(u(t_{n})-\mathcal{S}^{u}_{n}(u(t_{n}),p(t_{n})))\|^{2}\nonumber\\&\quad+\sum_{n=2}^{N}\tau\|(\partial_{t}-\bar\partial^{B})u(t_{n})\|^{2}. \end{align} (A.11) Now we are able to prove the pressure estimate (2.13). The inf–sup stability yields \begin{align*} \|P^{n}-p(t_{n})\| &\leq \|P^{n}-\mathcal{S}^{p}_{n}(u(t_{n}),p(t_{n}))\|+\|\mathcal{S}^{p}_{n}(u(t_{n}),p(t_{n}))- p(t_{n})\|\\ &\lesssim \sup_{v_{h}\in X_{h}}\frac{\langle P^{n}-\mathcal{S}^{p}_{n}(u(t_{n}),p(t_{n})),\operatorname{div} v_{h}\rangle}{\|\nabla v_{h}\|}+\|\mathcal{S}^{p}_{n}(u(t_{n}),p(t_{n}))- p(t_{n})\|\\ &= \sup_{v_{h}\in X_{h}}\frac{\langle \bar\partial^{B} U^{n}-\partial_{t} u(t_{n}),v_{n}\rangle+\langle\nabla\theta^{n},\nabla v_{h}\rangle}{\|\nabla v_{h}\|}+\|\mathcal{S}^{p}_{n}(u(t_{n}),p(t_{n}))- p(t_{n})\|\\ &\leq \|\bar\partial^{B}\theta^{n}\| + \|\bar\partial^{B}(u(t_{n})-\mathcal{S}^{u}_{n}(u(t_{n}),p(t_{n})))\|+\|(\bar\partial^{B}-\partial_{t})u(t_{n})\|\\ &\quad+ \|\nabla\theta^{n}\|+\|\mathcal{S}^{p}_{n}(u(t_{n}),p(t_{n}))- p(t_{n})\|. \end{align*} Square this inequality, multiply by τ and use equation (A.11) to get \begin{align*} \sum_{n=1}^{N}\tau\|P^{n}-p(t_{n})\|^{2} &\lesssim \|\nabla\theta^{1}\|^{2}+\tau\|P^{1}-p(t_{1})\|^{2}\\ &\quad+ \sum_{n=2}^{N}\tau\left(\|\bar\partial^{B}(u(t_{n})-\mathcal{S}^{u}_{n}(u(t_{n}),p(t_{n})))\|^{2}+\|(\partial_{t}-\bar\partial^{B})u(t_{n})\|^{2}\right.\\ &\qquad\qquad\quad\left.+\|\nabla\theta^{n}\|^{2}+\|\mathcal{S}^{p}_{n}(u(t_{n}),p(t_{n}))- p(t_{n})\|^{2}\right)\!. \end{align*} Then by using the initial estimate equations (A.1)–(A.4), the velocity estimate (A.10) and Lemma 2.3 the pressure estimate in equation (2.13) is proven. The $$\ell ^{\infty }(L^{2})$$ estimate for the gradients of the velocity is easily derived by equation (A.11) and Lemma 2.2. Appendix B. Discrete regularity In this section we derive regularity estimates for the solution of the semidiscrete instationary Stokes problem \begin{align} \partial_{t} Z(t) - \varDelta_{h} Z(t) + B_{h}^{\top} Q(t) = \tilde G (t),\qquad B_{h} Z(t)=0,\qquad Z(t)=V_{0}, \end{align} (B.1) for a sufficiently regular (in time) right-hand side $$\tilde G(t)\in X_{h}$$ and initial value V0 or, equivalently, by multiplying the above equation by $$\varPi _{J_{h}}$$, \begin{align} \partial_{t} Z(t) + \mathcal{A}_{h} Z(t) = F(t):=\varPi_{J_{h}}\tilde G(t),\qquad B_{h} Z(t)=0,\qquad Z(0)=V_{0}. \end{align} (B.2) Here, $$\varPi _{J_{h}}:X\to J_{h}$$ denotes the L2 projection onto the discrete solenoidal space Jh and \begin{align} \mathcal{A}_{h}:=\varPi_{J_{h}}(-\varDelta_{h}) \end{align} (B.3) the discrete Stokes operator. Regularity results for the full continuous system equation (2.1) can be found for instance in the study by Temam (1977, Chapter III), Sohr (2001, Chapter IV.2.7) and Heywood (1980). The following remark seems to be in order. The crucial difference between the continuous and the semidiscrete case is that in the latter case no compatibility conditions between the initial data and right-hand side are needed for getting higher regularity, since the domain of the discrete operator $$\mathcal{A}_{h}$$ is the whole of Jh. In contrast, in the continuous case higher regularity in time is always linked to such compatibility conditions, even if data are arbitrarily smooth. This can readily be seen for the heat equation: assume that u, △u, f and ∂tu are continuous on $$[0,T]\times \bar{\varOmega }$$, satisfying $$ \partial_{t}u -\varDelta u = f,\qquad u_{|\varGamma}=0,\qquad u(0)=u_{0}.$$ Since u(t, x)|Γ = 0 for t ∈ [0, T], we see that ∂tu(0)|Γ = 0 and therefore the additional condition for given data f, u0 must be satisfied: $$ \varDelta u_{0|\varGamma} = f(0)_{|\varGamma}.$$ Further compatibility conditions are required for higher order time derivatives $$\partial _{t}^{(m)} u$$, see for instance Evans (2010). For the Stokes problem the situation is even more involved because of the presence of the pressure gradient. Only if the pressure fulfills a Poisson equation with both, a Dirichlet and a Neumann boundary condition (and is thus overdetermined), higher time regularity can hold. This kind of compatibility condition cannot be checked in practice, see also Heywood & Rannacher (1982). To avoid these complicated conditions Sohr presents the following local results, cf. Sohr (2001, Theorem 2.7.2): Let $$m\in \mathbb{N}$$, let $$\varOmega \subset \mathbb{R}^{d}$$, d ≥ 2, be a uniform C2-domain, let $$0<T\leq \infty $$, $$1<s<\infty $$, $$u_{0}\in D\left (\mathcal{A}^{1-\frac{1}{s}}\right )\cap D\left (\mathcal{A}^{\frac{1}{2}}\right )$$, $$f\in L^{2}_{loc}\left ([0,T);L^{2}(\varOmega )^{d}\right )$$, and suppose u is a solution of the Stokes system (2.1) with data f, u0 together with an associated pressure p. Then $$ \partial_{t} f,\partial_{tt}f,\ldots,\partial_{t}^{(m)}f\in L^{s}_{loc}\left((0,T);L^{2}(\varOmega)^{d}\right)$$ implies $$ \partial_{t} u,\partial_{tt} u,\ldots,\partial_{t}^{(m+1)}u,\, \mathcal{A}\partial_{t} u,\mathcal{A}\partial_{tt} u,\ldots, \mathcal{A}\partial_{t}^{(m+1)}u\in L^{s}_{loc}((0,T);H),$$ and $$ \partial_{t} u,\partial_{tt} u,\ldots,\partial_{t}^{(m+1)}u \in L^{s}_{loc}\left((0,T);H^{2}(\varOmega)^{d}\right)\!.$$ Let us return to the problem of regularity for the semidiscrete problem. Since (B.2) is a linear, finite-dimensional evolution equation, it admits a unique solution for all times t > 0. Moreover, differentiability of u with respect to time directly follows from regularity of G(t). The following lemma gives an equivalent expression for the dual norm of Jh in terms of the discrete Stokes operator. Lemma B.1 For yh ∈ Jh there is a unique xh ∈ Jh, solution of the discrete Stokes problem $$\mathcal{A}_{h}x_{h}=y_{h}$$. Moreover, \begin{align} \|y_{h}\|_{J_{h}^{\ast}} = \|\nabla x_{h}\|. \end{align} (B.4) Proof. One has \begin{align} \langle \mathcal{A}_{h} v_{h},w_{h}\rangle = \langle\varPi_{J_{h}}(-\varDelta_{h} v_{h}),w_{h}\rangle = \langle -\varDelta_{h} v_{h}, w_{h}\rangle = \langle \nabla v_{h},\nabla w_{h}\rangle,\qquad \forall\ v_{h},\ w_{h}\in J_{h}. \end{align} (B.5) This shows that $$\mathcal{A}_{h}$$ is symmetric and coercive on Jh and in particular bijective. Using the above representation for $$\mathcal{A}_{h}$$, the lemma can be easily proven. First, $$ \|y_{h}\|_{J^{\ast}_{h}}=\sup_{z_{h}\in J_{h}}\frac{\langle y_{h},z_{h}\rangle}{\|\nabla z_{h}\|} =\sup_{z_{h}\in J_{h}}\frac{\langle \mathcal{A}_{h}x_{h},z_{h}\rangle}{\|\nabla z_{h}\|} =\sup_{z_{h}\in J_{h}}\frac{\langle -\varDelta_{h}x_{h},z_{h}\rangle}{\|\nabla z_{h}\|}\leq \|\nabla x_{h}\|. $$ Second, $$ \|\nabla x_{h}\| = \frac{\|\nabla x_{h}\|^{2}}{\|\nabla x_{h}\|}=\frac{\langle-\varDelta_{h}x_{h},x_{h}\rangle}{\|\nabla x_{h}\|} =\frac{\langle \mathcal{A}_{h}x_{h},x_{h}\rangle}{\|\nabla x_{h}\|}=\frac{\langle y_{h},x_{h}\rangle}{\|\nabla x_{h}\|} \leq\sup_{x_{h}\in J_{h}}\frac{\langle y_{h},x_{h}\rangle}{\|\nabla x_{h}\|}=\|y_{h}\|_{J^{\ast}_{h}}. $$ In the following theorem, regularity estimates for the solution of the semidiscrete system (B.2) in terms of the regularity of the right-hand side G and the initial data V0 are presented. Theorem B.2 Suppose that $$\partial _{t}^{(m)}G\in L^{2}((0,T),J_{h})$$ for all 1 ≤ m ≤ M. Then the solution Z(t) of equation (B.2) satisfies for all 1 ≤ m ≤ M \begin{align} \|\partial_{t}^{(m)} Z\|^{2}_{L^{\infty}(L^{2})} + \|\partial_{t}^{(m)} \nabla Z\|^{2}_{L^{2}(L^{2})} \lesssim \|\partial_{t}^{(m)} G\|^{2}_{L^{2}(J_{h}^{\ast})} + \|{Y^{m}_{0}}\|^{2}, \end{align} (B.6) \begin{align} \|\partial_{t}^{(m+1)}Z\|^{2}_{L^{2}(L^{2})} + \|\partial_{t}^{(m)} \nabla Z\|^{2}_{L^{\infty}(L^{2})} \lesssim \|\partial_{t}^{(m)} G\|^{2}_{L^{2}(L^{2})} + \|{Y^{m}_{0}}\|_{1}^{2}, \end{align} (B.7) \begin{align} \|\partial_{t}^{(m)}\nabla Z\|^{2}_{L^{\infty}(L^{2})} + \|\partial_{t}^{(m)}\mathcal{A}_{h} Z\|^{2}_{L^{2}(L^{2})} \lesssim \|\partial_{t}^{(m)} G\|^{2}_{L^{2}(L^{2})} + \|{Y^{m}_{0}}\|_{1}^{2}, \end{align} (B.8) \begin{align} \|\partial_{t}^{(m+1)}\nabla Z\|^{2}_{L^{2}(L^{2})} + \|\partial_{t}^{(m)} \mathcal{A}_{h} Z\|^{2}_{L^{\infty}(L^{2})} \lesssim \|\nabla\partial_{t}^{(m)} G\|^{2}_{L^{2}(L^{2})} + \|{Y^{m}_{0}}\|_{\mathcal{A}}^{2}, \end{align} (B.9) \begin{align} \|\partial_{t}^{(m)}Z\|^{2}_{L^{\infty}(J_{h}^{\ast})} + \|\partial_{t}^{(m)} Z\|^{2}_{L^{2}(L^{2})} \lesssim \|\partial_{t}^{(m)} G\|^{2}_{L^{1}(J_{h}^{\ast})} + \|{Y^{m}_{0}}\|_{J_{h}^{\ast}}^{2}, \end{align} (B.10) \begin{align} \|\partial_{t}^{(m+1)}Z\|^{2}_{L^{2}(J_{h}^{\ast})} + \|\partial_{t}^{(m)} Z\|^{2}_{L^{\infty}(L^{2})} \lesssim \|\partial_{t}^{(m)} G\|^{2}_{L^{2}(J_{h}^{\ast})} + \|{Y^{m}_{0}}\|^{2}, \end{align} (B.11) where \begin{align*} \|{Y^{m}_{0}}\|_{*}^{2} &:= \|\mathcal{A}_{h}^{m}Z(0)\|_{*}^{2}+\sum_{i=0}^{m-1}\|\partial_{t}^{(i)}\mathcal{A}_{h}^{m-1-i}G(0)\|_{*}^{2},\qquad *\in\{1,0,J_{h}^{\ast}\}, \\[-2pt] \|{Y^{m}_{0}}\|_{\mathcal{A}}^{2} &:= \|\mathcal{A}_{h}^{m+1}Z(0)\|^{2}+\sum_{i=0}^{m-1}\|\partial_{t}^{(i)}\mathcal{A}_{h}^{m-i}G(0)\|^{2}. \end{align*} Proof. Let 1 ≤ m ≤ M. Differentiating equation (B.2) m-times with respect to time leads to \begin{align} \partial_{t}^{(m+1)} Z(t) + \partial_{t}^{(m)} \mathcal{A}_{h} Z(t) = \partial_{t}^{(m)} G(t). \end{align} (B.12) First, to prove equation (B.6), test equation (B.12) with $$2\partial _{t}^{(m)} Z(t)\in J_{h}$$ to get $$ \frac{\mathrm{d}}{\mathrm{d}t}\|\partial_{t}^{(m)} Z(t)\|^{2}+\|\nabla \partial_{t}^{(m)} Z(t)\|^{2} \lesssim \|\partial_{t}^{(m)} G(t)\|^{2}_{J_{h}^{\ast}},$$ where we have used equation (B.5) for the second term on the left-hand side. Now, integrating the above equation from t = 0 to T yields $$ \|\partial_{t}^{(m)} Z\|^{2}_{L^{\infty}(L^{2})} + \|\partial_{t}^{(m)} \nabla Z\|^{2}_{L^{2}(L^{2})} \lesssim \|\partial_{t}^{(m)} G\|^{2}_{L^{2}(J_{h}^{\ast})} + \|\partial_{t}^{(m)} Z(0)\|^{2}.$$ Next, to prove equations (B.7)–(B.9), test equation (B.12) with $$v_{h} = \partial _{t}^{(m+1)} Z(t)$$, $$\partial _{t}^{(m)} \mathcal{A}_{h} Z(t)$$ and $$\partial _{t}^{(m+1)} \mathcal{A}_{h} Z(t)$$, and proceed exactly as above to get $$v_{h} = \partial _{t}^{(m+1)} Z(t)$$: $$\|\partial _{t}^{(m+1)}Z\|^{2}_{L^{2}(L^{2})} + \|\partial _{t}^{(m)}\nabla Z\|^{2}_{L^{\infty }(L^{2})} \lesssim \|\partial _{t}^{(m)} G\|^{2}_{L^{2}(L^{2})} + \|\partial _{t}^{(m)} \nabla Z(0)\|^{2}$$, $$v_{h}= \partial _{t}^{(m)} \mathcal{A}_{h} Z(t)$$: $$\|\partial _{t}^{(m)} \nabla Z\|^{2}_{L^{\infty }(L^{2})} + \|\partial _{t}^{(m)} \mathcal{A}_{h} Z\|^{2}_{L^{2}(L^{2})} \lesssim \|\partial _{t}^{(m)} G\|^{2}_{L^{2}(L^{2})} + \|\partial _{t}^{(m)} \nabla Z(0)\|^{2}$$, $$v_{h} = \partial _{t}^{(m+1)} \mathcal{A}_{h} Z(t)$$: $$\|\partial _{t}^{(m+1)}\nabla Z\|^{2}_{L^{2}(L^{2})} + \|\partial _{t}^{(m)} \mathcal{A}_{h} Z\|^{2}_{L^{\infty }(L^{2})} \lesssim \|\nabla \partial _{t}^{(m)} G\|^{2}_{L^{2}(L^{2})} + \|\partial _{t}^{(m)} \mathcal{A}_{h} Z(0)\|^{2}$$, respectively. For proving equations (B.10) and (B.11) we use a duality argument. We test equation (B.12) with vh, where vh is solution of the Stokes problem $$\mathcal{A}_{h} v_{h} = \partial _{t}^{(m)} Z(t)$$. For the left-hand side, one gets \begin{align*} \langle \partial_{t}^{(m+1)} Z(t),v_{h}(t)\rangle &= \langle \partial_{t} \mathcal{A}_{h} v_{h}(t),v_{h}(t)\rangle = \frac{1}{2}\frac{\mathrm{d}}{\mathrm{d}t}\|\nabla v_{h}(t)\|^{2},\\ \langle \partial_{t}^{(m)}\mathcal{A}_{h} Z(t),v_{h}\rangle &= \langle \partial_{t}^{(m)}-\varDelta_{h} Z(t),v_{h}\rangle = \langle \partial_{t}^{(m)} Z(t),\mathcal{A}_{h} v_{h}\rangle = \|\partial_{t}^{(m)} Z(t)\|^{2}, \end{align*} where for the second term equation (B.5) was used. For the right-hand side it follows $$\langle \partial _{t}^{(m)} G(t),v_{h}\rangle \leq \|\partial _{t}^{(m)} G(t)\|_{J^{\ast }_{h}}\|\nabla v_{h}\|$$ and then $$ \frac{1}{2}\frac{\mathrm{d}}{\mathrm{d}t}\|\nabla v_{h}(t)\|^{2}+ \|\partial_{t}^{(m)} Z(t)\|^{2} \leq \|\partial_{t}^{(m)} G(t)\|_{J^{\ast}_{h}}\|\nabla v_{h}\|.$$ Integrate from t = 0 to T and use the norm equivalence in Lemma B.1 to get $$ \|\partial_{t}^{(m)} Z\|^{2}_{L^{\infty}(J_{h}^{\ast})} + \|\partial_{t}^{(m)} Z\|^{2}_{L^{2}(L^{2})} \lesssim \|\partial_{t}^{(m)} G\|^{2}_{L^{1}(J_{h}^{\ast})} + \|\partial_{t}^{(m)} Z(0)\|^{2}_{J_{h}^{\ast}}. $$ Now let vh be solution of the Stokes problem $$\mathcal{A}_{h} v_{h} = \partial _{t}^{(m+1)} Z(t)$$. The same procedure as above yields $$ \|\partial_{t}^{(m+1)}Z\|^{2}_{L^{2}(J_{h}^{\ast})} + \|\partial_{t}^{(m)} Z\|^{2}_{L^{\infty}(L^{2})} \lesssim \|\partial_{t}^{(m)} G\|^{2}_{L^{2}(J_{h}^{\ast})} + \|\partial_{t}^{(m)} Z(0)\|^{2}.$$ It remains to bound the initial values $$\| \partial _{t}^{(m)}Z(0)\|^{2}$$, $$\| \partial _{t}^{(m)}\nabla Z(0)\|^{2}$$, $$\| \partial _{t}^{(m)} \mathcal{A}_{h} Z(0)\|^{2}$$ and $$\| \partial _{t}^{(m)} Z(0)\|^{2}_{J_{h}^{\ast }}$$. To this end we prove the following identity for m ≥ 1 \begin{align} \partial_{t}^{(m)}Z(0) = \displaystyle\sum_{i=0}^{m-1}(-1)^{i-1+m}\partial_{t}^{(i)}\mathcal{A}_{h}^{m-1-i}G(0) + (-1)^{m}\mathcal{A}_{h}^{m}Z(0). \end{align} (B.13) The cases m = 1 and m = 2 are easily shown by using equation (B.12) and taking the limit t ↘ 0. \begin{align*} \partial_{t} Z(0) &= G(0) - \mathcal{A}_{h}Z(0),\\ \partial_{tt} Z(0) &= \partial_{t} G(0) - \mathcal{A}_{h}\partial_{t}Z(0) = \partial_{t} G(0) - \mathcal{A}_{h}G(0) + \mathcal{A}_{h}^{2} Z(0). \end{align*} Now assume equation (B.13) holds for m ≥ 2. Then we have by another use of equation (B.12) \begin{align*} \partial_{t}^{(m+1)}Z(0) &= \partial_{t}^{(m)}G(0)-\mathcal{A}_{h}\partial_{t}^{(m)}Z(0)\\& = \partial_{t}^{(m)} G(0) - \mathcal{A}_{h}\sum_{i=0}^{m-1}(-1)^{i-1+m}\partial_{t}^{(i)}\mathcal{A}_{h}^{m-1-i}G(0) + (-1)^{m+1}\mathcal{A}_{h}^{m+1}Z(0)\\ &= \sum_{i=0}^{m}(-1)^{i+m}\partial_{t}^{(i)}\mathcal{A}_{h}^{m-i}G(0) + (-1)^{m+1}\mathcal{A}_{h}^{m+1}Z(0) \end{align*} and equation (B.13) is shown. Hence, the initial values can be easily bounded by equation (B.13) using the triangle inequality. © The Author(s) 2018. Published by Oxford University Press on behalf of the Institute of Mathematics and its Applications. All rights reserved. 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A posteriori estimates for the two-step backward differentiation formula and discrete regularity for the time-dependent Stokes equations

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Oxford University Press
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© The Author(s) 2018. Published by Oxford University Press on behalf of the Institute of Mathematics and its Applications. All rights reserved.
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0272-4979
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1464-3642
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10.1093/imanum/dry014
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Abstract

Abstract We derive optimal order residual-based a posteriori error estimates for fully discrete finite element approximations to the time-dependent Stokes equations. The time discretization uses the two-step backward differentiation formula, and the space discretization is based on inf–sup stable pairs of finite elements, which are allowed to change with time. We show that the time error estimators are of optimal order. This proof of optimality uses time regularity of the semidiscrete (discrete in space) time-dependent Stokes equations. Computational examples are given to confirm the theoretical findings. For completeness, a priori estimates are also presented. 1. Introduction Adaptive algorithms have become very popular and powerful tools for finite element approximations of partial differential equations. They are naturally linked to error control, which analysis is provided by a posteriori estimates. While for elliptic problems the theory and practice of adaptive methods has reached sound maturity, cf. Verfürth (1996) and Ainsworth & Oden (2011), and the references therein, error control for time-dependent problems is still a challenging area. We mention Makridakis & Nochetto (2003), Verfürth (2003), Akrivis et al. (2006), Lakkis & Makridakis (2006), Akrivis et al. (2009) and Lozinski et al. (2009) for some recent contributions to parabolic problems. For time-dependent problems involving elliptic constraints like the incompressible Navier–Stokes equations, there is even less known. In this paper we present a posteriori estimates for the time-dependent Stokes equations discretized in time by the two-step backward differentiation formula (BDF2) and in space by mixed finite elements, which are allowed to change in time. For linear parabolic problems discretized in time by BDF2 (and still continuous in space) we refer to the study by Akrivis & Chatzipantelidis (2010). To the best of our knowledge, however, there is no rigorous a posteriori analysis for the fully discrete Stokes equations using BDF2 in time so far. We round out our analysis by proving the optimality of our temporal error estimators (in case of nonchanging meshes). Here, optimality is understood in the sense that the a posteriori estimators are asymptotically of the same order of convergence as the corresponding a priori error estimates. Our main tool in deriving a posteriori estimates is an application of the elliptic reconstruction technique introduced by Makridakis & Nochetto (2003) for parabolic problems. A main advantage of this approach is that (in our case) any available a posteriori estimate for the stationary Stokes equations can be used to control the main part of the spatial error. The reason behind is that in deriving the error estimates an appropriate auxiliary function, called reconstruction, is constructed and the error between the exact and numerical solution is split into a part between error and reconstruction as well as an error of approximation type for the stationary problem. This approach is in some sense dual to using the Ritz projection (for parabolic problems) in the a priori analysis. More precisely, the reconstruction exhibits two fundamental properties: (i) its difference to the numerical solution can be estimated via known elliptic a posteriori results, and (ii) its solution of an error equation (structurally the same as the Stokes equation) with a right-hand side that can be bounded by a posteriori estimators in an optimal way. This technique has been used in many applications; let us mention Akrivis et al. (2006), Karakatsani & Makridakis (2007), Akrivis et al. (2009), Demlow et al. (2009), Georgoulis et al. (2011) and Bänsch et al. (2012). However, to the best of our knowledge there is no proof of optimality for the estimators for the corresponding fully discrete schemes for the Stokes equations. In the semidiscrete case (continuous in space) optimality theory is easily reduced to the regularity of the continuous solution and data, cf. Akrivis & Chatzipantelidis (2010, Chapter 3) for parabolic problems. For the fully discrete case, some additional technical effort has to be invested, for instance the semidiscrete solution (continuous in time, discrete in space) has to be considered as an intermediate step. Higher-order time derivatives of the semidiscrete solution have to be bounded and consequently we provide corresponding estimates, which we call discrete regularity, in Appendix B. However, due to technical difficulties we show the optimality result only for a constant mesh. For the ease of presentation we follow the study by Akrivis & Chatzipantelidis (2010) and restrict our presented analysis to the case of constant time steps. Variable time steps and an adaptive strategy for time step control are treated in Section 7. An adaptive strategy for-time dependent problems is still a challenging topic. A key problem is the required spatial mesh modifications with time. Their influences in the error estimates are often underestimated. Let us mention the classical article by Dupont (1982) showing that modifying the mesh too carelessly may lead to a wrong solution. In the context of Stokes equations the issue is even more complex. Using standard operators like interpolation or L2 projection for transferring the velocity from one mesh to the next may lead to devastating effects at least for the pressure, cf. Besier & Wollner (2011) and Brenner et al. (2014). From the theoretical point of view a solenoidal projection like Helmholtz or Stokes projection should be used, which is, however, from the computational point of view too expensive. Hence, we use the L2 projection for transferring the velocity from one mesh to the next and quantify the resulting error by our changing mesh estimators. The rest of this article is organized as follows: in Section 2 we introduce notation, recall some results for the Stokes projection and state corresponding a priori estimates. The main results, the a posteriori estimates and their optimality, are stated in Section 3. The corresponding proofs are given in Sections 4 and 5. The performance of the proposed error estimators are discussed in Section 6 by computational experiments. An extension for variable time steps will be presented in Section 7. Finally, for completeness a priori estimates are proved in Appendix A and the theory of discrete regularity is presented in Appendix B. 2. Notation and preliminaries Let $$\varOmega \subset \mathbb{R}^{d}$$ (d = 2, 3) be an open, connected and bounded domain with boundary $$\varGamma$$ := ∂Ω. We consider the incompressible time-dependent Stokes equations on a finite time interval ]0, T[: Find the velocity u and the pressure p fulfilling \begin{align} \partial_{t} u-\nu\varDelta u+\nabla p=f,\qquad \operatorname{div} u=0\;\textrm{in }\varOmega\times]0,T[, \qquad u_{|\varGamma}=0,\qquad u(t=0)=u_{0}\;\textrm{in }\varOmega. \end{align} (2.1) For the sake of simplicity we set ν = 1; u0 is the initial, sufficiently smooth velocity. In the following we assume that the unique solution (u, p) and the right-hand side f to the above system is sufficiently smooth in time and space (compare also the discussion in Appendix B regarding regularity for the continuous system). As usual the standard Sobolev spaces Hm(Ω) (m = 0, ±1, …) are used, whose norms are denoted by ∥⋅∥m. The norm and inner product of L2(Ω) = H0(Ω) is denoted by ∥⋅∥ and ⟨⋅, ⋅⟩, respectively. Moreover, define $${L^{2}_{0}}(\varOmega ):=\{q\in L^{2}(\varOmega ) \,|\, \int _{\varOmega } q(x)\,\mathrm{d}x=0\}$$. To account for the homogeneous Dirichlet boundary condition we introduce $${H_{0}^{1}}(\varOmega ):= \{v\in H^{1}(\varOmega )\: :\: v_{|\varGamma }=0 \}$$. Thanks to the Poincaré inequality, for $$v\in{H_{0}^{1}}(\varOmega )^{d}$$, the norms ∥∇v∥ and ∥v∥1 are equivalent, and thus in what follows ∥∇v∥ is taken as norm in $${H_{0}^{1}}(\varOmega )^{d}$$. Set $$X:={H_{0}^{1}}(\varOmega )^{d}$$ and $$Y:= {L^{2}_{0}}(\varOmega )$$, satisfying the continuous inf–sup condition: $$ \exists\ c>0,\quad\inf_{q\in Y}\sup_{v\in X}\frac{\langle q,\operatorname{div} v\rangle}{\|q\|\|\nabla v\|}\ \geq\ c.$$ Furthermore, define the spaces of solenoidal functions in X and L2(Ω)d, respectively: \begin{align*} J &:= \big\{v\in X\ :\ \operatorname{div} v=0\big\},\\ H &:= \left\{v \in L^{2}(\varOmega)^{d}\; :\operatorname{div} v=0,\ v\cdot n_{|\varGamma}=0\right\}\!. \end{align*} 2.1 Finite element spaces To discretize in time let 0 = t0 < t1 < ⋯ < tN = T be a partition of $$[0,T]$$, $$I_{n}:=(t_{n-1},t_{n}]$$ a time sub-interval and τn := tn − tn−1 the time step. In the following we assume constant time steps. An extension for variable time steps will be presented in Section 7. For the space discretization we use a standard finite element method and introduce a family $$\{\mathcal{T}_{n}\}_{n=0}^{N}$$ of conforming shape regular triangulations of the domain Ω. Each triangulation $$\mathcal{T}_{n}$$ corresponds to a time interval In. This allows arbitrary mesh changes. Let hn(x) be the mesh-size function for each given triangulation $$\mathcal{T}_{n}$$ defined by $$ h_{n}(x):=\operatorname{diam}(K),\quad K\in\mathcal{T}_{n},\ x\in K $$ and denote by $$h_{n}:=\max _{x\in \varOmega _{n}}h_{n}(x)$$. For each triangulation $$\mathcal{T}_{n}$$ we choose a pair of finite element spaces $$X_{n}:=X_{h_{n}}\subseteq X$$ and $$Y_{n}:=Y_{h_{n}}\subseteq Y$$ of X and Y. Assume for all n that Xn × Yn is of order $$k\in \mathbb{N}$$, i.e. for each v ∈ X ∩ Hk+1(Ω)d and q ∈ Y ∩ Hk(Ω) there exists approximations $$\mathcal{I}_{n} v\in X_{n}$$ and $$\mathcal{J}_{n} q\in Y_{n}$$ such that \begin{align} \|v-\mathcal{I}_{n} v\| + h_{n} \left\|\nabla(v-\mathcal{I}_{n} v)\right\|\leq C h_{n}^{k+1}\|v\|_{k+1}, \qquad \|q-\mathcal{J}_{n} q\|_{{L^{2}_{0}}(\varOmega)}\leq C {h_{n}^{k}}\|q\|_{k}. \end{align} (2.2) These are standard conditions satisfied for instance by Taylor–Hood elements $$\mathcal{P}_{k}-\mathcal{P}_{k-1}$$, k ≥ 2. To handle mesh changes let us denote by $$\varPi _{n}:=\varPi _{X_{n}}\ :X\to X_{n}$$ the L2 projection onto Xn. We introduce the linear operator −Δn : X → Xn, such that for all u ∈ X, vn ∈ Xn, −Δnu ∈ Xn with ⟨−Δnu, vn⟩ = ⟨∇u, ∇vn⟩, and the linear operator Bn : Xn → Yn and its transpose $$B_{n}^{\top }:Y_{n}\to X_{n}$$, so that for every couple (vn, qn) ∈ Xn × Yn the identities ⟨Bnvn, qn⟩ = −⟨div vn, qn⟩ and $$\langle v_{n},B_{n}^{\top } q_{n} \rangle =- \langle \operatorname{div} v_{n},q_{n} \rangle $$ hold. In the sequel we assume that the mixed approximation satisfies the discrete inf–sup condition \begin{align} \exists\ c>0,\quad\inf_{q_{n}\in Y_{n}}\sup_{v_{n}\in X_{n}}\frac{\langle B_{n} v_{n},q_{n} \rangle}{\|\nabla v_{n} \|\,\|q_{n} \|}\geq c \end{align} (2.3) with a constant c not depending on n. Remark 2.1 Our analysis is also valid for nonconforming finite elements spaces. In this case the definitions of the discrete operators has to be extended to \begin{align*} -\bigtriangleup_{n}:X\to X_{n}+X,\quad B_{n}: X_{n}+X\to Y_{n},\quad B_{n}^{\top} :Y_{n}\to X_{n}+X,\quad \varPi_{n}:X\to X_{n}+X, \end{align*} and we must require the equivalence of the norms ∥∇u∥ and ∥u∥1 for u ∈ Xn. This is fulfilled for instance for the Crouzeix–Raviart element of lowest order, cf. Crouzeix & Raviart (1973). The space of discretely divergence-free vector fields is denoted by $$ J_{n} := \left\{v_{n} \in X_{n}\: :\: \langle \operatorname{div} v_{n},q_{n}\rangle=0\;\;\forall\ q_{n}\in Y_{n}\right\} =\textrm{ker}\, B_{n} $$ and for u ∈ Hk+1(Ω)d ∩ X and p ∈ Hk(Ω) ∩ Y the discrete Stokes projection$$\mathcal{S}^{u}_{n}$$ is given by $$\mathcal{S}_{n}(u,p)=(\mathcal{S}^{u}_{n}(u,p),\mathcal{S}^{p}_{n}(u,p))\in X_{n}\times Y_{n}$$, solution of the problem \begin{align} \left\langle \nabla \mathcal{S}^{u}_{n}(u,p),\nabla v_{n}\right\rangle -\left\langle \mathcal{S}^{p}_{n}(u,p),\operatorname{div} v_{n}\right\rangle &= \langle \nabla u,\nabla v_{n}\rangle+\langle \nabla p,v_{n}\rangle & &\forall\:v_{n}\in X_{n},\nonumber\\ \left\langle \operatorname{div} \mathcal{S}^{u}_{n}(u,p),q_{n}\right\rangle &=0 & &\forall\:q_{n}\in Y_{n}. \end{align} (2.4) Note that in the case of a discrete pair $$(V_{n},Q_{n})\in X_{n}\times Y_{n}$$ one has $$(\mathcal{S}^{u}_{n} (V_{n},Q_{n}),\mathcal{S}^{p}_{n}(V_{n},Q_{n})) = (V_{n},Q_{n})$$. In what follows C will denote a generic constant (possibly changing values at different places) that is independent of h and τ, but may depend on the solution and the domain. Furthermore, we write $$a\lesssim b$$ for two functions or quantities a, b, whenever there is a generic constant C, such that a ≤ C b. In the study by Heywood & Rannacher (1988) the following estimates are proved: Lemma 2.2 Let $$m_{0}\in \mathbb{N}$$ and $$\partial _{t}^{(m)}u\in H^{k+1}(\varOmega )^{d}\cap X$$ and $$\partial _{t}^{(m)}p\in H^{k}(\varOmega )\cap Y$$ for 0 ≤ m ≤ m0. Then \begin{align*} \left\|u-\mathcal{S}^{u}_{n}(u,p)\right\|+h_{n}\left\|\nabla(u-\mathcal{S}^{u}_{n}(u,p))\right\| &\lesssim h_{n}^{k+1}\left(\|u\|_{k+1}+\|\,p\|_{k}\right)\!,\\ \left\|\partial_{t}^{(m)}\left(u-\mathcal{S}^{u}_{n}(u,p)\right)\right\|+h_{n}\left\|\partial_{t}^{(m)}\nabla\left(u-\mathcal{S}^{u}_{n}(u,p)\right)\right\| &\lesssim h_{n}^{k+1}\left(\left\|\partial_{t}^{(m)}u\right\|_{k+1}+\left\|\partial_{t}^{(m)}p\right\|_{k}\right)\!. \end{align*} Additionally we need an estimate for the associated pressure. Lemma 2.3 Let u ∈ Hk+1(Ω)d ∩ X and p ∈ Hk(Ω) ∩ Y. Then the following pressure estimate holds: $$ \left\|\mathcal{S}^{p}_{n}(u,p)-p\right\|\:\leq \: {h_{n}^{k}}\left(\|u\|_{k+1}+\|\,p\|_{k}\right)\!.$$ Proof. Use the inf–sup stability (2.3) and the interpolation operator $$\mathcal{J}_{h}$$ to obtain \begin{align*} \left\|\mathcal{S}^{p}_{n}(u,p)-p\right\| &\leq \left\|\mathcal{S}^{p}_{n}(u,p)-\mathcal{J}_{n}p\right\| + \left\|(I-\mathcal{J}_{n})p\right\|\\ &\lesssim \sup_{\substack{v_{n}\in X_{n}\\\|\nabla v_{n}\|=1}}\left\langle \mathcal{S}^{p}_{n}(u,p)-\mathcal{J}_{n}p,\operatorname{div} v_{n}\right\rangle + \left\|(I-\mathcal{J}_{n})p\right\|\\ &= \sup_{\substack{v_{n}\in X_{n}\\\|\nabla v_{n}\|=1}}\left\langle \mathcal{S}^{p}_{n}(u,p)-p,\operatorname{div} v_{n}\right\rangle+ \sup_{\substack{v_{n}\in X_{n}\\\|\nabla v_{n}\|=1}}\left\langle(I-\mathcal{J}_{n})p,\operatorname{div} v_{n}\right\rangle+\left\|(I-\mathcal{J}_{n})p\right\|\\ &\lesssim \left\|\nabla \left(u-\mathcal{S}^{u}_{n}(u,p)\right)\right\|+\left\|(I-\mathcal{J}_{n})p\right\|\!, \end{align*} where in the last step equation (2.4) was used. Then the lemma is concluded by equation (2.2) and Lemma 2.2. Let us define the Stokes operator \begin{align} \mathcal{A} u:=-\varPi_{H}\!\bigtriangleup\! u\quad\forall\ u\in D(\mathcal{A})=J\cap H^{2}(\varOmega)^{d}, \end{align} (2.5) where ΠH : L2(Ω)d → H is the Helmholtz projection, see for instance Sohr (2001, III.2). For a given g ∈ L2(Ω)d, $$w=\mathcal{A}^{-1}g$$ is solution of the Stokes system \begin{align} -\varDelta w+\nabla r=g,\;\operatorname{div} w=0\quad\textrm{in }\varOmega,\qquad w=0\quad\textrm{on }\partial\varOmega. \end{align} (2.6) In what follows, we assume that the Stokes problem is H2 × H1-regular. More precisely: Assumption 2.4 Let (w, r) be the solution of equation (2.6). Then (w, r) ∈ H2(Ω)d ∩ X × H1(Ω) ∩ Y and \begin{align} \|w\|_{2}+\|\nabla r\|\lesssim\|g\|. \end{align} (2.7) 2.2 Time discretization For the time discretization we introduce the finite difference operators $$\bar \partial ^{k}$$. For a sequence $$(v^{n})_{n=0}^{N}$$ with vn ∈ Xn we set recursively $$ \bar\partial v^{n}:=\frac{v^{n}-v^{n-1}}{\tau},\qquad \bar\partial^{2} v^{n}:=\bar\partial(\bar\partial v^{n}), \qquad \bar\partial^{k} v^{n}=\bar\partial(\bar\partial^{k-1}v^{n}),$$ and the BDF2 time discretization operator by \begin{align} \bar\partial^{B} v^{n}:= \frac{\tau}{2}\bar\partial^{2} v^{n} +\bar\partial v^{n} =\frac{3v^{n} - 4v^{n-1} + v^{n-2}}{2\tau}. \end{align} (2.8) An extension of $$\bar \partial ^{B}$$ to arbitrary time steps will be presented in Section 7. Note that in general $$\bar \partial v^{n}\notin X_{n}$$. Hence, define $$ {\bar\partial_{n}^{k}} v^{n}:=\varPi_{n}\bar\partial^{k} v^{n}\ \in\ X_{n},\qquad{\bar\partial_{n}^{B}} v^{n} :=\varPi_{n}\bar\partial^{B} v^{n}\ \in\ X_{n}.$$ For any Banach space Z, $$L^{p}((t_{1},t_{2});Z)$$ is the space of Lebesgue measurable Z-valued functions u : (t1, t2) → Z with $$ \|u\|_{L^{p}((t_{1},t_{2});Z)} := \left(\int_{t_{1}}^{t_{2}}\left\|u(t)\right\|^{p}_{Z}\mathrm{d}t\right)^{1/p},\qquad \|u\|_{L^{\infty}((t_{1},t_{2});Z)} := \textrm{ess} \sup_{t_{1}<t<t_{2}} \left\|u(t)\right\|_{Z}\!, $$$$1\leq p<\infty $$, and $$\|\cdot \|_{L^{p}(Z)}:=\|\cdot \|_{L^{p}((0,T);Z)}$$. Furthermore, for a sequence $$v^{i_{1}},\ldots ,v^{i_{m}}\in Z$$, $$i_{1},\ldots ,i_{m}\in \mathbb{N}$$, i1 ≤⋯ ≤ im, let $$ \|u\|_{\ell^{p}(i_{1},i_{m};Z)} := \left(\sum_{n=i_{1}}^{i_{m}}\tau\|u^{n}\|^{p}_{Z}\right)^{1/p}, \qquad \|u\|_{\ell^{\infty}(i_{1},i_{m};Z)} := \max_{i_{1}\leq n\leq i_{m}} \|u^{n}\|_{Z}, $$$$1\leq p<\infty $$, and set $$\|u\|_{\ell ^{p}(Z)}:=\|u\|_{\ell ^{p}(1,N;Z)}$$. Throughout this paper we denote space discrete quantities by capitals and continuous quantities by small letters. 2.3 The scheme: BDF2 for the fully discrete Stokes equation In order to discretize equation (2.1) define a sequence of approximate velocities Un ∈ Xn and pressures Pn ∈ Yn fulfilling: let X0 = X1 = X2 and U0 ∈ X0, U1 ∈ X1 be given; for n ≥ 2 determine $$\big (U^{n},P^{n}\big )\in X_{n}\times Y_{n}$$ from \begin{align} \boxed{{\bar\partial_{n}^{B}} U^{n}-\varDelta_{n} U^{n} + B_{n}^{\top} P^{n}= F^{n},\qquad B_{n} U^{n}=0,} \end{align} (2.9) where Fn := Πnf(tn). To initialize the scheme, values for U0, P0 and U1 are needed. To this end determine p0 := p(0) from u0 by taking the divergence of the first equation of equation (2.1) for t = 0 resulting in a Poisson problem for p0 with, say, Neumann boundary values obtained by testing the first equation of equation (2.1) (evaluated on the boundary) with the normal vector. Set $$(U^{0},P^{0}):=\mathcal{S}_{0}(u_{0},p_{0})=(\mathcal{S}_{0}^{u}(u_{0},p_{0}),\mathcal{S}_{0}^{p} (u_{0},p_{0} ))$$ and calculate $$(U^{1},P^{1})\in X_{1}\times Y_{1}$$ by the trapezoidal scheme: \begin{align} \bar\partial_{1} U^{1}-\varDelta_{1} U^{1/2}+B_{1}^{\top} P^{1/2}=F^{1/2},\qquad B_{1} U^{1} =0, \end{align} (2.10) where $$U^{1/2}:=\frac{1}{2}(U^{1}+U^{0})$$. The above discretization yields the following a priori estimates for constant meshes: Theorem 2.5 (A priori estimates for the time-dependent Stokes equations) Suppose the exact solution (u, p) of equation (2.1) is sufficiently regular. Let Xn ≡ Xh ⊆ X for all n ≥ 0 with mesh size h and (Un, Pn) be the solution of equation (2.9) initialized by (2.10). Then \begin{align} \|U-u\|^{2}_{\ell^{\infty}(L^{2})} \lesssim h^{2(k+1)}(A^{0}+A^{1}) \;+\; \tau^{4} \left\|{\partial_{t}^{3}} u\right\|^{2}_{L^{1}(H^{-1})}\!, \end{align} (2.11) \begin{align} {\hskip-22pt}\left\|\nabla(U-u)\right\|^{2}_{\ell^{2}(L^{2})} \lesssim h^{2k}(A^{0}+h^{2} A^{1}) \;+\; \tau^{4} \left\|{\partial_{t}^{3}} u\right\|^{2}_{L^{1}(H^{-1})}\!, \end{align} (2.12) \begin{align} {\hskip-100pt}\left\|\nabla(U-u)\right\|^{2}_{\ell^{\infty}(L^{2})}+\|P-p\|^{2}_{\ell^{2}(L^{2})} \lesssim h^{2k}(A^{0}+h^{2} A^{1}) \;+\; \tau^{4} \left\|{\partial_{t}^{3}} u\right\|^{2}_{L^{1}(L^{2})}\!, \end{align} (2.13) where $$A^{i} := \|{\partial ^{i}_{t}} u\|^{2}_{L^{\infty }(H^{k+1})}+\|{\partial ^{i}_{t}} p\|^{2}_{L^{\infty }(H^{k})}$$, i = 0, 1. The proof is outlined in Appendix A. 3. A posteriori estimates for the fully discrete scheme In this section we introduce a posteriori error estimators and state corresponding a posteriori estimates in Theorem 3.4. The proof of the theorem is outlined in Section 4. Furthermore, in Theorem 3.6 we state that our time estimators are optimal, i.e. that they have the same asymptotic behavior as the a priori estimates. The theorem is proven in Section 5 and utilizes discrete regularity estimates, which are provided in Appendix B. In order to derive optimal a posteriori estimates we need an approximation U(t) to u(t) ∀ t ∈ ]0, T[ to the right order. To this end we introduce a piecewise linear function, interpolating a sequence {Un}n=0, …, N at the time nodes, by \begin{align} U(t):=U^{n}+(t-t_{n})\bar\partial U^{n},\quad t\in I_{n},\; n=1,\ldots,N. \end{align} (3.1) Even though the interpolant U(t), defined in equation (3.1), is a second-order approximation, its residual is of first order only, cf. the study by Akrivis & Chatzipantelidis (2010) for parabolic problems. We follow the study by Akrivis & Chatzipantelidis (2010) and modify the interpolant U(t) using the piecewise quadratic three-point reconstruction$$\hat{U}(t)$$, interpolating a sequence {Un}n=0, …, N in tn−2, tn−1, tn: \begin{align} \hat{U}(t) := U(t)+\frac{1}{2}(t-t_{n})(t-t_{n-1})\bar{\partial}^{2} U^{n},\quad t\in I_{n},\; n=2,\ldots,N. \end{align} (3.2) Its time derivative is given by \begin{align} \partial_{t}\hat{U}(t) =\bar\partial^{B} U^{n}+(t-t_{n})\bar\partial^{2} U^{n},\quad t\in I_{n},\; n=2,\ldots,N. \end{align} (3.3) This three-point reconstruction was also proposed in the study by Lozinski et al. (2009) for the Crank–Nicolson scheme. Our main tool in deriving estimates is an application of the elliptic reconstruction technique introduced by Makridakis & Nochetto (2003) for parabolic problems and extended to the Stokes equations in the study by Karakatsani & Makridakis (2007). To this end define the Stokes Reconstruction$$\mathcal{R}_{u}^{n}$$ of a discrete function $$(U^{n},P^{n})\in X_{n}\times Y_{n}$$ by $$ \mathcal{R}_{u}^{n}:=\mathcal{A}^{-1}\left(-\varDelta_{n} U^{n}+B_{n}^{\top} P^{n}-F^{n}+f(t_{n})\right)\!,$$ i.e. there is a unique $$\mathcal{R}_{p}^{n}\in Y$$ such that $$(\mathcal{R}_{u}^{n},\mathcal{R}_{p}^{n})\in X\times Y$$ is the continuous solution of \begin{align} -\varDelta \mathcal{R}_{u}^{n}+\nabla \mathcal{R}_{p}^{n}=-\varDelta_{n} U^{n}+B_{n}^{\top} P^{n}-F^{n}+f(t_{n}),\qquad\operatorname{div} \mathcal{R}_{u}^{n}=0,\qquad{\mathcal{R}_{u}^{n}}{|_{\varGamma}}=0. \end{align} (3.4) The next lemma, which is straightforward, though crucial for our analysis, shows that the finite element approximation of $$(\mathcal{R}_{u}^{n},\mathcal{R}_{p}^{n})$$ in equation (3.4) is nothing else but $$(U^{n},P^{n})$$ itself. Lemma 3.1 Let $$ (V^{n},R^{n})\in X_{n}\times Y_{n}$$ be the finite element solution of equation (3.4), namely $$ -\varDelta_{n} V^{n}+B_{n}^{\top} R^{n}=-\varDelta \mathcal{R}_{u}^{n} + \nabla \mathcal{R}_{p}^{n} = -\varDelta_{n} U^{n}+B_{n}^{\top} P^{n}-F^{n}+f(t_{n}),\qquad B_{n} V^{n}=0. $$ Then, $$ V^{n} = U^{n}\qquad\textrm{and}\qquad R^{n} = P^{n}.$$ Proof. See Karakatsani & Makridakis (2007, Lemma 1.3). The above lemma implies that the difference of a given finite element function and its Stokes reconstruction can be estimated by well-known estimates for the stationary Stokes system. We thus make the assumption on the availability of such a posteriori error estimates. Assumption 3.2 Let $$(\mathcal{R}_{u}^{n},\mathcal{R}_{p}^{n})\in X\times Y$$ and $$(U^{n},P^{n})\in X_{n}\times Y_{n}$$ be as in equation (3.4). We assume that there exist a posteriori estimator functions $${\eta ^{n}_{u}} = {\eta ^{n}_{u}}(U^{n};Z)$$, $${\eta ^{n}_{p}}={\eta ^{n}_{p}}(P^{n};Y)$$, such that \begin{align} \begin{aligned} \left\|U^{n}-\mathcal{R}_{u}^{n}\right\|_{Z}^{2} &\leq{\eta^{n}_{u}}(U^{n};Z),\qquad Z\in\left\{J^{\ast},L^{2},{H_{0}^{1}}\right\}\!,\\ \left\|P^{n}-\mathcal{R}_{p}^{n}\right\|^{2} &\leq{\eta^{n}_{p}}(P^{n};Y). \end{aligned} \end{align} (3.5 ) Additionally there exists an estimator function $$\eta ^{0}(Z)$$ of the initial approximation error, such that $$ \left\|u_{0}-\mathcal{S}^{u}_{0}(u_{0},p_{0})\right\|^{2}_{Z} \leq \eta^{0}(Z).$$ Note that for a large class of finite element spaces Xn, Yn such estimators are available, see for example Verführt (2013). In the study by Bank & Welfert (1990) a comparison of several a posteriori error estimates for the stationary Stokes problem is given. Let us now summarize the notation. First we denote space discrete quantities by capitals and continuous quantities by small letters. Un ∈ Xn denotes the solution of equation (2.9) at discrete time tn, whereas U(t) and $$\hat{U}(t)$$ are interpolants of Un and Un−1 in the corresponding time interval In. Equally we apply this notation for the Stokes reconstruction $$\mathcal{R}_{u}^{n}$$, which is defined in equation (3.4) at discrete time tn and $$\mathcal{R}_{u}(t)$$ or $$\hat{\mathcal{R}}_{u}(t)$$ are the interpolation, respectively three-point reconstruction, in the time interval In. Now we are able to define the error estimators. Definition 3.3 (Error estimators). We assume X2 = X1 = X0 and set to simplify the presentation $$U^{-1}:=U^{1}-2\tau _{0}(\varDelta _{0} U^{0}-B_{0}^{\top } P^{0}+F^{0})\in X_{0}$$ and $$U^{-2}:=-6\tau _{0}(\varDelta _{0} U^{0}-B_{0}^{} P^{0}+F^{0})-3U^{0}+4U^{1}\in X_{0}$$, cf. Lemma 4.5. Let us introduce the time error estimators \begin{align*} \mathcal{E}_{1} &:= \sum_{n=2}^{N}\tau\left\|{\bar\partial_{n}^{2}} U^{n}\right\|^{2}, & \mathcal{E}_{2} &:= \sum_{n=2}^{N}\frac{1}{\tau}\left\|{\bar\partial_{n}^{2}} U^{n}-\bar\partial^{2}_{n-1}U^{n-1}\right\|^{2}_{J^{\ast}}, & \mathcal{E}_{3} &:= \max_{2\leq n\leq N}\left\|{\bar\partial_{n}^{2}} U^{n}\right\|^{2},\\ \mathcal{E}_{6} &:= \sum_{n=2}^{N}\frac{1}{\tau}\left\|{\bar\partial_{n}^{2}} U^{n}-\bar\partial^{2}_{n-1}U^{n-1}\right\|^{2}, & \mathcal{E}_{7} &:= \max_{2\leq n\leq N}\left\|{\nabla\bar\partial_{n}^{2}} U^{n}\right\|^{2}, & & \end{align*} >the Stokes reconstruction error estimators $$ \mathcal{E}_{4} := \sum_{n=2}^{N}\tau\left\|\bar\partial^{2}f(t_{n})-{\bar\partial^{2}_{n}}\bar\partial^{B} U^{n}\right\|^{2}_{J^{\ast}},\qquad \mathcal{E}_{5} := \sum_{n=2}^{N}\tau\left\|\bar\partial^{2}f(t_{n})-{\bar\partial^{2}_{n}}\bar\partial^{B} U^{n}\right\|^{2}, $$ the space error estimators \begin{align*} \varLambda_{1} &:= \sum_{n=0}^{N}\tau\ {\eta_{u}^{n}}(U^{n};L^{2}), & \varLambda_{2} &:= \max_{0\leq n\leq N}{\eta_{u}^{n}}(U^{n};L^{2}),\\ \varLambda_{3} &:= \sum_{n=1}^{N}\tau\ {\eta_{u}^{n}}\left(\bar\partial_{n} U^{n};J^{\ast}\right)\!, & \varLambda_{4} &:= \sum_{n=1}^{N}\tau\ {\eta_{u}^{n}}\left(U^{n};{H^{1}_{0}}\right)\!,\\ \varLambda_{5} &:= \max_{0\leq n\leq N}{\eta_{u}^{n}}\left(U^{n};{H_{0}^{1}}\right)\!, & \varLambda_{6} &:= \sum_{n=1}^{N}\tau\ {\eta_{u}^{n}}\left(\bar\partial_{n} U^{n};L^{2}\right)\!,\\ \varLambda_{7} &:= \sum_{n=0}^{N}\tau\ {\eta_{p}^{n}}(P^{n};Y), & \end{align*} the changing mesh estimators \begin{align*} \varXi_{1} &:= \sum_{n=2}^{N}\tau \xi_{B}^{n,n}, & \varXi_{4} &:= \sum_{n=2}^{N}\frac{{h_{n}^{2}}}{\tau}\left(\xi_{B}^{n,n}+\xi_{B}^{n-1,n} +\xi_{B}^{n-2,n} \right)\!, & \varXi_{6} &:= \sum_{n=2}^{N}\frac{1}{\tau}\xi_{B}^{n,n}, \\ \varXi_{2} &:= \sum_{n=2}^{N}\frac{{h_{n}^{2}}}{\tau}\xi_{B}^{n,n}, & \varXi_{5} &:= \sum_{n=2}^{N}\frac{1}{\tau}\left(\xi_{B}^{n,n}+\xi_{B}^{n-1,n} +\xi_{B}^{n-2,n} \right)\!, & \varXi_{7} &:= \tau^{4}\max_{2\leq n\leq N}\left\|\nabla\left(\bar\partial^{2}-{\bar\partial_{n}^{2}}\right)U^{n}\right\|^{2}\!, \\ \varXi_{3} &:= \max_{2\leq n\leq N}\xi_{B}^{n,n}, & & & \end{align*} where the local BDF2-changing mesh operator $$\xi _{B}^{a,b}$$ is defined by \begin{align} \xi_{B}^{a,b}:= \left(\|U^{a}\|^{2}-\|\varPi_{b} U^{a}\|^{2}\right) + \left(\|U^{a-1}\|^{2}-\|\varPi_{b} U^{a-1}\|^{2}\right) + \left(\|U^{a-2}\|^{2}-\|\varPi_{b} U^{a-2}\|^{2}\right)\!, \end{align} (3.6) and the data approximation error estimators \begin{align*} \mathcal{F}_{1} :=&\, \sum_{n=1}^{N}\int_{t_{n-1}}^{t_{n}}\!\left\|\,f(t_{n})+(t-t_{n})\bar\partial f(t_{n})-f(t)\right\|^{2}_{J^{\ast}}\mathrm{d}t,\quad \mathcal{F}_{2} :=\sum_{n=1}^{N}\int_{t_{n-1}}^{t_{n}}\!\left\|\,f(t_{n})+(t-t_{n})\bar\partial f(t_{n})-f(t)\right\|^{2}\!,\\ {\textit{I}}_{1} :=&\, \tau^{3}\|\varPsi_{1}\|^{2}_{J^{\ast}},\qquad\qquad \qquad\qquad\qquad\qquad\qquad\quad\ {\textit{I}}_{2} := \tau^{3}\|\varPsi_{1}\|^{2}, \end{align*} where $$\varPsi _{1}:= \bar \partial ^{2} U^{2}-\varDelta _{1} \bar \partial U^{1}+ B_{1}^{\top }\bar \partial P^{1}-\bar \partial F^{1}$$. In the following theorem a posteriori estimates in various norms are stated. Theorem 3.4 (A posteriori error estimates for the time-dependent Stokes system). Let $$(U,P)$$ be the solution of equation (2.9) initialized by equation (2.10) and $$(u,p)$$ the exact solution of equation (2.1). Then \begin{align} {\hskip-56pt}\|U-u\|^{2}_{L^{2}(L^{2})} \lesssim \eta^{0}(J^{\ast}) + \tau^{4}(\mathcal{E}_{1} + \mathcal{E}_{2}) + \varLambda_{1} +\varXi_{1}+\varXi_{2} + \mathcal{F}_{1} + {\textit{I}}_{1}, \end{align} (3.7) \begin{align} {\hskip32pt}\|U-u\|^{2}_{L^{\infty}(L^{2})} \lesssim \eta^{0}(L^{2}) + \tau^{4}(\mathcal{E}_{2}+\mathcal{E}_{3}+\mathcal{E}_{4}) + \varLambda_{2} + \varLambda_{3} + \varXi_{2}+\varXi_{3}+\varXi_{4} + \mathcal{F}_{1}+ {\textit{I}}_{1}, \end{align} (3.8) \begin{align} {\hskip-45pt}\|\nabla(U-u)\|^{2}_{L^{2}(L^{2})} \lesssim \eta^{0}(L^{2}) + \tau^{4}(\mathcal{E}_{2}+\mathcal{E}_{4}) + \varLambda_{3} + \varLambda_{4} + \varXi_{2}+\varXi_{4} + \mathcal{F}_{1}+ {\textit{I}}_{1}, \end{align} (3.9) \begin{align} \|\nabla(U-u)(t)\|^{2}_{L^{\infty}(L^{2})} \lesssim \eta^{0}\left({H_{0}^{1}}\right) \!+\! \tau^{4}(\mathcal{E}_{5}\!+\!\mathcal{E}_{6}\!+\!\mathcal{E}_{7}) \!+\! \varLambda_{5}\!+\!\varLambda_{6} \!+\! \varXi_{5}\!+\!\varXi_{6}\!+\!\varXi_{7} \!+\! \mathcal{F}_{2}\!+\! {\textit{I}}_{2}, \!\!\!\qquad\end{align} (3.10) \begin{align} {\hskip-22pt}\|P-p\|^{2}_{L^{2}(L^{2})} \lesssim \eta^{0}\left({H_{0}^{1}}\right) + \tau^{4}(\mathcal{E}_{5}+\mathcal{E}_{6}) + \varLambda_{6} + \varLambda_{7} + \varXi_{5}+\varXi_{6}+ \mathcal{F}_{2}+ {\textit{I}}_{2}, \end{align} (3.11) where the estimators are defined in Definition 3.3 and $$\eta ^{0}(Z)$$ fulfills Assumption 3.2. Remark 3.5 (Negative norms). The time error estimator $$\mathcal{E}_{2}$$, Stokes reconstruction estimator $$\mathcal{E}_{4}$$ and data approximation error estimator $$\mathcal{F}_{1}$$ are defined with respect to a negative norm, as usual for error estimates of BDF2 time discretizations, cf. Akrivis & Chatzipantelidis (2010). From a computational point of view this is completely prohibitive, since its evaluation may be computationally more expensive than solving the discrete scheme. However, there are possibilities for estimating it. The simplest way is to bound $$\|\cdot \|_{J^{\ast }}$$ by ∥⋅∥, which can be pessimistic in practice. Another way is to approximate the J* norm in an adequate way. In our case the estimator $$\mathcal{E}_{4}$$ (for the sake of convenience assume constant meshes) can be handled further in the following way. We approximate the norm J* by the discrete norm $$J^{\ast }_{h}$$: $$ \left\|\bar\partial^{2} f(t_{n})-\bar\partial^{2}\bar\partial^{B}U^{n}\right\|_{J^{\ast}} \approx \left\|\bar\partial^{2} f(t_{n})-\bar\partial^{2}\bar\partial^{B}U^{n}\right\|_{J_{h}^{\ast}}\!.$$ Now we use the triangle inequality and insert the scheme to get \begin{align*} \left\|\bar\partial^{2} f(t_{n})-\bar\partial^{2}\bar\partial^{B}U^{n}\right\|_{J_{h}^{\ast}} &\leq \left\|\bar\partial^{2}\left(\,f(t_{n})-F^{n}\right)\right\|_{J_{h}^{\ast}}+\left\|\bar\partial^{2}(F^{n}-\bar\partial^{B} U^{n})\right\|_{J_{h}^{\ast}}\\ &= \left\|\bar\partial^{2}\left(\,f(t_{n})-F^{n}\right)\right\|_{J_{h}^{\ast}}+\left\|\bar\partial^{2}(-\varDelta_{h} U^{n})\right\|_{J_{h}^{\ast}}\\ &\leq \left\|\bar\partial^{2}\left(\,f(t_{n})-F^{n}\right)\right\|_{J_{h}^{\ast}}+\|\nabla\bar\partial^{2} U^{n}\|. \end{align*} For the handling of the data approximation errors in negative norms let us mention the discussion in the study by Ern et al. (2016). Our next theorem states the optimality of the time error estimators $$\mathcal{E}_{i}$$. We understand optimality in the sense that the time error estimators have the same asymptotic behavior as corresponding a priori estimates. In view of Theorem 3.4 the time error estimators are of second order, which corresponds to the a priori estimates in Theorem 2.5. However, this requires that the estimators are independent of the time step and mesh size, i.e. there exists a constant C independent of τ and h, such that $$\mathcal{E}_{i}\ \leq \ C$$. Theorem 3.6 (Stability of the time estimators). Suppose the exact solution (u, p) and data f, u0 are sufficiently regular. Let $$(X_{h},Y_{h})\subseteq X\times Y$$ be a mixed finite element approximation on a quasi-uniform triangulation $$\mathcal{T}_{h}$$ of order k ≥ 2 and let (Un, Pn) be solution of equation (2.9) with $$(X_{n},Y_{n})= (X_{h},Y_{h} )$$ ∀ n = 0, …, N. Then the time error estimators $$\mathcal{E}_{1}$$–$$\mathcal{E}_{7}$$ defined in (3.3) are stable, i.e. $$ \mathcal{E}_{1}+\mathcal{E}_{2}+\mathcal{E}_{3}+\mathcal{E}_{4} + \mathcal{E}_{5} + \mathcal{E}_{6} + \mathcal{E}_{7} \leq C(u,p,f),$$ where C is independent of the time step τ and the mesh size h. The proof is outlined in Section 5. Due to its technical difficulties we have to restrict our approximations in equation (2.9) to a constant, quasi-uniform mesh. This assumption, of course, does not go together with spatial adaptivity. However, this restriction is only necessary for the theoretical result of stability, not for the proof of the a posteriori estimates in Theorem 3.4. 4. Proofs of the a posteriori estimates In this section the proof of the a posteriori error estimates in Theorem 3.4 is presented. Let us start with some auxiliary statements. In the first step we deduce the error equation using the three-point reconstruction $$\hat{U}(t)$$: Lemma 4.1 (Error equation). Let n ≥ 2 and t ∈ In. Then we have the following identity in H−1: \begin{align} \partial_{t}(\hat{U}-u)(t)-\varDelta (U-u)(t) + \nabla (P-p)(t) =-\varDelta( U-\mathcal{R}_{u})(t) + \nabla(P-\mathcal{R}_{p})(t)+\Upsilon(t), \end{align} (4.1) where for t ∈ In \begin{align} \Upsilon(t) := f(t_{n})-f(t)+ (t-t_{n})\left(\bar\partial^{2} U^{n} -\varDelta\bar\partial \mathcal{R}_{u}^{n} + \nabla\bar\partial \mathcal{R}_{p}^{n}\right)+\left(\bar\partial^{B}-{\bar\partial^{B}_{n}}\right)U^{n}. \end{align} (4.2) Proof. Using the definitions of $$\hat{U}(t)$$ in equation (3.2) and the Stokes reconstruction inequation (3.4) one has for n ≥ 2, \begin{align*} \partial_{t}\hat{U}(t)-\varDelta U(t)+\nabla P(t)=&\ \bar\partial^{B} U^{n} + (t-t_{n})\bar\partial^{2} U^{n} -\varDelta(U-\mathcal{R}_{u})(t) +\nabla (P-\mathcal{R}_{p})(t) -\varDelta \mathcal{R}_{u}^{n} + \nabla \mathcal{R}_{p}^{n}\\ &+ (t-t_{n})\left(-\varDelta\bar\partial \mathcal{R}_{u}^{n} + \nabla\bar\partial \mathcal{R}_{p}^{n}\right)\\ =&\ {\bar\partial_{n}^{B}} U^{n} -\varDelta_{n} U^{n} + B_{n}^{\top} P^{n} -F^{n}+f\big(t_{n}\big) + (t-t_{n})\left(\bar\partial^{2} U^{n} -\varDelta\bar\partial \mathcal{R}_{u}^{n} + \nabla\bar\partial \mathcal{R}_{p}^{n}\right)\\ &-\varDelta(U-\mathcal{R}_{u})(t) +\nabla (P-\mathcal{R}_{p})(t)+\left(\bar\partial^{B}-{\bar\partial^{B}_{n}}\right)U^{n}\\ =&\ f(t_{n})+ (t-t_{n})\left(\bar\partial^{2} U^{n} -\varDelta\bar\partial \mathcal{R}_{u}^{n} + \nabla\bar\partial \mathcal{R}_{p}^{n}\right)-\varDelta(U-\mathcal{R}_{u})(t) \\ &+\nabla (P-\mathcal{R}_{p})(t)+\left(\bar\partial^{B}-{\bar\partial^{B}_{n}}\right)U^{n}.\\ \end{align*} Now subtract equation (2.1) to finish the proof. 4.1 Estimating initial errors Next we present upper bounds for the initial errors. Lemma 4.2 (Initial a posteriori error estimates). Let $$(u,p)$$ be the exact solution of equation (2.1), X2 = X1 = X0 and $$ (U^{0},P^{0}):=(\mathcal{S}^{u}_{n}(u_{0},p_{0}),\mathcal{S}^{p}_{n}(u_{0},p_{0}))$$ the solution of equation (2.4). Let $$(U^{1},P^{1})$$ be given by equation (2.10) and $$(U^{2},P^{2})$$ by equation (2.9). Then \begin{align} &\|\hat{U}-u\|^{2}_{L^{\infty}((0,t_{1});J^{\ast})}+\|U-u\|^{2}_{L^{2}((0,t_{1});L^{2})}\nonumber\\ &\quad\lesssim \eta^{0}(J^{\ast})+\int_{0}^{t_{1}}\left(\left\|(\hat{U}-U)(t)\right\|^{2} +\left\|(U-\mathcal{R}_{u})(t)\right\|^{2}+ \left\|\Upsilon(t)\right\|^{2}_{J^{\ast}}\right)\mathrm{d}t, \end{align} (4.3) \begin{align} &\|U-u\|^{2}_{L^{\infty}((0,t_{1});L^{2})}\nonumber\\ &\quad\lesssim \eta^{0}(L^{2})+ \max_{t\in[0,t_{1}]}\left\|(U-\hat{U})(t)\right\|^{2}+\max_{t\in[0,t_{1}]}\left\|(\hat{\mathcal{R}}_{u}-\hat{U})(t)\right\|^{2}\nonumber \\ &\qquad+\int_{0}^{t_{1}}\left(\left\|\partial_{t}(\hat{\mathcal{R}}_{u}-\hat{U})(t)\right\|^{2}_{J^{\ast}} +\left\|\nabla(\mathcal{R}_{u}-\hat{\mathcal{R}}_{u})(t)\right\|^{2}+\left\|\Upsilon(t)\right\|^{2}_{J^{\ast}}\right)\mathrm{d}t, \end{align} (4.4) \begin{align} &\left\|\nabla(U-u)\right\|^{2}_{L^{2}((0,t_{1});L^{2})}\nonumber\\ &\quad\lesssim \eta^{0}(L^{2})+\int_{0}^{t_{1}}\left\|\nabla(U-\mathcal{R}_{u})(t)\right\|^{2}\mathrm{d}t\nonumber\\ &\qquad+\int_{0}^{t_{1}}\left(\left\|\partial_{t}(\hat{\mathcal{R}}_{u}-\hat{U})(t)\right\|^{2}_{J^{\ast}} +\left\|\nabla(\mathcal{R}_{u}-\hat{\mathcal{R}}_{u})(t)\right\|^{2}+\left\|\varUpsilon(t)\right\|^{2}_{J^{\ast}}\right)\mathrm{d}t. \end{align} (4.5) \begin{align} &\left\|\nabla(U-u)\right\|^{2}_{L^{\infty}((0,t_{1});L^{2})}\nonumber\\ &\quad\lesssim \eta^{0}\left({H^{1}_{0}}\right) + \max_{t\in[0,t_{1}]}\left\|\nabla(U-\hat{U})(t)\right\|^{2}+\max_{t\in[0,t_{1}]}\left\|\nabla(\hat{\mathcal{R}}_{u}-\hat{U})(t)\right\|^{2}\nonumber\\ &\qquad+\int_{0}^{t_{1}}\left(\left\|\partial_{t}(\hat{\mathcal{R}}_{u}-\hat{U})(t)\right\|^{2} +\left\|\varDelta(\mathcal{R}_{u}-\hat{\mathcal{R}}_{u})(t)\right\|^{2}+\left\|\varUpsilon(t)\right\|^{2}\right)\mathrm{d}t, \end{align} (4.6) \begin{align} &\|P-p\|^{2}_{L^{2}((0,t_{1});L^{2})}\nonumber\\ &\lesssim \eta^{0}\left({H^{1}_{0}}\right)+\int_{0}^{t_{1}}\left\|(P-\mathcal{R}_{p})(t)\right\|^{2}\mathrm{d}t\nonumber\\ &\qquad+\int_{0}^{t_{1}}\left(\left\|\partial_{t}(\hat{\mathcal{R}}_{u}-\hat{U})(t)\right\|^{2} +\left\|\varDelta(\mathcal{R}_{u}-\hat{\mathcal{R}}_{u})(t)\right\|^{2}+\left\|\varUpsilon(t)\right\|^{2}\right)\mathrm{d}t, \end{align} (4.7) where η0 is defined in Assumption 3.2 and \begin{align} \varUpsilon := (t-t_{1/2})\left(\bar\partial^{2}U^{2} - \varDelta_{1}\bar\partial U^{1}+ B_{1}^{\top} \bar\partial P^{1} -\bar\partial F^{1}\right)+ f(t_{1})+(t-t_{1})\bar\partial f(t_{1})-f(t),\qquad t\in I_{1}. \end{align} (4.8) Proof. For t ∈ I1 = (0, t1] define the three-point reconstruction \begin{align} \hat{U}(t) = U(t)+\frac{1}{2}(t-t_{0})(t-t_{1})\bar\partial^{2} U^{2},\qquad \partial_{t}\hat{U}(t)=\bar\partial U^{1}+(t-t_{1/2})\bar\partial^{2} U^{2}, \end{align} (4.9) where the identity $$t-t_{1/2}=\frac{\tau }{2}+(t-t_{1})$$ was used. The interpolant $$\hat{U}(t)$$ fulfills (in H−1) \begin{align*} &\partial_{t} \hat{U}(t)-\varDelta U(t)+\nabla P(t) \\ &\quad=\bar\partial U^{1}+(t-t_{1/2})\bar\partial^{2} U^{2}-\varDelta(U-\mathcal{R}_{u})(t)+\nabla (P-\mathcal{R}_{p})(t) \\ &\qquad-\varDelta \mathcal{R}_{u}^{1}+\nabla \mathcal{R}_{p}^{1} + (t-t_{1})\left(-\varDelta\bar\partial \mathcal{R}_{u}^{1}+\nabla \bar\partial \mathcal{R}_{p}^{1}\right)\\ &\quad=\varDelta_{1} U^{1/2}-B_{1}^{\top} P^{1/2}+F^{1/2}+(t-t_{1/2})\bar\partial^{2} U^{2}-\varDelta(U-\mathcal{R}_{u})(t)+\nabla (P-\mathcal{R}_{p})(t) \\ &\qquad-\varDelta_{1} U^{1}+B_{1}^{\top} P^{1}-F^{1}+f(t_{1}) + (t-t_{1})\bar\partial\left(-\varDelta_{1} U^{1}+B_{1}^{\top} P^{1}-F^{1}+f(t_{1})\right)\\ &\quad=(t-t_{1/2})\left(\bar\partial^{2} U^{2}-\varDelta_{1}\bar\partial U^{1}+ B_{1}^{\top}\bar\partial P^{1}-\bar\partial F^{1}\right)+f(t_{1})+(t-t_{1})\bar\partial f(t_{1})\\ &\qquad-\varDelta(U-\mathcal{R}_{u})(t)+\nabla (P-\mathcal{R}_{p})(t). \end{align*} Note we have used the fact that X2 = X1 = X0. Subtract equation (2.1) from the above identity to get the error equation for t ∈ I1 \begin{align} \partial_{t}(\hat{U}-u)(t)-\varDelta (U-u)(t)+\nabla (P-p)(t) =\varUpsilon(t)-\varDelta(U-\mathcal{R}_{u})(t)+\nabla (P-\mathcal{R}_{p})(t), \end{align} (4.10) where $$\varUpsilon $$ is defined inequation (4.8). To prove equation (4.3) test equation (4.10) by $$w(t):=\mathcal{A}^{-1}(\hat{U}-u)(t)$$, i.e. (w(t), q(t)) is solution of $$ -\varDelta w(t)+\nabla q(t)=(\hat{U}-u)(t),\quad \textrm{div}\, w(t)=0,\quad w(t)|_{\varGamma}=0. $$ For the first term on the left-hand side, one gets $$ \left\langle\partial_{t}(\hat{U}-u)(t),w(t)\right\rangle = \frac{1}{2}\frac{d}{\mathrm{d}t}\left\|\nabla w(t)\right\|^{2}\!. $$ The second term can be treated with the help of the identity \begin{align} 2\langle a-b,c-b\rangle=\|a-b\|^{2}+\|c-b\|^{2}-\|a-c\|^{2} \end{align} (4.11) and the fact that the interpolant of the Stokes reconstruction $$\mathcal{R}_{u}(t)$$ is divergence free: \begin{align*} \left\langle\nabla(U-u)(t),\nabla w(t)\right\rangle &= \left\langle(U-u)(t),-\varDelta w(t)\right\rangle=\left\langle(U-u)(t),(\hat{U}-u)(t)\right\rangle-\left\langle(U-u)(t),\nabla q(t)\right\rangle\\ &= \frac{1}{2}\left\|(U\!-u)(t)\right\|^{2}\!+\frac{1}{2}\left\|(\hat{U}\!-u)(t)\right\|^{2}\!-\frac{1}{2}\left\|(\hat{U}\!-U)(t)\right\|^{2}\!-\left\langle(U\!-\mathcal{R}_{u})(t),\nabla q(t)\right\rangle\\ &\geq \frac{1}{2}\left\|(U-u)(t)\right\|^{2}+\frac{1}{2}\left\|(\hat{U}-u)(t)\right\|^{2}-\frac{1}{2}\left\|(\hat{U}-U)(t)\right\|^{2}\\ &\quad- C\left\|(U-\mathcal{R}_{u})(t)\right\|\,\left\|(\hat{U}-u)(t)\right\|\!. \end{align*} For the last inequality the regularity estimate (2.7) for ∇q(t) has been used. For the terms on the right-hand side of equation (4.10), we obtain \begin{align*} \left\langle-\varDelta (U-\mathcal{R}_{u})(t)+ \varUpsilon(t),w(t)\right\rangle &= \left\langle (U-\mathcal{R}_{u})(t),-\varDelta w(t)\right\rangle+\left\langle\varUpsilon(t),w(t)\right\rangle\\ &= \left\langle(U-\mathcal{R}_{u})(t),-\nabla q(t)+(\hat{U}-u)(t)\right\rangle+\left\langle\varUpsilon(t),w(t)\right\rangle\\ &\lesssim \left\|(U-\mathcal{R}_{u})(t)\right\|^{2} + \left\|(\hat{U}-u)(t)\right\| + \left\|\varUpsilon(t)\right\|_{J^{\ast}}\left\|\nabla w(t)\right\|\!. \end{align*} Now, integrating from t = 0 to t1, yields \begin{align} &\max_{t\in[0,t_{1}]}\left\|(\hat{U}-u)(t)\right\|^{2}_{J^{\ast}}+\int_{0}^{t_{1}}\left(\left\|(U-u)(t)\right\|^{2}+\left\|(\hat{U}-u)(t)\right\|^{2}\right)\mathrm{d}t\nonumber\\ &\quad\lesssim \left\|(\hat{U}-u)(0)\right\|^{2}_{J^{\ast}}+\int_{0}^{t_{1}}\left(\left\|(\hat{U}-U)(t)\right\|^{2}+ \left\|(U-\mathcal{R}_{u})(t)\right\|^{2}+ \left\|\varUpsilon(t)\right\|^{2}_{J^{\ast}}\right)\mathrm{d}t, \end{align} (4.12) with $$\|(\hat{U}-u)(0)\|^{2}_{J^{\ast }} = \|U^{0}-u_{0}\|^{2}_{J^{\ast }}=\|\mathcal{S}^{u}_{n}(u_{0},p_{0})-u_{0}\|^{2}_{J^{\ast }}\leq \eta ^{0}(J^{\ast })$$. Rearrange equation (4.10) to get for t ∈ I1 \begin{align} \partial_{t}(\hat{\mathcal{R}}_{u} -u)(t)-\varDelta\left(\mathcal{R}_{u}-u\right)\!(t) +\nabla (\mathcal{R}_{p}-p)(t) =\partial_{t}(\hat{\mathcal{R}}_{u}-\hat{U})(t)+\varUpsilon(t). \end{align} (4.13) Test equation (4.13) with $$\left (\mathcal{R}_{u}-u\right )\!(t)\in J$$, use identity (4.11) and integrate from t = 0 to t1 to obtain \begin{align} &\max_{t\in[0,t_{1}]}\left\|(\hat{\mathcal{R}}_{u}-u)(t)\right\|^{2}+\int_{0}^{t_{1}}\left(\left\|\nabla(\hat{\mathcal{R}}_{u}-u)(t)\right\|^{2}+\left\|\nabla(\mathcal{R}_{u}-u)(t)\right\|^{2}\right)\mathrm{d}t\nonumber\\ &\quad\lesssim \left\|(\hat{\mathcal{R}}_{u}-u)(0)\right\|^{2}+\int_{0}^{t_{1}}\left(\left\|\partial_{t}(\hat{\mathcal{R}}_{u}-\hat{U})(t)\right\|^{2}_{J^{\ast}} +\left\|\nabla(\mathcal{R}_{u}-\hat{\mathcal{R}}_{u})(t)\right\|^{2}+\left\|\varUpsilon(t)\right\|^{2}_{J^{\ast}}\right)\mathrm{d}t. \end{align} (4.14) Finally, the triangle inequality yields \begin{align*} \max_{t\in[0,t_{1}]}\left\|(U-u)(t)\right\|^{2} \leq&\, \max_{t\in[0,t_{1}]}\left\|(U-\hat{U})(t)\right\|^{2} +\max_{t\in[0,t_{1}]}\left\|(\hat{\mathcal{R}}_{u}-\hat{U})(t)\right\|^{2}+\max_{t\in[0,t_{1}]}\left\|(\hat{\mathcal{R}}_{u}-u)(t)\right\|^{2}\\ \lesssim&\, \eta^{0}(L^{2})+ \max_{t\in[0,t_{1}]}\left\|(U-\hat{U})(t)\right\|^{2}+\max_{t\in[0,t_{1}]}\left\|(\hat{\mathcal{R}}_{u}-\hat{U})(t)\right\|^{2}\\ &+\int_{0}^{t_{1}}\left(\left\|\partial_{t}(\hat{\mathcal{R}}_{u}-\hat{U})(t)\right\|^{2}_{J^{\ast}} +\left\|\nabla(\mathcal{R}_{u}-\hat{\mathcal{R}}_{u})(t)\right\|^{2}+\left\|\varUpsilon(t)\right\|^{2}_{J^{\ast}}\right)\mathrm{d}t. \end{align*} To prove equation (4.5) the error is split into $$ \int_{0}^{t_{1}}\left\|\nabla(U-u)(t)\right\|^{2}\mathrm{d}t \leq \int_{0}^{t_{1}}\left\|\nabla(U-\mathcal{R}_{u})(t)\right\|^{2} \mathrm{d}t + \int_{0}^{t_{1}}\left\|\nabla(\mathcal{R}_{u}-u)(t)\right\|^{2}$$ and then equation (4.14) is used. The proofs of equations (4.6) and (4.7) are omitted, since they can be achieved in exactly the same way as the proofs of equations (3.10) and (3.11) in Theorem 3.4 using the error equation (4.13). 4.2 Error estimator In the following lemma we study how to handle mesh changes within the a posteriori framework. One fundamental technical difficulty arises from the fact that because of mesh changes Un ∈ Xn≠Xn−1 ∋ Un−1 in general, and thus terms like $$\bar \partial U^{n}$$ are not well defined as a finite element function. Lemma 4.3 (Changing meshes). For an arbitrary function Ws ∈ Xs, s = 1, …, N and for arbitrary $$n=1,\dots , N$$ as well m = 1, …, k + 1, we have \begin{align} {\hskip-138pt}\left\|(I-\varPi_{n})W^{s}\right\|^{2} = \|W^{s}\|^{2}-\|\varPi_{n} W^{s}\|^{2}, \end{align} (4.15) \begin{align} {\hskip-113pt}\left\|(I-\varPi_{n})W^{s}\right\|^{2}_{-m} \lesssim \ h^{2m}_{n}\left(\|W^{s}\|^{2}-\|\varPi_{n} W^{s}\|^{2}\right)\!, \end{align} (4.16) \begin{align} \|\bar\partial W^{s}\|^{2}_{-m} \lesssim \|\bar\partial_{s} W^{s}\|^{2}_{-m} +C\frac{h_{n}^{2m}}{\tau^{2}} \left(\|W^{s-1}\|^{2}-\|\varPi_{n} W^{s-1}\|^{2}\right)\!, \end{align} (4.17) \begin{equation} {\hskip-186pt}\left\|(I-\varPi_{n})\bar\partial^{B} W^{s}\right\|_{-m}^{2} \lesssim \frac{h^{2m}_{n}}{\tau^{2}}\xi_{B}^{s,n}(W), \end{equation} (4.18) \begin{align} {\hskip-89pt}\|\bar\partial^{2} W^{s}\|^{2} \leq \left\|{\bar\partial_{s}^{2}} W^{s}\right\|^{2} + C\frac{1}{\tau^{4}}\xi_{B}^{s,s}(W), \end{align} (4.19) where the local BDF2-changing mesh operator $$\xi _{B}^{s,n}(W)$$ is defined in equation (3.6). Proof. By definition of the L2 projection $$ \left\langle(I-\varPi_{n})W^{s},v_{n}\right\rangle=0\qquad \forall\ v_{n}\in X_{n}$$ and the identity \begin{align} 2\langle a-b, a\rangle = \|a\|^{2}-\|b\|^{2}+\|a-b\|^{2} \end{align} (4.20) it follows $$ 0=2\left\langle(I-\varPi_{n})W^{s},\varPi_{n} W^{s}\right\rangle = -\|\varPi_{n}W^{s}\|^{2}+\|W^{s}\|^{2}-\left\|(I-\varPi_{n})W^{s}\right\|^{2}\!, $$ which implies equation (4.15). To prove equation (4.16) observe that $$ \left\|(I\!-\varPi_{n})W^{s}\right\|_{-m}=\!\sup_{v\in{H_{0}^{1}}\cap H^{m}}\!\frac{\left\langle(I-\varPi_{n})W^{s},v\right\rangle}{\|v\|_{H^{m}}} =\!\sup_{v\in{H_{0}^{1}}\cap H^{m}}\!\frac{\left\langle(I-\varPi_{n})W^{s},v-\mathcal{I}_{n}v\right\rangle}{\|v\|_{H^{m}}} \!\leq\! C\ {h^{m}_{n}} \left\|(I-\varPi_{n})W^{s}\right\|\!. $$ Now using equation (4.15) yields the result. To derive equation (4.17) use the estimate $$ \|\bar\partial W^{s}\|^{2}_{-m} \lesssim \|\bar\partial_{s} W^{s}\|^{2}_{-m} + \left\|(I-\varPi_{s})\bar\partial W^{s}\right\|^{2}_{-m}= \|\bar\partial_{s} W^{s}\|^{2}_{-m} + \frac{1}{\tau^{2}}\left\|(I-\varPi_{s}) W^{s-1}\right\|^{2}_{-m} $$ and apply equation (4.16) to the last term on the right-hand side. Equation (4.18) follows from the definition of the $$\bar \partial ^{B}$$ operator and equation (4.16): $$ \left\|(I-\varPi_{n})\bar\partial^{B} W^{s}\right\|^{2}_{-m} \lesssim \frac{1}{\tau^{2}}\left( \left\|(I-\varPi_{n})W^{s}\right\|^{2}_{-m} + \left\|(I-\varPi_{n})W^{s-1}\right\|^{2}_{-m} + \left\|(I-\varPi_{n})W^{s-2}\right\|^{2}_{-m} \right)\!. $$ To prove equation (4.19) rewrite $$\bar \partial ^{2} W^{s}$$ as $$ \bar\partial^{2} W^{s} = \frac{2}{\tau}(\bar\partial^{B} W^{s}-\bar\partial W^{s}).$$ Then one obtains by using equations (4.15) and (4.18) \begin{align*} \|\bar\partial^{2} W^{s}\|^{2} &= \left\|{\bar\partial^{2}_{s}} W^{s}\right\|^{2} + \left\|(I-\varPi_{s})\bar\partial^{2} W^{s}\right\|^{2}\\ &\leq \left\|{\bar\partial^{2}_{s}} W^{s}\right\|^{2} + C\frac{1}{\tau^{2}}\left(\left\|(I-\varPi_{s})\bar\partial^{B} W^{s}\right\|^{2} + \frac{1}{\tau^{2}}\left\|(I-\varPi_{s}) W^{s-1}\right\|^{2}\right)\\ &\leq \left\|{\bar\partial^{2}_{s}} W^{s}\right\|^{2} + C\frac{1}{\tau^{4}}\xi_{B}^{s,s}(W). \end{align*} Note that the difference ∥(I − Πs)Ws−1∥2 is already contained in the definition of $$\xi _{B}^{s,s}$$. Below we derive the spatial and reconstruction estimators. We start with bounds of the error between the discrete solution and the reconstruction. Lemma 4.4 Let $$\left (U^{n}, P^{n}\right )$$ be the discrete solution of equation (2.1) and equation (2.10) and $$(\mathcal{R}_{u}^{n},\mathcal{R}_{p}^{n})$$ be the reconstruction defined in equation (3.2). Then we have the following estimates for the interpolants \begin{align}{\hskip-11pt} {\int_{0}^{T}}\left\|(U-\mathcal{R}_{u})(t)\right\|^{2} \lesssim \ \varLambda_{1}, \end{align} (4.21) \begin{align} {\hskip-22pt}\max_{t\in[0,T]}\left\|(\hat{U}-\hat{\mathcal{R}}_{u})(t)\right\|^{2} \lesssim \ \varLambda_{2}, \end{align} (4.22) \begin{align} {\int_{0}^{T}}\left\|\partial_{t}(\hat{U}-\hat{\mathcal{R}}_{u})(t)\right\|^{2}_{J^{\ast}}\lesssim \ \varLambda_{3}+\varXi_{2}, \end{align} (4.23) \begin{align} {\hskip-21pt}{\int_{0}^{T}}\left\|\nabla(U-\mathcal{R}_{u})(t)\right\|^{2} \lesssim \ \varLambda_{4}, \end{align} (4.24) \begin{align} {\hskip-30pt}\max_{t\in[0,T]}\left\|\nabla(\hat{U}-\hat{\mathcal{R}}_{u})(t)\right\|^{2} \lesssim \ \varLambda_{5}, \end{align} (4.25) \begin{align}{\hskip3pt} {\int_{0}^{T}}\left\|\partial_{t}(\hat{U}-\hat{\mathcal{R}}_{u})(t)\right\|^{2} \lesssim \ \varLambda_{6}+\varXi_{6}, \end{align} (4.26) \begin{align} {\hskip-13pt}{\int_{0}^{T}}\left\|(P-\mathcal{R}_{p})(t)\right\|^{2} \lesssim \ \varLambda_{7}, \end{align} (4.27) where the estimators are defined in Definition 3.3. Proof. Due to the definitions of the interpolant in equation (3.1) and the reconstruction in equation (3.2) and equation (4.9), it holds \begin{align*} \left\|(U-\mathcal{R}_{u})(t)\right\| \ &\lesssim \ \left\|U^{n}-\mathcal{R}_{u}^{n}\right\| + \left\|U^{n-1}-\mathcal{R}_{u}^{n-1}\right\|\!, & &t\in I_{n}, \ n\geq 1,\\ \left\|(\hat{U}- \hat{\mathcal{R}}_{u})(t)\right\| \ &\lesssim \ \left\|U^{n}-\mathcal{R}_{u}^{n}\right\| + \left\|U^{n-1}-\mathcal{R}_{u}^{n-1}\right\|+ \left\|U^{n-2}-\mathcal{R}_{u}^{n-2}\right\|\!, & &t\in I_{n}, \ n\geq 2,\\ \left\|(\hat{U}- \hat{\mathcal{R}}_{u})(t)\right\| \ &\lesssim \ \left\|U^{2}-\mathcal{R}_{u}^{2}\right\| + \left\|U^{1}-\mathcal{R}_{u}^{1}\right\|+ \left\|U^{0}-\mathcal{R}_{u}^{0}\right\|\!, & &t\in I_{1}. \end{align*} Hence equations (4.21), (4.22), (4.24), (4.25) and (4.27) follow directly by applying the above inequalities and integrating over time. To prove equation (4.23) and equation (4.26), the definition of the three-point reconstruction yields \begin{align*} {\int_{0}^{T}}\left\|\partial_{t}(\hat{U}-\hat{\mathcal{R}}_{u})(t)\right\|^{2}_{*} \lesssim&\, \sum_{n=1}^{N} \tau \left\|\bar\partial\left(U^{n}-\mathcal{R}_{u}^{n}\right)\right\|^{2}_{*}\\ \lesssim&\, \sum_{n=1}^{N} \tau \left\|\bar\partial_{n}\big(U^{n}-\bar\partial \mathcal{R}_{u}^{n}\big)\right\|^{2}_{*} + \sum_{n=2}^{N}\tau \left\|(\bar\partial-\bar\partial_{n})U^{n}\right\|^{2}_{*} ,\quad *\in\{L^{2},J^{\ast}\}, \end{align*} respectively. This proves equations (4.23) and (4.26), where the last term on the right-hand side was absorbed in the estimators Ξ2 and Ξ6, respectively. In the next lemma, upper bounds for the Stokes reconstruction are given. Lemma 4.5 Let Un be the discrete solution of equation (2.1) and equation (2.10) and $$\mathcal{R}_{u}^{n}$$ the corresponding reconstruction defined in equation (3.2). Further let X2 = X1 = X0 and set $$U^{-1}:= U^{1}-2\tau (\varDelta _{0} U^{0}-B_{0}^{\top } P^{0}+F^{0} )\in X_{0}$$ and $$U^{-2}:=-6\tau (\varDelta _{0} U^{0}-B_{0}^{\top } P^{0}+F^{0} )-3U^{0}+4U^{1}\in X_{0}$$ to simplify the presentation. Then \begin{align} {\int_{0}^{T}}\left\|\nabla(\mathcal{R}_{u}-\hat{\mathcal{R}}_{u})(t)\right\|^{2}\mathrm{d}t \lesssim \ \tau^{4}\mathcal{E}_{4} + \varXi_{4}, \end{align} (4.28) \begin{align} {\int_{0}^{T}}\left\|\varDelta(\mathcal{R}_{u}-\hat{\mathcal{R}}_{u})(t)\right\|^{2}\mathrm{d}t \lesssim \ \tau^{4}\mathcal{E}_{5} + \varXi_{5}, \end{align} (4.29) where the estimators are defined in Definition 3.3. Proof. By definition of the reconstruction in equation (3.2) one has for n ≥ 2 \begin{align*} -\varDelta\bar\partial^{2} \mathcal{R}_{u}^{n}+\nabla\bar\partial^{2} \mathcal{R}_{p}^{n} =&\, \frac{1}{\tau^{2}}\left( \left(-\varDelta_{n} U^{n} +B_{n}^{\top} P^{n}-F^{n}\right) -2\left(-\varDelta_{n-1}U^{n-1}+B_{n-1}^{\top} P^{n-1}-F^{n-1}\right)\right.\\ &\quad\left.+ \left(-\varDelta_{n-2}U^{n-2}+B_{n-2}^{\top} P^{n-2}-F^{n-2}\right)\right) + \bar\partial^{2}f^{n}. \end{align*} By applying equation (2.9), we get \begin{align*} -\varDelta\bar\partial^{2} \mathcal{R}_{u}^{n}+\nabla\bar\partial^{2} \mathcal{R}_{p}^{n} &= \frac{1}{\tau^{2}}\left(-{\bar\partial^{B}_{n}} U^{n}+2\bar\partial^{B}_{n-1}U^{n-1}-\bar\partial^{B}_{n-2}U^{n-2}\right) + \bar\partial^{2} f^{n}\\ &= -{\bar\partial^{2}_{n}}\bar\partial^{B} U^{n} + \bar\partial^{2} f^{n} +\frac{1}{\tau^{2}}\left(2\left(\bar\partial^{B}_{n-1}-{\bar\partial^{B}_{n}}\right)U^{n-1}-\left(\bar\partial^{B}_{n-2}-{\bar\partial^{B}_{n}}\right)U^{n-2}\right)\!. \end{align*} Note that in the case n = 2, 3 it holds $${\bar \partial ^{B}_{1}} U^{1}=\varDelta _{1} U^{1}+F^{1}$$ and $${\bar \partial ^{B}_{0}} U^{0} = \varDelta _{0} U^{0} +F^{0}$$ due to the definition of U−1 and U−2. Now we use the definition of the three-point reconstruction in equation (3.2) and equation (4.9) to get \begin{align*} {\int_{0}^{T}}\left\|\nabla(\mathcal{R}_{u}-\hat{\mathcal{R}}_{u})(t)\right\|^{2}\mathrm{d}t \lesssim&\, \sum_{n=2}^{N}\tau^{5}\left\|\nabla\bar\partial^{2} \mathcal{R}_{u}^{n}\right\|^{2}\\ \lesssim&\, \sum_{n=2}^{N}\tau^{5}\left\|\bar\partial^{2}f^{n}-{\bar\partial^{2}_{n}}\bar\partial^{B} U^{n} \right\|^{2}_{J^{\ast}}\\ &+\sum_{n=2}^{N}\tau\left(\underbrace{\left\|\left(\bar\partial^{B}_{n-1}-{\bar\partial^{B}_{n}}\right)U^{n-1}\right\|_{J^{\ast}}^{2}+\left\|\left(\bar\partial^{B}_{n-2}-{\bar\partial^{B}_{n}}\right)U^{n-2}\right\|^{2}_{J^{\ast}}}_{=(*)}\right)\!. \end{align*} The last term on the right-hand side is handled by using equation (4.18): \begin{align*} (*) &\lesssim \left\|(I\!-\!\varPi_{n-1})\bar\partial^{B}U^{n-1}\right\|^{2}_{J^{\ast}}\!+\!\left\|(I\!-\!\varPi_{n})\bar\partial^{B}U^{n-1}\right\|^{2}_{J^{\ast}} + \left\|(I\!-\!\varPi_{n-2})\bar\partial^{B} U^{n-2}\right\|^{2}_{J^{\ast}} \!+\!\left\|(I\!-\!\varPi_{n})\bar\partial^{B} U^{n-2}\right\|_{J^{\ast}}^{2}\\ &\lesssim \frac{h_{n-1}^{2}}{\tau^{2}}\xi_{B}^{n-1,n-1}(U) + \frac{{h_{n}^{2}}}{\tau^{2}}\xi_{B}^{n-1,n}(U) +\frac{h_{n-2}^{2}}{\tau^{2}}\xi_{B}^{n-2,n-2}(U)+\frac{{h_{n}^{2}}}{\tau^{2}}\xi_{B}^{n-2,n}(U). \end{align*} Thus, we obtain $$ {\int_{0}^{T}}\left\|\nabla(\mathcal{R}_{u}-\hat{\mathcal{R}}_{u})(t)\right\|^{2}\mathrm{d}t \lesssim \mathcal{E}_{4} + \varXi_{4}.$$ In the same way equation (4.29) is shown by using Assumption 2.4. Now we bound the residual $$\varUpsilon $$ of the error equation (4.1) and the difference between the linear interpolant (3.1) and three-point reconstruction (3.2). Lemma 4.6 Let Un be the discrete solution of equation (2.1) and equation (2.10), respectively. Then we have \begin{align} {\hskip-78pt}{\int_{0}^{T}}\left\|(\hat{U}-U)(t)\right\|^{2} \lesssim \ \tau^{4}\mathcal{E}_{1} + \varXi_{1}, \end{align} (4.30) \begin{align} {\int_{0}^{T}}\left\|\varUpsilon(t)\right\|_{J^{\ast}}^{2} \lesssim \ \tau^{4}\mathcal{E}_{2} + \varXi_{2} + \mathcal{F}_{1}(t) + {\textit{I}}_{1}, \end{align} (4.31) \begin{align} {\hskip-86pt}\max_{t\in[0,T]}\left\|(\hat{U}-U)(t)\right\|^{2} \lesssim \ \tau^{4}\mathcal{E}_{3} + \varXi_{3}, \end{align} (4.32) \begin{align} {\hskip-5pt}{\int_{0}^{T}}\left\|\varUpsilon(t)\right\|^{2}\mathrm{d}t \lesssim \ \tau^{4}\mathcal{E}_{6} + \varXi_{6} + \mathcal{F}_{2}(t)+ {\textit{I}}_{2}, \end{align} (4.33) \begin{align} {\hskip-93pt}\max_{t\in[0,T]}\left\|\nabla(\hat{U}-U)(t)\right\|^{2} \lesssim \ \tau^{4}\mathcal{E}_{7} + \varXi_{7}, \end{align} (4.34) where the estimators are defined in Definition 3.3. Proof. Equations (4.30), (4.32) and (4.34) are easily shown by the definition of the reconstruction, cf. equations (3.2) and (4.9), and equation (4.19): $$ \left\|(\hat{U}-U)(t)\right\|^{2}\ \lesssim\ \tau^{4}\|\bar\partial^{2} U^{n}\|^{2}\ \lesssim \ \tau^{4}\left\|{\bar\partial^{2}_{n}} U^{n}\right\|^{2} + \xi_{B}^{n,n}(U).$$ For the proof of equations (4.31) and (4.33) we reformulate $$\varUpsilon{(t)}$$ . In the time interval In, n ≥ 2, we have \begin{align*} \varUpsilon(t) &= f(t_{n})-f(t)+(t-t_{n})\left(\bar\partial^{2}U^{n} - \varDelta\bar\partial \mathcal{R}_{u}^{n}+\nabla\bar\partial \mathcal{R}_{p}^{n}\right)+\left(\bar\partial^{B}-{\bar\partial^{B}_{n}}\right)U^{n}\\ &=f(t_{n})-f(t)+\left(\bar\partial^{B}-{\bar\partial^{B}_{n}}\right)U^{n}+(t-t_{n})\underbrace{\left(\bar\partial^{2} U^{n} +\frac{1}{\tau}\left(-\varDelta \mathcal{R}_{u}^{n}+\nabla \mathcal{R}_{p}^{n} +\varDelta \mathcal{R}_{u}^{n-1}-\nabla \mathcal{R}_{p}^{n-1}\right)\right)}_{=(*)}\!. \end{align*} Let us simplify the last bracket further by using equation (2.9). \begin{align*} (*) &= \bar\partial^{2} U^{n} + \frac{1}{\tau}\left(-\varDelta_{n} U^{n} + B_{n}^{\top} P^{n}-F^{n}+f(t_{n}) - \left(-\varDelta_{n-1} U^{n-1} + B_{n-1}^{\top} P^{n-1}-F^{n-1}+f(t_{n-1})\right) \right)\\ &= \bar\partial f(t_{n}) + \frac{1}{\tau}\left(\bar\partial U^{n} - \bar\partial U^{n-1}-{\bar\partial^{B}_{n}} U^{n} + \bar\partial^{B}_{n-1} U^{n-1}\right)\\ &= \bar\partial f(t_{n}) - \left({\bar\partial^{2}_{n}} U^{n}-\bar\partial^{2}_{n-1} U^{n-1}\right) + \frac{1}{\tau}(\bar\partial-\bar\partial_{n}) U^{n} - \frac{1}{\tau}(\bar\partial-\bar\partial_{n-1})U^{n-1}. \end{align*} Note that we have used the definition of U−1 in Lemma 4.5. Then the triangle inequality yields \begin{align*} \|\varUpsilon(t)\|_{*} \ \lesssim& \ \left\|\,f(t_{n})+(t-t_{n})\bar\partial f(t_{n})-f(t)\right\|_{*} + \left\|(\bar\partial-{\bar\partial^{B}_{n}})U^{n}\right\|_{*} \\ &+ \tau\left\|{\bar\partial^{2}_{n}} U^{n}-\bar\partial^{2}_{n-1}U^{n-1}\right\|_{*} + \left\|(\bar\partial-\bar\partial_{n})U^{n}\right\|_{*}+\left\|(\bar\partial-\bar\partial_{n-1})U^{n-1}\right\|_{*},\quad *\in\{L^{2},J^{\ast}\}. \end{align*} For t ∈ I1, we have cf. equation (4.8), $$ \left\|\varUpsilon(t)\right\|_{*} \ \lesssim \ \left\|\,f(t_{1})+(t-t_{1})\bar\partial f(t_{1})-f(t)\right\|_{*} + \tau\|\varPsi_{1}\|_{*},\quad *\in\{L^{2},J^{\ast}\}.$$ Together we obtain \begin{align*} {\int_{0}^{T}}\left\|\varUpsilon(t)\right\|^{2}_{*} \ \lesssim&\ \tau^{3}\|\varPsi_{1}\|^{2}_{*} + {\int_{0}^{T}}\left\|\,f(t_{n})+(t-t_{n})\bar\partial f(t_{n})-f(t)\right\|_{*}\mathrm{d}t \\ &+ \sum_{n=2}^{N}\tau^{3}\left\|{\bar\partial^{2}_{n}} U^{n}-\bar\partial^{2}_{n-1}U^{n-1}\right\|_{*}^{2} +\sum_{n=2}^{N} \tau \left\|\left(\bar\partial-{\bar\partial^{B}_{n}}\right)U^{n}\right\|_{*}^{2}+\sum_{n=1}^{N}\tau\left\|(\bar\partial-\bar\partial_{n})U^{n}\right\|_{*}^{2}. \end{align*} Now apply equation (4.18) to the fourth term on the right-hand side to prove equations (4.31) and (4.33). Note that the last term on the right-hand side can be absorbed by the fourth term. 4.3 Proof of Theorem 3.4 Proof. (Theorem 3.4) First we prove equation (3.7). To this end we test equation (4.1) with y ∈ J to get $$ \left\langle\partial_{t}(\hat{U}-u)(t),y\right\rangle+\left\langle\nabla(U-u)(t),\nabla y\right\rangle =\left\langle-\varDelta (U-\mathcal{R}_{u})(t),y\right\rangle+\left\langle \varUpsilon(t),y\right\rangle\!,\qquad t\in I_{n},\ n\geq2. $$ Next, set $$y=w(t):=\mathcal{A}^{-1}(\hat{U}-u)(t)$$, i.e. (w(t), q(t)) is solution of $$ -\varDelta w(t)+\nabla q(t)=(\hat{U}-u)(t),\quad \textrm{div}\, w(t)=0,\quad w(t)|_{\varGamma}=0. $$ For the first term on the left-hand side, one gets $$ \langle\partial_{t}(\hat{U}-u)(t),w(t)\rangle = \frac{1}{2}\frac{\mathrm{d}}{\mathrm{d}t}\|\nabla w(t)\|^{2}. $$ The second term is handled by the use of regularity estimate (2.7) for ∇q(t), the identity equation (4.11) and the fact that the interpolant of the Stokes reconstruction $$\mathcal{R}_{u}(t)$$ is divergence free: \begin{align*} \left\langle\nabla(U-u)(t),\nabla w(t)\right\rangle &= \left\langle(U-u)(t),-\varDelta w(t)\right\rangle=\left\langle(U-u)(t),(\hat{U}-u)(t)\right\rangle-\left\langle(U-u)(t),\nabla q(t)\right\rangle\\ &= \frac{1}{2}\left\|(U-u)(t)\right\|^{2}\!+\!\frac{1}{2}\left\|(\hat{U}-u)(t)\right\|^{2}\!-\!\frac{1}{2}\left\|(\hat{U}-U)(t)\right\|^{2}\!-\!\left\langle(U-\mathcal{R}_{u})(t),\nabla q(t)\right\rangle\\ &\geq \frac{1}{2}\left\|(U-u)(t)\right\|^{2}+\frac{1}{2}\left\|(\hat{U}-u)(t)\right\|^{2}-\frac{1}{2}\left\|(\hat{U}-U)(t)\right\|^{2}\\ &\quad-\left\|(U-\mathcal{R}_{u})(t)\right\|\left\|(\hat{U}-u)(t)\right\|\!. \end{align*} The terms on the right-hand side are treated as follows \begin{align*} \left\langle-\varDelta (U-\mathcal{R}_{u})(t)+ \varUpsilon(t),w(t)\right\rangle &= \left\langle (U-\mathcal{R}_{u})(t),-\varDelta w(t)\right\rangle+\left\langle\varUpsilon(t),w(t)\right\rangle\\ &= \left\langle(U-\mathcal{R}_{u})(t),-\nabla q(t)+(\hat{U}-u)(t)\right\rangle+\left\langle\varUpsilon(t),w(t)\right\rangle\\ &\lesssim \left\|(U-\mathcal{R}_{u})(t)\right\|^{2} + \frac{1}{8}\left\|(\hat{U}-u)(t)\right\| + \left\|\varUpsilon(t)\right\|_{J^{\ast}}\left\|\nabla w(t)\right\|\!. \end{align*} Collecting these results, integrating from t = t1 to T, using Gronwall’s Lemma and Young’s inequality one arrives at \begin{align*} &\max_{t\in[t_{1},T]}\left\|\nabla w(t)\right\|^{2}+\int_{t_{1}}^{T}\left(\left\|(U-u)(t)\right\|^{2}+\left\|(\hat{U}-u)(t)\right\|^{2}\right)\mathrm{d}t\\ &\quad\lesssim \left\|\nabla w(t_{1})\right\|^{2}+\int_{t_{1}}^{T}\left(\left\|(\hat{U}-U)(t)\right\|^{2} + \left\|(U-\mathcal{R}_{u})(t)\right\|^{2} + \left\|\varUpsilon(t)\right\|_{J^{\ast}}^{2}\right)\mathrm{d}t. \end{align*} This combined with the initial estimate (4.3) yields $$ {\int_{0}^{T}}\left\|(U-u)(t)\right\|^{2}\mathrm{d}t \ \lesssim \ \eta^{0}(J^{\ast})+{\int_{0}^{T}}\left(\left\|(\hat{U}-U)(t)\right\|^{2} + \left\|(U-\mathcal{R}_{u})(t)\right\|^{2} + \left\|\varUpsilon(t)\right\|_{J^{\ast}}^{2}\right)\mathrm{d}t.$$ Now use equations (4.21), (4.30) and (4.31) to prove equation (3.7). To show equations (3.8) and (3.9), rearrange terms in equation (4.1) to get the following error equation for t ∈ In, n ≥ 2, \begin{align} \partial_{t}(\hat{\mathcal{R}}_{u}-u)(t)-\varDelta(\mathcal{R}_{u}-u)(t)+\nabla(\mathcal{R}_{p}-p)(t) = \partial_{t}(\hat{\mathcal{R}}_{u}-\hat{U})(t)+\varUpsilon(t). \end{align} (4.35) Test equation (4.35) by $$(\hat{\mathcal{R}}_{u}-u)(t)\in J$$, use identity (4.11) and Young’s inequality to get \begin{align*} &\frac{\mathrm{d}}{\mathrm{d}t}\left\|(\hat{\mathcal{R}}_{u}-u)(t)\right\|^{2}+\left\|\nabla(\hat{\mathcal{R}}_{u}-u)(t)\right\|^{2}+\left\|\nabla(\mathcal{R}_{u}-u)(t)\right\|^{2}\\ &\quad\lesssim\left\|\partial_{t}(\hat{\mathcal{R}}_{u}-\hat{U})(t)\right\|^{2}_{J^{\ast}}+\left\|\nabla(\mathcal{R}_{u}-\hat{\mathcal{R}}_{u})(t)\right\|^{2}+\|\varUpsilon\|^{2}_{J^{\ast}}. \end{align*} Integrating from t = t1 to T yields \begin{align} &\max_{t\in[t_{1},T]}\left\|(\hat{\mathcal{R}}_{u}-u)(t)\right\|^{2}+\int_{t_{1}}^{T}\left(\left\|\nabla(\hat{\mathcal{R}}_{u}-u)(t)\right\|^{2}+\left\|\nabla(\mathcal{R}_{u}-u)(t)\right\|^{2}\right)\mathrm{d}t\nonumber \\ &\quad\lesssim \left\|(\hat{\mathcal{R}}_{u}-u)(t_{1})\right\|^{2}+\int_{t_{1}}^{T}\left(\left\|\partial_{t}(\hat{\mathcal{R}}_{u}-\hat{U})(t)\right\|^{2}_{J^{\ast}}+\left\|\nabla(\mathcal{R}_{u}-\hat{\mathcal{R}}_{u})(t)\right\|^{2} +\left\|\varUpsilon(t)\right\|^{2}_{J^{\ast}}\right)\mathrm{d}t. \end{align} (4.36) Now combine this estimate with equation (4.4) and apply the triangle inequality to get \begin{align*} \max_{t\in[0,T]}\left\|(U-u)(t)\right\|^{2} \lesssim& \max_{t\in[0,t_{1}]}\left\|(U-u)(t)\right\|^{2}+ \max_{t\in[t_{1},T]}\left\|(U-\hat{U})(t)\right\|^{2}\\ &+ \max_{t\in[t_{1},T]}\left\|(\hat{\mathcal{R}}_{u}-\hat{U})(t)\right\|^{2}+ \max_{t\in[t_{1},T]}\left\|(\hat{\mathcal{R}}_{u}-u)(t)\right\|^{2}\\ \lesssim& \eta^{0}(L^{2})+ \max_{t\in[0,T]}\left\|(U-\hat{U})(t)\right\|^{2}+ \max_{t\in[0,T]}\left\|(\hat{\mathcal{R}}_{u}-\hat{U})(t)\right\|^{2}\\ &+{\int_{0}^{T}}\left(\left\|\partial_{t}(\hat{\mathcal{R}}_{u}-\hat{U})(t)\right\|^{2}_{J^{\ast}}+\left\|\nabla(\mathcal{R}_{u}-\hat{\mathcal{R}}_{u})(t)\right\|^{2}+\left\|\varUpsilon(t)\right\|^{2}_{J^{\ast}}\right)\mathrm{d}t. \end{align*} With equations (4.22), (4.23), (4.28), (4.31) and (4.32), the proof of equation (3.8) is finished. In a similar way equation (3.9) is bounded. Use the triangle inequality and the initial estimate (4.5) to get \begin{align*} {\int_{0}^{T}}\left\|\nabla(U-u)(t)\right\|^{2} \lesssim&\, \int_{0}^{t_{1}}\left\|\nabla(U-u)(t)\right\|^{2}+\int_{t_{1}}^{T}\left\|\nabla(U-\mathcal{R}_{u})(t)\right\|^{2}+\int_{t_{1}}^{T}\left\|\nabla(\mathcal{R}_{u}-u)(t)\right\|^{2}\\ \lesssim&\, \eta^{0}(L^{2})+ {\int_{0}^{T}}\left\|\nabla(U-\mathcal{R}_{u})(t)\right\|^{2}\\ &+{\int_{0}^{T}}\left(\left\|\partial_{t}(\hat{\mathcal{R}}_{u}-\hat{U})(t)\right\|^{2}_{J^{\ast}}+\left\|\nabla(\mathcal{R}_{u}-\hat{\mathcal{R}}_{u})(t)\right\|^{2}+\left\|\varUpsilon(t)\right\|^{2}_{J^{\ast}}\right)\mathrm{d}t, \end{align*} and then use equations (4.23), (4.24), (4.28) and (4.31). To prove equations (3.10) and (3.11), test equation (4.35) by $$\partial _{t}(\hat{\mathcal{R}}_{u}-u)(t)\in J$$ to obtain for t ∈ In, n ≥ 2, $$ \left\|\partial_{t}(\hat{\mathcal{R}}_{u}-u)(t)\right\|^{2}+\left\langle-\varDelta(\mathcal{R}_{u}-u)(t),\partial_{t}(\hat{\mathcal{R}}_{u}-u)(t)\right\rangle \lesssim \left\|\partial_{t}(\hat{\mathcal{R}}_{u}-\hat{U})(t)\right\|^{2}+\left\|\varUpsilon(t)\right\|^{2}\!, $$ where the inner product is treated as follows: $$ \left\langle-\varDelta(\mathcal{R}_{u}-u)(t),\partial_{t}(\hat{\mathcal{R}}_{u}-u)(t)\right\rangle = \left\langle\nabla(\hat{\mathcal{R}}_{u}-u)(t),\nabla\partial_{t}(\hat{\mathcal{R}}_{u}-u)(t)\right\rangle + \left\langle-\varDelta(\mathcal{R}_{u}-\hat{\mathcal{R}}_{u}),\partial_{t}(\hat{\mathcal{R}}_{u}-u)\right\rangle\!.$$ Then, integrate from t = t1 to T to get \begin{align} &\int_{t_{1}}^{T}\left\|\partial_{t}(\hat{\mathcal{R}}_{u}-u)(t)\right\|^{2}+\max_{t\in[t_{1},T]}\left\|\nabla(\hat{\mathcal{R}}_{u}-u)\right\|^{2}\nonumber \\ &\quad\lesssim \left\|\nabla(\hat{\mathcal{R}}_{u}-u)(t_{1})\right\|^{2} +\int_{t_{1}}^{T}\left(\left\|\partial_{t}(\hat{\mathcal{R}}_{u}-\hat{U})(t)\right\|^{2} + \left\|\varDelta(\mathcal{R}_{u}-\hat{\mathcal{R}}_{u})(t)\right\|^{2}+\left\|\varUpsilon(t)\right\|^{2}\right)\mathrm{d}t. \end{align} (4.37) This estimate combined with equation (4.4) yields by using the triangle inequality \begin{align*} \max_{t\in[0,T]}\left\|\nabla(U-u)(t)\right\|^{2} \lesssim& \max_{t\in[0,t_{1}]}\left\|\nabla(U-u)(t)\right\|^{2}+ \max_{t\in[t_{1},T]}\left\|\nabla(U-\hat{U})(t)\right\|^{2}\\ &+ \max_{t\in[t_{1},T]}\left\|\nabla(\hat{\mathcal{R}}_{u}-\hat{U})(t)\right\|^{2}+ \max_{t\in[t_{1},T]}\left\|\nabla(\hat{\mathcal{R}}_{u}-u)(t)\right\|^{2}\\ \lesssim&\, \eta^{0}\left({H_{0}^{1}}\right)+ \max_{t\in[0,T]}\left\|\nabla(U-\hat{U})(t)\right\|^{2}+ \max_{t\in[0,T]}\left\|\nabla(\hat{\mathcal{R}}_{u}-\hat{U})(t)\right\|^{2}\\ &+{\int_{0}^{T}}\left(\left\|\partial_{t}(\hat{\mathcal{R}}_{u}-\hat{U})(t)\right\|^{2}+\left\|\varDelta(\mathcal{R}_{u}-\hat{\mathcal{R}}_{u})(t)\right\|^{2}+\left\|\varUpsilon(t)\right\|^{2}\right)\mathrm{d}t. \end{align*} Then equation (3.10) is shown by equations (4.25), (4.26), (4.29), (4.33) and (4.34). It remains to prove equation (3.11). The inf–sup condition and equation (4.37) yield for t ∈ In \begin{align*} \left\|(\mathcal{R}_{p}-p)(t)\right\|&\leq \sup_{v\in X,\|\nabla v\|=1}\left\langle \nabla(\mathcal{R}_{p}-p)(t),v\right\rangle\\ &=\sup_{v\in X,\|\nabla v\|=1}\left\langle-\partial_{t}(\hat{\mathcal{R}}_{u}-u)(t)+\varDelta(\mathcal{R}_{u}-u)(t)+\partial_{t}(\hat{\mathcal{R}}_{u}-\hat{U})(t)+\varUpsilon(t),v\right\rangle\\ &\lesssim \left\|\partial_{t}(\hat{\mathcal{R}}_{u}-u)(t)\right\|_{-1}+\left\|\nabla(\mathcal{R}_{u}-u)(t)\right\|+\left\|\partial_{t}(\hat{\mathcal{R}}_{u}-\hat{U})(t)\right\|_{-1}+\left\|\varUpsilon(t)\right\|_{-1}\\ &\lesssim \left\|\partial_{t}(\hat{\mathcal{R}}_{u}-u)(t)\right\|+\left\|\nabla(\mathcal{R}_{u}-u)(t)\right\|+\left\|\partial_{t}(\hat{\mathcal{R}}_{u}-\hat{U})(t)\right\|_{-1}+\left\|\varUpsilon(t)\right\|\!. \end{align*} Upon squaring this equation, integrating from t = t1 to T and using equations (4.36) and (4.37), one gets \begin{align} \int_{t_{1}}^{T}\!\left\|(\mathcal{R}_{p}-p)(t)\right\|^{2} \!\lesssim\! \left\|\nabla(\hat{\mathcal{R}}_{u}-u)(t_{1})\right\|^{2} \!+\!\int_{t_{1}}^{T}\!\left(\left\|\partial_{t}(\hat{\mathcal{R}}_{u}-\hat{U})(t)\right\|^{2} \!+\! \left\|\varDelta(\mathcal{R}_{u}-\hat{\mathcal{R}}_{u})(t)\right\|^{2}\!+\!\left\|\varUpsilon(t)\right\|^{2}\right)\!\mathrm{d}t. \end{align} (4.38) Next, one gets by the triangle inequality, equation (4.7) and equation (4.38) \begin{align*} {\int_{0}^{T}}\left\|(P-p)(t)\right\|^{2} &\lesssim \int_{0}^{t_{1}}\left\|(P-p)(t)\right\|^{2}+\int_{t_{1}}^{T}\left\|(P-\mathcal{R}_{p})(t)\right\|^{2}+\int_{t_{1}}^{T}\left\|(\mathcal{R}_{p}-p)(t)\right\|^{2}\\ &\lesssim \ \eta^{0}\left({H_{0}^{1}}\right)+{\int_{0}^{T}}\left\|(P-\mathcal{R}_{p})(t)\right\|^{2} \\ &\quad+{\int_{0}^{T}}\left(\left\|\partial_{t}(\hat{\mathcal{R}}_{u}-\hat{U})(t)\right\|^{2} + \left\|\varDelta(\mathcal{R}_{u}-\hat{\mathcal{R}}_{u})(t)\right\|^{2}+\left\|\varUpsilon(t)\right\|^{2}\right)\mathrm{d}t. \end{align*} Then using equations (4.26), (4.27), (4.29) and (4.33) to complete the proof. 5. Proof of optimality, Theorem 3.6 In this section we show that the time error estimators $$\mathcal{E}_{1}$$–$$\mathcal{E}_{7}$$ in Theorem 3.4 are bounded independently of the time step τ and the mesh size h, and in turn the a posteriori estimates yield the same asymptotic convergence rates in time as the a priori estimates. Due to the complexity we assume constant meshes. The proof is subdivided into four steps. First we bound the estimators by difference quotients of the initial values in Lemma 5.1. Hence, for proving optimality of the estimators, corresponding difference quotients of the initial values have to be bounded. To the best of our knowledge such a result is available only for the semidiscrete case, i.e. continuous in space, see for instance the study by Akrivis & Chatzipantelidis (2010) and Bänsch & Brenner (2016). Let u be the exact solution and w be the corresponding semidiscrete solution. In the study by Akrivis & Chatzipantelidis (2010), error estimates for semidiscrete parabolic problems using BDF2 for time discretization are considered, and the resulting error estimator is stable if the third-order difference quotient $$\bar \partial ^{3} w^{3}$$ of initial values can be bounded. This is easily done by inserting the exact solution u and estimating $$ \|\bar\partial^{3} w^{3}\|\leq \left\|\bar\partial^{3} \left(w^{3}-u(t_{3})\right)\right\| + \left\|\bar\partial^{3} u(t_{3})\right\|\leq \frac{1}{\tau^{3}}\sum_{i=0}^{3}\left\|w^{i}-u(t_{i})\right\|+\left\|\bar\partial^{3} u(t_{3})\right\|\!.$$ Since the initial errors ∥wi − u(ti)∥ are of order $$\mathcal{O}(\tau ^{3})$$, the third-order difference quotient $$\bar \partial ^{3} w^{3}$$ is bounded independently of τ. However, in our case of the fully discrete solution Un of equation (2.9) the initial errors Ui − u(ti), i = 0, 1, 2, 3, fulfill $$ \left\|U^{i} - u(t_{i})\right\| \leq \ C(\tau^{3} + h^{k+1})$$ and applying the above technique directly would result in a condition on the minimal size of the time step depending on h. Therefore, we introduce as a second step in Lemma 5.2 the corresponding time-dependent semidiscrete problem, and estimate the error between a fully and semidiscrete solution. As a result we are able to bound the initial errors by corresponding time derivatives of this semidiscrete solution. In the third step the discrete regularity results of Appendix B are used in Lemma 5.3. This allows us to bound the semidiscrete values by interpolation estimates, and regularity of the exact solution and given data. Finally, in the fourth step, we combine these results to prove Theorem 3.6. Let us first state two stability estimates. Lemma 5.1 (Stability). Assume Xn = Xh ∀ n = 0, …, N. Then the scheme defined in equation (2.9) fulfills the following stability results for all i ≥ 0: \begin{align} {\hskip-40pt}\max_{2+i\leq n\leq N}\|\bar\partial^{i} U^{n}\|^{2} + \sum_{n=2+i}^{N}\tau\|\nabla\bar\partial^{i} U^{n}\|^{2} \lesssim \sum_{n=2+i}^{N}\tau \|\bar\partial^{i} F^{n}\|^{2}_{J_{h}^{\ast}}+\|\bar\partial^{i} U^{1+i}\|^{2} + \|\bar\partial^{i} U^{i}\|^{2}, \end{align} (5.1) \begin{align} \max_{2+i\leq n\leq N}\|\nabla\bar\partial^{i} U^{n}\|^{2} + \sum_{n=2+i}^{N}\tau\|\bar\partial^{i+1} U^{n}\|^{2} \lesssim \sum_{n=2+i}^{N}\tau \|\bar\partial^{i} F^{n}\|^{2}+\|\nabla \bar\partial^{i} U^{1+i}\|^{2} + \tau \|\bar\partial^{i+1} U^{i+1}\|^{2}. \end{align} (5.2) Proof. Apply the time difference operator $$\bar \partial ^{i}$$ to equation (2.9) to obtain for n ≥ i + 2, i ≥ 0, \begin{align} \bar\partial^{i}\bar\partial^{B} U^{n} - \varDelta_{h}\bar\partial^{i} U^{n} + B_{h}^{\top} \bar\partial^{i} P^{n} = \bar\partial^{i} F^{n}. \end{align} (5.3) To prove equation (5.1), test equation (5.3) with $$4\tau \bar \partial ^{i} U^{n}$$. The first term on the left-hand side results in \begin{align*} 4\tau\langle \bar\partial^{i}\bar\partial^{B} U^{n},\bar\partial^{i} U^{n}\rangle =&\, 2\langle3\bar\partial^{i} U^{n}-4\bar\partial^{i} U^{n-1}+ \bar\partial^{i} U^{n-2},\bar\partial^{i} U^{n}\rangle\\ =&\,\|\bar\partial^{i} U^{n}\|^{2}-\|\bar\partial^{i} U^{n-1}\|^{2} + \tau^{4}\|\bar\partial^{i+2}U^{n}\|^{2} + \|2\bar\partial^{i} U^{n}-\bar\partial^{i} U^{n-1}\|^{2}\\ &-\|2\bar\partial^{i} U^{n-1}-\bar\partial^{i} U^{n-2}\|^{2}, \end{align*} where we have used identity (A.8) in the last step. Use Young’s inequality on the right-hand side to get \begin{align*} &\|\bar\partial^{i} U^{n}\|^{2}-\|\bar\partial^{i} U^{n-1}\|^{2} + \tau^{4}\|\bar\partial^{i+2}U^{n}\|^{2} + \|2\bar\partial^{i} U^{n}-\bar\partial^{i} U^{n-1}\|^{2}-\|2\bar\partial^{i} U^{n-1}-\bar\partial^{i} U^{n-2}\|^{2}\\ &\quad+ 4\tau\|\nabla\bar\partial^{i} U^{n}\|^{2} \leq C\tau\|\bar\partial^{i} F^{n}\|^{2}_{J^{\ast}_{h}} + \tau\|\nabla\bar\partial^{i} U^{n}\|^{2}. \end{align*} Summing up from n = 2 + i to N yields equation (5.1). For proving equation (5.2) we test equation (5.3) with $$4\tau \bar \partial ^{i+1}U^{n}$$. By using identity equation (4.20), we obtain for the left-hand side \begin{align*} 4\tau\langle \bar\partial^{i}\bar\partial^{B} U^{n},\bar\partial^{i+1} U^{n}\rangle =&\, 4\tau\left\langle \frac{\tau}{2}\bar\partial^{i+2}U^{n} + \bar\partial^{i+1}U^{n},\bar\partial^{i+1} U^{n}\right\rangle\\ =&\,\tau\left(\|\bar\partial^{i+1}U^{n}\|^{2}-\|\bar\partial^{i+1}U^{n-1}\|^{2}+\tau^{2}\|\bar\partial^{i+2} U^{n}\|^{2} + 4\|\bar\partial^{i+1} U^{n}\|^{2}\right)\!. \end{align*} Use again equation (4.11) for the second term on the left-hand side and Young’s inequality for the right-hand side of equation (5.3) to obtain the error inequality $$ \tau\left(\|\bar\partial^{i+1}U^{n}\|^{2}-\|\bar\partial^{i+1}U^{n-1}\|^{2}\right)+\tau\|\bar\partial^{i+1} U^{n}\|^{2} + \|\nabla\bar\partial^{i}U^{n}\|^{2}-\|\nabla\bar\partial^{i} U^{n-1}\|^{2} \lesssim \tau\|\bar\partial^{i} F^{n}\|^{2}. $$ Summing up from n = 2 + i to N yields equation (5.2). In the next step we present a priori estimates of the corresponding semidiscrete solution of equations (2.9) and (2.10). Lemma 5.2 Let (W(t), Q(t)) ∈ Xh × Yh be solution of the semidiscrete scheme \begin{align} \partial_{t} W(t)-\varDelta_{h} W(t) + B_{h}^{\top} Q(t) = \varPi_{h}\, f(t),\qquad B_{h}W=0,\qquad W(0)= U^{0}. \end{align} (5.4) Then for i = 1, 2, 3 one can estimate $$ \left\|U^{i}-W(t_{i})\right\|+\tau^{1/2}\left\|\nabla\left(U^{i}-W(t_{i})\right)\right\| \! \lesssim\! \tau^{5/2}\!\left(\left\|\partial_{t}^{(3)}W(t)\right\|_{L^{2}((0,3\tau);L^{2}(\varOmega)^{d})} +\!\left\|\partial_{t}^{(2)}f(t)\right\|_{L^{2}((0,\tau);L^{2}(\varOmega)^{d})}\right)\!.$$ Proof. Set Θi := Ui − W(ti) and $$\varPi _{J_{h}}:X\to J_{h}$$ the L2 projection on Jh. The error equation in $$J_{h}^{\ast }$$ corresponding to equation (2.10) reads $$ \varTheta^{1}-\frac{\tau}{2}\varDelta_{h} \varTheta^{1} = \tau R^{1},$$ where $$R^{1}:=(\partial _{t}-\bar \partial )W(t_{1})-\varDelta _{h}(W(t_{1})-W(t_{1/2}))+\varPi _{J_{h}}(f(t_{1})-f(t_{1/2}))$$ fulfilling $$ \|R^{1}\| \ \lesssim \ \tau^{3/2} \left(\left\|\partial_{t}^{(3)}W(t)\right\|_{L^{2}((0,\tau);L^{2}(\varOmega)^{d})} +\left\|\partial_{t}^{(2)}f(t)\right\|_{L^{2}((0,\tau);L^{2}(\varOmega)^{d})}\right)\!. $$ Test this equation by Θ1 to get $$ \|\varTheta^{1}\|^{2} + \tau\left\|\nabla{e_{1}^{1}}\right\|^{2} \lesssim \tau^{2} \|R^{1}\|^{2}.$$ To bound the error Θ2, set $$R^{2}:=(\partial _{t}-\bar \partial ^{B})W(t_{2})$$ fulfilling $$\|R^{2}\| \ \lesssim \ \tau ^{3/2} \|\partial _{t}^{(3)}W(t)\|_{L^{2}((\tau ,2\tau );L^{2}(\varOmega )^{d})}$$. This gives $$ 3\varTheta^{2}-2\tau\varDelta_{h} \varTheta^{2} = 4\varTheta^{1} - 2\tau R^{2}.$$ The same procedure as for Θ1 then yields the estimate for Θ2 and also for Θ3. The above lemma bounds the initial errors Ui − W(ti) in terms of the third-order time derivative of the semidiscrete velocity W. In the next lemma the results from Appendix B are used to further estimate this term. In what follows we have to make use of the inverse estimate \begin{align} \|\nabla V\|\lesssim \frac{1}{h}\|V\|\qquad\forall\ V\in X_{h}. \end{align} (5.5) This is why a condition of quasi-uniform meshes is required. In view of adaptive meshes we would like to drop this condition and use a local variant of equation (5.5). However, the proof of the next lemma needs a global inverse estimate due to the global definition of the operator $$\mathcal{A}_{h}$$ and the Stokes projection $$\mathcal{S}^{u}_{n}$$. Lemma 5.3 Assume that the exact solution u of equation (2.1) is sufficiently regular. Let W be as in Lemma 5.2. Then \begin{align*} \left\|{\partial_{t}^{3}}W\right\|_{L^{2}((0,3\tau);L^{2}(\varOmega)^{d})} \lesssim&\, \left\|{\partial_{t}^{3}}u-\mathcal{S}^{u}_{n}\left({\partial_{t}^{3}}u,{\partial_{t}^{3}}p\right)\right\|_{L^{2}((0,3\tau);L^{2}(\varOmega)^{d})} + \left\|{\partial_{t}^{3}}u\right\|_{L^{2}((0,3\tau);L^{2}(\varOmega)^{d})}\\ &+\frac{1}{h^{3}}\left\|u_{0}-\mathcal{S}^{u}_{n}\left(u_{0},p_{0}\right)\right\| + \frac{1}{h}\left\|\partial_{t}u_{0}-\mathcal{S}^{u}_{n}(\partial_{t} u_{0},\partial_{t} p_{0})\right\|\!. \end{align*} Proof. To simplify the notation we set $$ \mathcal{S}^{u}_{n}(u,p)(t):=\mathcal{S}^{u}_{n}\left(u(t),p(t)\right)\qquad\textrm{and}\qquad{\partial_{t}^{l}}\mathcal{S}^{u}_{n}(u,p)(t) := \mathcal{S}^{u}_{n}\left({\partial_{t}^{l}}u(t),{\partial_{t}^{l}}p(t)\right)\!.$$ The triangle inequality yields $$ \left\|{\partial_{t}^{3}}W\right\|\leq \left\|{\partial_{t}^{3}}(W-\mathcal{S}^{u}_{n}(u,p))\right\|+\left\|{\partial_{t}^{3}}(u-\mathcal{S}^{u}_{n}(u,p))\right\|+\left\|{\partial_{t}^{3}}u\right\|\!.$$ It remains to bound the term $$\|{\partial _{t}^{3}}(W-\mathcal{S}^{u}_{n}(u,p))\|_{L^{2}((0,3\tau );L^{2}(\varOmega )^{d})}$$. To this end we use the results on discrete regularity provided in Appendix B. Subtract the continuous system (2.1) from the semidiscrete system (5.4) and combine the resulting equation with the Stokes projection (2.4) to get $$ \partial_{t}\left(W-\mathcal{S}^{u}_{n}(u,p)\right)(t) + \mathcal{A}_{h}\left(W-\mathcal{S}^{u}_{n}(u,p)\right)(t)= \varPi_{J_{h}}\partial_{t}\left(u(t)-\mathcal{S}^{u}_{n}(u,p)\right)(t),$$ where $$\varPi _{J_{h}}$$ is the L2 projection onto Jh and $$\mathcal{A}_{h}$$ the discrete Stokes operator defined in equation (B.3). For the initial error one has $$ W(0)-\mathcal{S}^{u}_{n}(u_{0},p_{0}) = U^{0} - \mathcal{S}^{u}_{n}(u_{0},p_{0}) = \mathcal{S}^{u}_{n}(u_{0},p_{0})-\mathcal{S}^{u}_{n}(u_{0},p_{0})=0.$$ We use equation (B.7) of Theorem B.2 in the case k = 2 applied to $$ Z(t)=W(t)-\mathcal{S}^{u}_{n}(u,p)(t),\; G=\varPi_{J_{h}}\partial_{t}\left(u-\mathcal{S}^{u}_{n}(u,p)\right)(t)\;\textrm{ and }Z(0)=0.$$ Hence we get \begin{align*} \left\|{\partial_{t}^{3}}\left(W-\mathcal{S}^{u}_{n}(u,p)\right)\right\|_{L^{2}((0,3\tau);L^{2}(\varOmega)^{d})} \lesssim&\, \left\|{\partial_{t}^{3}}\left(u-\mathcal{S}^{u}_{n}( u,p)\right)\right\|_{L^{2}((0,3\tau);L^{2}(\varOmega)^{d}) }\\ &+ \left\|\nabla \varPi_{J_{h}}\left(u_{0}-\mathcal{S}^{u}_{n}(u_{0},p_{0})\right)\right\| + \left\|\nabla \varPi_{J_{h}}\partial_{t}\left(u_{0}-\mathcal{S}^{u}_{n}(u_{0},p_{0})\right)\right\|\\ &+ \left\|\nabla \mathcal{A}_{h} \varPi_{J_{h}}\left(u_{0}-\mathcal{S}^{u}_{n}(u_{0},p_{0})\right)\right\|\!. \end{align*} The second and third term on the right-hand side can be bounded with the help of the inverse estimate (5.5) and the stability of the L2 projection: \begin{align*} \left\|\nabla \varPi_{J_{h}}\left(u_{0}-\mathcal{S}^{u}_{n}(u_{0},p_{0})\right)\right\| &\lesssim \frac{1}{h}\left\| \varPi_{J_{h}}\left(u_{0}-\mathcal{S}^{u}_{n}(u_{0},p_{0})\right)\right\| \leq \frac{1}{h} \left\|\left(u_{0}-\mathcal{S}^{u}_{n}(u_{0},p_{0})\right)\right\|\!,\\ \left\|\nabla \varPi_{J_{h}}\partial_{t}\left(u_{0}-\mathcal{S}^{u}_{n}(u_{0},p_{0})\right)\right\| &\lesssim \frac{1}{h} \left\|\partial_{t}\left(u_{0}-\mathcal{S}^{u}_{n}(u_{0},p_{0})\right)\right\|\!. \end{align*} For the last term on the right-hand side we apply the inverse estimate (5.5) once again and use the definition of the discrete Stokes operator, cf. (B.12): \begin{align*} \left\|\nabla \mathcal{A}_{h} \varPi_{J_{h}}\left(u_{0}-\mathcal{S}^{u}_{n}(u_{0},p_{0})\right)\right\|^{2} &\lesssim \frac{1}{h^{2}}\left\|\mathcal{A}_{h} \varPi_{J_{h}}\left(u_{0}-\mathcal{S}^{u}_{n}(u_{0},p_{0})\right)\right\|^{2} \\ &=\frac{1}{h^{2}}\left\langle \nabla \varPi_{J_{h}}\left(u_{0}-\mathcal{S}^{u}_{n}(u_{0},p_{0})\right),\nabla \mathcal{A}_{h}\varPi_{J_{h}}\left(u_{0}-\mathcal{S}^{u}_{n}(u_{0},p_{0})\right)\right\rangle \\ &\leq \frac{1}{h^{3}}\left\|\left(u_{0}-\mathcal{S}^{u}_{n}(u_{0},p_{0})\right)\right\|\left\|\nabla \mathcal{A}_{h}\varPi_{J_{h}}\left(u_{0}-\mathcal{S}^{u}_{n}(u_{0},p_{0})\right)\right\|\!. \end{align*} Dividing the last equation by $$\|\nabla \mathcal{A}_{h} \varPi _{J_{h}}(u_{0}-\mathcal{S}^{u}_{n}(u_{0},p_{0}))\|$$ finally together with the above estimates concludes the lemma. Now we have all the ingredients to prove Theorem 3.6. Proof. (Theorem 3.6) Use equation (5.1) for i = 2 to bound the estimators $$\mathcal{E}_{1}$$ and $$\mathcal{E}_{3}$$ $$ \mathcal{E}_{1}\lesssim \mathcal{E}_{3} \lesssim \sum_{n=4}^{N}\tau\|\bar\partial^{2} F^{n}\|^{2}_{J_{h}^{\ast}} + \|\bar\partial^{2} U^{2}\|^{2} + \|\bar\partial^{2} U^{3}\|^{2}$$ and use equation (5.2) for i = 2 to bound the estimators $$\mathcal{E}_{2}$$ and $$\mathcal{E}_{6}$$–$$\mathcal{E}_{7}$$: $$ \mathcal{E}_{2} + \mathcal{E}_{7} \leq \mathcal{E}_{6} + \mathcal{E}_{7} \lesssim \sum_{n=4}^{N}\tau\|\bar\partial^{2} F^{n}\|^{2} + \|\nabla\bar\partial^{2} U^{2}\|^{2} + \|\nabla\bar\partial^{2} U^{3}\|^{2} +\tau\|\bar\partial^{3}U^{2}\|+\tau\|\bar\partial^{3}U^{3}\|.$$ The estimators $$\mathcal{E}_{4}$$ and $$\mathcal{E}_{5}$$ can easily be handled by the triangle inequality: $$ \mathcal{E}_{4}\leq \mathcal{E}_{5} \lesssim \sum_{n=2}^{N}\tau\left\|\bar\partial^{2} f(t_{n})\right\|^{2} + \tau \|\bar\partial^{3} U^{1}\|^{2} +\mathcal{E}_{6}.$$ It is clear that the term $$\sum _{n=4}^{N}\tau \|\bar \partial ^{2} F^{n}\|^{2}$$ is bounded by regularity of f. It remains to bound the quantities $$\|\bar \partial ^{2} U^{2}\|^{2}$$, $$\|\bar \partial ^{2} U^{3}\|^{2}$$, $$\|\nabla \bar \partial ^{2} U^{2}\|^{2}$$, $$\|\nabla \bar \partial ^{2} U^{3}\|^{2}$$ and $$\tau \|\bar \partial ^{3} U^{3}\|$$. Note that the terms $$\tau \|\bar \partial ^{3} U^{1}\|$$ and $$\tau \|\bar \partial ^{3} U^{2}\|$$ can be absorbed into the data approximation error estimators $$ {\textit{I}}_{1}$$ and $$ {\textit{I}}_{2}$$ and in lower-order difference quotients. Let W(ti) be the semidiscrete solution of equation (5.4). Then for instance the term $$\tau \|\bar \partial ^{3} U^{3}\|^{2}$$ can be split by the triangle inequality: $$ \sqrt{\tau}\|\bar\partial^{3} U^{3}\| \leq \sqrt{\tau}\left\|\bar\partial^{3} \big(U^{3}-W(t_{3})\big)\right\| + \sqrt{\tau}\left\|\bar\partial^{3} W(t_{3})\right\|\!.$$ The first term on the right-hand side can be estimated with the help of Lemma 5.2: \begin{align*} \sqrt{\tau}\left\|\bar\partial^{3} \left(U^{3}-W(t_{3})\right)\right\| &\lesssim \tau^{-5/2}\left(\left\|U^{1}-W(t_{1})\right\|+\left\|U^{2}-W(t_{2})\right\|+\left\|U^{3}-W(t_{3})\right\|\right)\\ &\lesssim \left\|\partial_{t}^{(3)}W(t)\right\|_{L^{2}((0,3\tau);L^{2}(\varOmega)^{d})}+\left\|\partial_{t}^{(2)}f(t)\right\|_{L^{2}((0,\tau);L^{2}(\varOmega)^{d})}\!. \end{align*} It is easily shown that the second term fulfills $$ \sqrt{\tau }\|\bar \partial ^{3} W(t_{3})\|\lesssim \|\partial _{t}^{(3)}W(t)\|_{L^{2}((0,3\tau );L^{2}(\varOmega )^{d})}$$. Collect these results and use Lemma 5.3 to get \begin{align*} \sqrt{\tau}\|\bar\partial^{3} U^{3}\| \lesssim&\, \left\|\partial_{t}^{(3)}u-\mathcal{S}^{u}_{n}\left(\partial_{t}^{(3)}u,\partial_{t}^{(3)}p\right)\right\|_{L^{2}((0,3\tau);L^{2}(\varOmega)^{d})} + \left\|\partial_{t}^{(3)}u\right\|_{L^{2}((0,3\tau);L^{2}(\varOmega)^{d})}\\ &+\frac{1}{h^{3}}\left\|u_{0}-\mathcal{S}^{u}_{n}(u_{0},p_{0})\right\| + \frac{1}{h}\left\|\partial_{t}u_{0} - \mathcal{S}^{u}_{n}(\partial_{t} u_{0},\partial_{t} p_{0})\right\| +\left\|\partial_{t}^{(2)}f(t)\right\|_{L^{2}((0,\tau);L^{2}(\varOmega)^{d})}\!. \end{align*} Finally use Lemma 2.2 to bound the approximation errors $$\|u(\cdot )-\mathcal{S}^{u}_{n}(u(\cdot ),p(\cdot ))\|$$ to conclude the proof of the error bound of $$\sqrt{\tau }\|\bar \partial ^{3} U^{3}\|$$. Note that at this point we need at least order k = 2 elements for the velocity space Xh to bound the term $$\frac{1}{h^{3}}\|u_{0}-\mathcal{S}^{u}_{n}(u_{0},p_{0})\|$$. The other terms $$\|\bar \partial ^{2} U^{2}\|^{2}$$, $$\|\bar \partial ^{2} U^{3}\|^{2}$$, $$\|\nabla \bar \partial ^{2} U^{2}\|^{2}$$ and $$\|\nabla \bar \partial ^{2} U^{3}\|^{2}$$ can be treated in a similar way. 6. Computational results In this section computational examples are presented to numerically verify our analysis. The numerical simulations are calculated by NAVIER (Bänsch, 1998) and FENICS (Alnæs et al., 2015). 6.1 Asymptotic behavior of the temporal estimators In this first example we show the asymptotic behavior of the time and Stokes reconstruction error estimators as stated in Theorem 3.4. We chose Ω =]0, 1[2 and \begin{align*} u(t, x) &= \left(\pi\sin(t)\sin(2\pi x_{2})\sin^{2}(\pi x_{1}),\: -\pi\sin(t)\sin(2\pi x_{1})\sin^{2}(\pi x_{2})\right)^{\top}\! , \\ p(t, x) &= \sin(t)\cos(\pi x_{1})\sin(\pi x_{2}), \end{align*} with a right-hand side f, such that the above pair (u, p) is a solution to equation (2.1). The time interval was [0, 1]. As we are only interested here in the time estimators, we use a constant finite element $$\mathcal{P}_{2}-\mathcal{P}_{1}$$ Taylor–Hood discretization to compute the numerical solutions and a fine grid consisting of 16641 vertices. Consequently, the error due to space discretization was very small. For nonchanging meshes the estimators of Definition 3.3 read \begin{align*} \mathcal{E}_{1} &:= \sum_{n=2}^{N}\tau\|\bar\partial^{2} U^{n}\|^{2}, & \mathcal{E}_{4} &:= \sum_{n=2}^{N}\tau\left\|\bar\partial^{2}\left(f(t_{n})-\bar\partial^{B} U^{n}\right)\right\|^{2}_{J^{\ast}}, & \mathcal{E}_{7} &:= \max_{2\leq n\leq N}\|\nabla\bar\partial^{2} U^{n}\|^{2},\\ \mathcal{E}_{2} &:= \sum_{n=3}^{N}\tau\|\bar\partial^{3} U^{n}\|^{2}_{J^{\ast}}, & \mathcal{E}_{5} &:= \sum_{n=2}^{N}\tau\left\|\bar\partial^{2}\left(\,f(t_{n})-\bar\partial^{B} U^{n}\right)\right\|^{2}\, & &\\ \mathcal{E}_{3} &:= \max_{2\leq n\leq N}\|\bar\partial^{2} U^{n}\|^{2}, & \mathcal{E}_{6} &:= \sum_{n=3}^{N}\tau\|\bar\partial^{3} U^{n}\|^{2}, & & \end{align*} and we set \begin{align*} {\varTheta^{2}_{1}} &:= \tau^{4}(\mathcal{E}_{1}+\mathcal{E}_{2}), & {\varTheta^{2}_{2}} &:= \tau^{4}(\mathcal{E}_{2}+\mathcal{E}_{3}+\mathcal{E}_{4}), & {\varTheta^{2}_{3}} &:= \tau^{4}(\mathcal{E}_{2}+\mathcal{E}_{4}),\\{\varTheta^{2}_{4}} &:= \tau^{4}(\mathcal{E}_{5}+\mathcal{E}_{6}+\mathcal{E}_{7}), & {\varTheta^{2}_{5}} &:= \tau^{4}(\mathcal{E}_{5}+\mathcal{E}_{6}). & & \end{align*} With this notation Theorem 3.4 reads \begin{align*} \|U-u\|^{2}_{L^{2}(L^{2})} &\lesssim{\varTheta^{2}_{1}}+ \eta^{0}(J^{\ast}) + \varLambda_{1} + \mathcal{F}_{1} + {\textit{I}}_{1},\\ \|U-u\|^{2}_{L^{\infty}(L^{2})} &\lesssim{\varTheta^{2}_{2}} + \eta^{0}(L^{2}) + \varLambda_{2} + \varLambda_{3} + \mathcal{F}_{1}+ {\textit{I}}_{1},\\ \left\|\nabla(U-u)\right\|^{2}_{L^{2}(L^{2})} &\lesssim{\varTheta^{2}_{3}} + \eta^{0}(L^{2}) + \varLambda_{3} + \varLambda_{4} + \mathcal{F}_{1}+ {\textit{I}}_{1},\\ \left\|\nabla(U-u)(t)\right\|^{2}_{L^{\infty}(L^{2})} &\lesssim{\varTheta^{2}_{4}} + \eta^{0}\left({H_{0}^{1}}\right) + \varLambda_{5}+\varLambda_{6} + \mathcal{F}_{2}+ {\textit{I}}_{2},\\ \|P-p\|^{2}_{L^{2}(L^{2})} &\lesssim{\varTheta^{2}_{5}} +\eta^{0}\left({H_{0}^{1}}\right) + \varLambda_{6} + \varLambda_{7} +\mathcal{F}_{2}+ {\textit{I}}_{2}. \end{align*} For simplicity only the Θ- terms are presented, since the other terms are less interesting in the present context. Figure 1 shows the asymptotic orders of the estimators for velocity and pressure. As can be seen, the order of estimators and errors are in perfect agreement as long as the time error is dominant. For smaller time step sizes the curves for the errors eventually start to stagnate, since then the spatial discretization error becomes dominant. Fig. 1. View largeDownload slide BDF2: errors and (scaled) estimators vs. time step size. Triangles have slope 2.0. Fig. 1. View largeDownload slide BDF2: errors and (scaled) estimators vs. time step size. Triangles have slope 2.0. 6.2 Robustness of the error estimators In this second computation we want to test the estimator of Definition 3.3 for robustness in a more complex example whose solution is nonseparable with respect to the time and space variable. We chose the time interval [0, 1], Ω =]0, 1[2 and \begin{align*} u(t, x) &= \Big(x_{2}\sin\!\left(2\pi\ t\left({x_{1}^{2}}+ {x_{2}^{2}}\right)\right),\: -x_{1}\sin\!\left(2\pi\ t\left({x_{1}^{2}} +{x_{2}^{2}}\right)\right)\Big)^{\top}, \end{align*} \begin{align*} p(t, x) &= x_{1}x_{2}\sin\!\left(2\pi\ t\left({x_{1}^{2}}+{x_{2}^{2}}\right)\right)\!,\end{align*} with a right-hand side f, such that (u, p) is a solution to equation (2.1). For the spatial estimators we follow Verführt (2013) and set \begin{align*} {\eta_{u}^{n}}\left(U^{n};{H_{0}^{1}}\right)&:=\sum_{T\in\mathcal{T}_{n}}\left({h_{T}^{2}}\left\|\varDelta_{n} U^{n} + F^{n}- B_{n}^{\top} P^{n}\right\|^{2}_{T} + \|\!\operatorname{div} U^{n}\|^{2}\right) +\sum_{e\in\mathcal{E}_{\mathcal{T}}}h_{e}\left\|\left[n_{e}\cdot(\nabla U^{n}-P^{n}I)\right]\right\|^{2}_{e},\\{\eta_{u}^{n}}(U^{n};L^{2}) &:=\sum_{T\in\mathcal{T}_{n}}\left({h_{T}^{4}}\left\|\varDelta_{n} U^{n} + F^{n}- B_{n}^{\top} P^{n}\right\|^{2}_{T} + {h_{T}^{2}}\|\!\operatorname{div} U^{n}\|^{2}\right) +\sum_{e\in\mathcal{E}_{\mathcal{T}}}{h_{e}^{3}}\left\|\left[n_{e}\!\cdot\!(\nabla U^{n}-P^{n}I)\right]\right\|^{2}_{e}. \end{align*} To shorten the presentation we present only the spatial estimators$$\varLambda_{1} $$, $$\varLambda_{2} $$, $$\varLambda_{4} $$ and $$\varLambda_{5} $$. Table 1 shows the values of the estimators on a constant finite element $$\mathcal{P}_{2}-\mathcal{P}_{1}$$ Taylor–Hood discretization with 16641 vertices (h = 0.011). As claimed in Theorem 3.6 the error estimators $$\mathcal{E}_{1}$$–$$\mathcal{E}_{7}$$ are independent of the time step τ. Also the space estimators do not depend on the time step size. Table 1 Temporal and space estimators for different time step sizes τ on a constant mesh with h = 0.011 τ $$\mathcal{E}_{1}^{1/2}$$ $$\mathcal{E}_{2}^{1/2}$$ $$\mathcal{E}_{3}^{1/2}$$ $$\mathcal{E}_{4}^{1/2}$$ $$\mathcal{E}_{5}^{1/2}$$ $$\mathcal{E}_{6}^{1/2}$$ $$\mathcal{E}_{7}^{1/2}$$ $$\varLambda _{1}^{1/2}$$ $$\varLambda _{2}^{1/2}$$ $$\varLambda _{4}^{1/2}$$ $$\varLambda _{5}^{1/2}$$ 0.2 6.49 0.15 8.88 1.68 263 32 54 0.000251 0.000288 0.0227 0.0261 0.1 7.47 0.17 9.60 2.18 374 35 58 0.000244 0.000305 0.0221 0.0276 0.05 7.28 0.16 9.83 2.01 331 34 59 0.000224 0.000313 0.0203 0.0283 0.025 7.35 0.16 9.88 2.05 340 33 59 0.000223 0.000316 0.0202 0.0286 0.0125 7.44 0.16 9.90 2.10 353 33 59 0.000225 0.000316 0.0204 0.0286 0.00625 7.42 0.16 9.90 2.10 351 33 59 0.000224 0.000317 0.0203 0.0287 τ $$\mathcal{E}_{1}^{1/2}$$ $$\mathcal{E}_{2}^{1/2}$$ $$\mathcal{E}_{3}^{1/2}$$ $$\mathcal{E}_{4}^{1/2}$$ $$\mathcal{E}_{5}^{1/2}$$ $$\mathcal{E}_{6}^{1/2}$$ $$\mathcal{E}_{7}^{1/2}$$ $$\varLambda _{1}^{1/2}$$ $$\varLambda _{2}^{1/2}$$ $$\varLambda _{4}^{1/2}$$ $$\varLambda _{5}^{1/2}$$ 0.2 6.49 0.15 8.88 1.68 263 32 54 0.000251 0.000288 0.0227 0.0261 0.1 7.47 0.17 9.60 2.18 374 35 58 0.000244 0.000305 0.0221 0.0276 0.05 7.28 0.16 9.83 2.01 331 34 59 0.000224 0.000313 0.0203 0.0283 0.025 7.35 0.16 9.88 2.05 340 33 59 0.000223 0.000316 0.0202 0.0286 0.0125 7.44 0.16 9.90 2.10 353 33 59 0.000225 0.000316 0.0204 0.0286 0.00625 7.42 0.16 9.90 2.10 351 33 59 0.000224 0.000317 0.0203 0.0287 Table 1 Temporal and space estimators for different time step sizes τ on a constant mesh with h = 0.011 τ $$\mathcal{E}_{1}^{1/2}$$ $$\mathcal{E}_{2}^{1/2}$$ $$\mathcal{E}_{3}^{1/2}$$ $$\mathcal{E}_{4}^{1/2}$$ $$\mathcal{E}_{5}^{1/2}$$ $$\mathcal{E}_{6}^{1/2}$$ $$\mathcal{E}_{7}^{1/2}$$ $$\varLambda _{1}^{1/2}$$ $$\varLambda _{2}^{1/2}$$ $$\varLambda _{4}^{1/2}$$ $$\varLambda _{5}^{1/2}$$ 0.2 6.49 0.15 8.88 1.68 263 32 54 0.000251 0.000288 0.0227 0.0261 0.1 7.47 0.17 9.60 2.18 374 35 58 0.000244 0.000305 0.0221 0.0276 0.05 7.28 0.16 9.83 2.01 331 34 59 0.000224 0.000313 0.0203 0.0283 0.025 7.35 0.16 9.88 2.05 340 33 59 0.000223 0.000316 0.0202 0.0286 0.0125 7.44 0.16 9.90 2.10 353 33 59 0.000225 0.000316 0.0204 0.0286 0.00625 7.42 0.16 9.90 2.10 351 33 59 0.000224 0.000317 0.0203 0.0287 τ $$\mathcal{E}_{1}^{1/2}$$ $$\mathcal{E}_{2}^{1/2}$$ $$\mathcal{E}_{3}^{1/2}$$ $$\mathcal{E}_{4}^{1/2}$$ $$\mathcal{E}_{5}^{1/2}$$ $$\mathcal{E}_{6}^{1/2}$$ $$\mathcal{E}_{7}^{1/2}$$ $$\varLambda _{1}^{1/2}$$ $$\varLambda _{2}^{1/2}$$ $$\varLambda _{4}^{1/2}$$ $$\varLambda _{5}^{1/2}$$ 0.2 6.49 0.15 8.88 1.68 263 32 54 0.000251 0.000288 0.0227 0.0261 0.1 7.47 0.17 9.60 2.18 374 35 58 0.000244 0.000305 0.0221 0.0276 0.05 7.28 0.16 9.83 2.01 331 34 59 0.000224 0.000313 0.0203 0.0283 0.025 7.35 0.16 9.88 2.05 340 33 59 0.000223 0.000316 0.0202 0.0286 0.0125 7.44 0.16 9.90 2.10 353 33 59 0.000225 0.000316 0.0204 0.0286 0.00625 7.42 0.16 9.90 2.10 351 33 59 0.000224 0.000317 0.0203 0.0287 Table 2 Temporal and space estimators for different mesh sizes h and fixed τ = 0.02 h $$\mathcal{E}_{1}^{1/2}$$ $$\mathcal{E}_{2}^{1/2}$$ $$\mathcal{E}_{3}^{1/2}$$ $$\mathcal{E}_{4}^{1/2}$$ $$\mathcal{E}_{5}^{1/2}$$ $$\mathcal{E}_{6}^{1/2}$$ $$\mathcal{E}_{7}^{1/2}$$ $$\varLambda _{1}^{1/2}$$ $$\varLambda _{2}^{1/2}$$ $$\varLambda _{4}^{1/2}$$ $$\varLambda _{5}^{1/2}$$ 0.3535 7.39 0.16 9.90 2.03 337 34 59 0.85383 1.90111 2.415 5.377 0.1767 7.37 0.16 9.89 2.06 342 33 59 0.12283 0.26697 0.694 1.510 0.0883 7.37 0.16 9.89 2.06 342 33 59 0.01962 0.03554 0.222 0.402 0.0441 7.37 0.16 9.89 2.06 342 33 59 0.00393 0.00541 0.089 0.122 0.0220 7.37 0.16 9.89 2.06 342 33 59 0.00091 0.00126 0.041 0.057 0.0110 7.37 0.16 9.89 2.06 342 33 59 0.00022 0.00031 0.020 0.028 h $$\mathcal{E}_{1}^{1/2}$$ $$\mathcal{E}_{2}^{1/2}$$ $$\mathcal{E}_{3}^{1/2}$$ $$\mathcal{E}_{4}^{1/2}$$ $$\mathcal{E}_{5}^{1/2}$$ $$\mathcal{E}_{6}^{1/2}$$ $$\mathcal{E}_{7}^{1/2}$$ $$\varLambda _{1}^{1/2}$$ $$\varLambda _{2}^{1/2}$$ $$\varLambda _{4}^{1/2}$$ $$\varLambda _{5}^{1/2}$$ 0.3535 7.39 0.16 9.90 2.03 337 34 59 0.85383 1.90111 2.415 5.377 0.1767 7.37 0.16 9.89 2.06 342 33 59 0.12283 0.26697 0.694 1.510 0.0883 7.37 0.16 9.89 2.06 342 33 59 0.01962 0.03554 0.222 0.402 0.0441 7.37 0.16 9.89 2.06 342 33 59 0.00393 0.00541 0.089 0.122 0.0220 7.37 0.16 9.89 2.06 342 33 59 0.00091 0.00126 0.041 0.057 0.0110 7.37 0.16 9.89 2.06 342 33 59 0.00022 0.00031 0.020 0.028 Table 2 Temporal and space estimators for different mesh sizes h and fixed τ = 0.02 h $$\mathcal{E}_{1}^{1/2}$$ $$\mathcal{E}_{2}^{1/2}$$ $$\mathcal{E}_{3}^{1/2}$$ $$\mathcal{E}_{4}^{1/2}$$ $$\mathcal{E}_{5}^{1/2}$$ $$\mathcal{E}_{6}^{1/2}$$ $$\mathcal{E}_{7}^{1/2}$$ $$\varLambda _{1}^{1/2}$$ $$\varLambda _{2}^{1/2}$$ $$\varLambda _{4}^{1/2}$$ $$\varLambda _{5}^{1/2}$$ 0.3535 7.39 0.16 9.90 2.03 337 34 59 0.85383 1.90111 2.415 5.377 0.1767 7.37 0.16 9.89 2.06 342 33 59 0.12283 0.26697 0.694 1.510 0.0883 7.37 0.16 9.89 2.06 342 33 59 0.01962 0.03554 0.222 0.402 0.0441 7.37 0.16 9.89 2.06 342 33 59 0.00393 0.00541 0.089 0.122 0.0220 7.37 0.16 9.89 2.06 342 33 59 0.00091 0.00126 0.041 0.057 0.0110 7.37 0.16 9.89 2.06 342 33 59 0.00022 0.00031 0.020 0.028 h $$\mathcal{E}_{1}^{1/2}$$ $$\mathcal{E}_{2}^{1/2}$$ $$\mathcal{E}_{3}^{1/2}$$ $$\mathcal{E}_{4}^{1/2}$$ $$\mathcal{E}_{5}^{1/2}$$ $$\mathcal{E}_{6}^{1/2}$$ $$\mathcal{E}_{7}^{1/2}$$ $$\varLambda _{1}^{1/2}$$ $$\varLambda _{2}^{1/2}$$ $$\varLambda _{4}^{1/2}$$ $$\varLambda _{5}^{1/2}$$ 0.3535 7.39 0.16 9.90 2.03 337 34 59 0.85383 1.90111 2.415 5.377 0.1767 7.37 0.16 9.89 2.06 342 33 59 0.12283 0.26697 0.694 1.510 0.0883 7.37 0.16 9.89 2.06 342 33 59 0.01962 0.03554 0.222 0.402 0.0441 7.37 0.16 9.89 2.06 342 33 59 0.00393 0.00541 0.089 0.122 0.0220 7.37 0.16 9.89 2.06 342 33 59 0.00091 0.00126 0.041 0.057 0.0110 7.37 0.16 9.89 2.06 342 33 59 0.00022 0.00031 0.020 0.028 Table 2 presents the values of the estimators for refining meshes with halving mesh size h and constant time step τ = 0.02. The time estimators $$\mathcal{E}_{1}$$–$$\mathcal{E}_{7}$$ are independent of the mesh size h. This shows the robustness of the temporal estimators with respect to the spatial discretization. The space estimators behave as expected. We have quadratic convergence for the L2 estimators $$\varLambda _{1}^{1/2}$$ and $$\varLambda _{2}^{1/2}$$ and linear convergence for the H1 estimators $$\varLambda _{4}^{1/2}$$ and $$\varLambda _{5}^{1/2}$$. In our next computation we want to show the robustness of our time error estimator with respect to the end time T. To this end we consider a solution of equation (2.1), which is periodic in time: $$ u(t, x) = \Big(\sin(t)\sin(2\pi(x_{1}+ x_{2})),\: -\sin(t)\sin(2\pi(x_{1} +x_{2}))\Big)^{\top},\qquad p(t, x) = 0, $$ with a corresponding right-hand side f in the time interval [0, 4π] on Ω =]0, 1[2. We set τ = 0.05 and use the same spatial discretization as in the example before. In Fig. 2 we compare the local contributions $$\|\bar \partial ^{2} U^{n}\|$$ and $$\|\bar \partial ^{3} U^{n}\|$$ of the time estimators with the time derivatives $$\|{\partial _{t}^{2}} u(t_{n})\|$$ and $$\|{\partial _{t}^{3}} u(t_{n})\|$$ of the exact solution in view of a dependence on the final time T. As can be seen the difference quotients of the discrete solution are in perfect agreement with the continuous time derivatives, and we conclude the robustness of our estimators with respect to the end time T. Fig. 2. View largeDownload slide Time versus local contributions $$\|\bar \partial ^{2} U^{n}\|$$ (left), $$\|\bar \partial ^{3} U^{n}\|$$ (right) of the time estimator and $$\|{\partial _{t}^{2}} u(t_{n})\|$$ (left), $$\|{\partial _{t}^{3}} u(t_{n})\|$$ (right) of the exact solution for end time T = 4π. Fig. 2. View largeDownload slide Time versus local contributions $$\|\bar \partial ^{2} U^{n}\|$$ (left), $$\|\bar \partial ^{3} U^{n}\|$$ (right) of the time estimator and $$\|{\partial _{t}^{2}} u(t_{n})\|$$ (left), $$\|{\partial _{t}^{3}} u(t_{n})\|$$ (right) of the exact solution for end time T = 4π. Fig. 3. View largeDownload slide Example with mesh change. L2 pressure error versus time (initial mesh with 1024 elements, τ=1.25e-3). Fig. 3. View largeDownload slide Example with mesh change. L2 pressure error versus time (initial mesh with 1024 elements, τ=1.25e-3). 6.3 Mesh changes With this example we study the effect of mesh modification. The domain here is Ω = ] − 0.5, 0.5[2, T = 1.5 and the exact solution is \begin{align} \begin{aligned} u(x,y,t)& = \pi\sin(t+0.25)(\sin(2\pi y)\sin(\pi x)^{2},\sin(2\pi x)\sin(\pi y)^{2})^{T},\\ p(x,y,t)&= \sin(t+0.25)\cos(\pi x)\sin(\pi y). \end{aligned} \end{align} (6.1) In order to focus on the mesh change, the setting is as follows. The computations are started with different uniform meshes with mesh size h0 (corresponding to 256, 1024, 4096 and 16384 elements, respectively) and uniform time step sizes τ. The initial mesh is created by uniformly refining a given macro triangulation with the bisection method (Bänsch, 1991b). At the time instant tjump = 1.0 the mesh is derefined, resulting in a coarsened mesh with mesh size 2 h0. This scenario is as in the study by Besier & Wollner (2012). Since the pressure is the unknown most sensitive to mesh changes, we study the effect of mesh modification for this variable. In the study by Besier & Wollner (2012), it was shown that a derefinement of the mesh results in a jump of the pressure error of the order 1/τ. This behavior is confirmed by our experiments. In Fig. 3 the L2 error of the pressure is shown as a function of time. The huge jump in the error at tjump is clearly visible. More quantitative information is given in Tables 3 and 4. In Table 3 the jumps δEp in the pressure error are listed for various τ and h0. Here, δEp is defined as the ‘additional’ contribution to the pressure error by the mesh change: \begin{align} \delta E_{p} := \| P^{n_{0}}-p(t_{\textrm{jump}}) \| - \| P^{n_{0}-1}-p(t_{\textrm{jump}}-\tau) \|, \end{align} (6.2) Table 3 Example with mesh change: ‘additional’ pressure error δEp equation (6.2) for various meshes (nt = no. of initial elements) and time step sizes τ τ 2.e−2 1.e−2 5.e−3 2.5e−3 1.25e−3 nt= 256 8.3534e−02 1.5880e−01 3.0628e−01 5.9729e−01 1.1755e+00 nt=1024 5.6903e−03 1.0481e−02 2.0098e−02 3.9038e−02 7.6344e−02 nt=4096 5.9363e−04 8.1377e−04 1.3692e−03 2.5847e−03 5.0541e−03 nt=16384 1.2083e−04 1.2638e−04 1.4346e−04 1.9564e−04 3.3151e−04 τ 2.e−2 1.e−2 5.e−3 2.5e−3 1.25e−3 nt= 256 8.3534e−02 1.5880e−01 3.0628e−01 5.9729e−01 1.1755e+00 nt=1024 5.6903e−03 1.0481e−02 2.0098e−02 3.9038e−02 7.6344e−02 nt=4096 5.9363e−04 8.1377e−04 1.3692e−03 2.5847e−03 5.0541e−03 nt=16384 1.2083e−04 1.2638e−04 1.4346e−04 1.9564e−04 3.3151e−04 View Large Table 3 Example with mesh change: ‘additional’ pressure error δEp equation (6.2) for various meshes (nt = no. of initial elements) and time step sizes τ τ 2.e−2 1.e−2 5.e−3 2.5e−3 1.25e−3 nt= 256 8.3534e−02 1.5880e−01 3.0628e−01 5.9729e−01 1.1755e+00 nt=1024 5.6903e−03 1.0481e−02 2.0098e−02 3.9038e−02 7.6344e−02 nt=4096 5.9363e−04 8.1377e−04 1.3692e−03 2.5847e−03 5.0541e−03 nt=16384 1.2083e−04 1.2638e−04 1.4346e−04 1.9564e−04 3.3151e−04 τ 2.e−2 1.e−2 5.e−3 2.5e−3 1.25e−3 nt= 256 8.3534e−02 1.5880e−01 3.0628e−01 5.9729e−01 1.1755e+00 nt=1024 5.6903e−03 1.0481e−02 2.0098e−02 3.9038e−02 7.6344e−02 nt=4096 5.9363e−04 8.1377e−04 1.3692e−03 2.5847e−03 5.0541e−03 nt=16384 1.2083e−04 1.2638e−04 1.4346e−04 1.9564e−04 3.3151e−04 View Large Table 4 Example with mesh change: estimator $$\sqrt{\tilde{\xi }_{B}^{n}}$$ for various meshes (nt = no. of initial elements) and time step sizes τ τ 2.e−2 1.e−2 5.e−3 2.5e−3 1.25e−3 nt=256 9.2077e−01 1.8415e+00 3.6831e+00 7.3662e+00 1.4732e+01 nt=1024 1.3887e−01 2.7773e−01 5.5546e−01 1.1109e+00 2.2218e+00 nt=4096 1.8840e−02 3.7681e−02 7.5361e−02 1.5072e−01 3.0144e−01 nt=16384 2.4194e−03 4.8389e−03 9.6779e−03 1.9356e−02 3.8712e−02 τ 2.e−2 1.e−2 5.e−3 2.5e−3 1.25e−3 nt=256 9.2077e−01 1.8415e+00 3.6831e+00 7.3662e+00 1.4732e+01 nt=1024 1.3887e−01 2.7773e−01 5.5546e−01 1.1109e+00 2.2218e+00 nt=4096 1.8840e−02 3.7681e−02 7.5361e−02 1.5072e−01 3.0144e−01 nt=16384 2.4194e−03 4.8389e−03 9.6779e−03 1.9356e−02 3.8712e−02 View Large Table 4 Example with mesh change: estimator $$\sqrt{\tilde{\xi }_{B}^{n}}$$ for various meshes (nt = no. of initial elements) and time step sizes τ τ 2.e−2 1.e−2 5.e−3 2.5e−3 1.25e−3 nt=256 9.2077e−01 1.8415e+00 3.6831e+00 7.3662e+00 1.4732e+01 nt=1024 1.3887e−01 2.7773e−01 5.5546e−01 1.1109e+00 2.2218e+00 nt=4096 1.8840e−02 3.7681e−02 7.5361e−02 1.5072e−01 3.0144e−01 nt=16384 2.4194e−03 4.8389e−03 9.6779e−03 1.9356e−02 3.8712e−02 τ 2.e−2 1.e−2 5.e−3 2.5e−3 1.25e−3 nt=256 9.2077e−01 1.8415e+00 3.6831e+00 7.3662e+00 1.4732e+01 nt=1024 1.3887e−01 2.7773e−01 5.5546e−01 1.1109e+00 2.2218e+00 nt=4096 1.8840e−02 3.7681e−02 7.5361e−02 1.5072e−01 3.0144e−01 nt=16384 2.4194e−03 4.8389e−03 9.6779e−03 1.9356e−02 3.8712e−02 View Large where n0 corresponds to the time instant tjump = n0τ. Notice that our theory does not provide error control for δEp. This term is used in our experiments only as a tool to quantify the observed jump in the pressure. The comparison then to our coarsening estimator is indicative only in order to access whether or not our estimator can detect the jump in pressure. Since we perform only one mesh change, a simplified mesh change estimator $$\tilde{\xi }_{B}^{n}$$ is used: $$ \tilde{\xi}_{B}^{n} := \| \tau^{-1}(\varPi_{n} - I)U^{n-1}\|^{2}. $$ In Table 4, the coarsening estimator terms $$\sqrt{\tilde{\xi }_{B}^{n}}$$ are reported. Note that because of orthogonality the relation $$ \tilde{\xi}_{B}^{n} =\| \tau^{-1}U^{n-1}\|^{2} - \| \tau^{-1}\varPi^{n}U^{n-1}\|^{2}$$ holds, simplifying the computation of this term. Notice that in this example $$\tilde{\xi }_{B}^{n}=0$$ for n≠n0. The projection ΠnUn−1 can easily be computed without the need of (inexact) interpolation or quadrature, see Bänsch, 1991a for details. From Table 3 one deduces the following behavior of δEp. If the mesh is fine enough, the ‘extra contribution’ of the error is negligibly small. However, from a certain small time step size τ (depending on h0), the extra contribution to the error behaves like 1/τ (reading the rows of Table 3). In contrast, fixing a time step size and reading the columns, δEp is decreasing. A similar behavior can be observed for the coarsening estimator $$\tilde{\xi }_{B}^{n}$$. However, the 1/τ increase is valid for all h0. For fixed time step size, $$\sqrt{\tilde{\xi }_{B}^{n}}$$ decreases approximately like $${h_{0}^{3}}$$, which is the order one would expect for $$\mathcal{P}_{2}-\mathcal{P}_{1}$$ Taylor–Hood element. We conclude that, in this experiment, our estimator has the right qualitative behavior and captures the significant increase of the error introduced by the mesh change. 7. Variable time steps In this section we present a generalization of the analysis of this paper to the case of variable time steps. For the ease of presentation we restrict ourselves to a constant mesh and present only the estimate in the L2–L2 norm. The other a posteriori estimates can be obtained in an analogous manner. In contrast, it is an open problem whether the optimality proof in Section 5 carries over to the case of variable time step sizes. At least we expect an assumption on the ratio of successive time steps, see Emmrich (2005) and the literature cited there. 7.1 A posteriori estimates For a partition 0 = t0 < t1 < ⋯ < tN = T define the time step τn := tn − tn−1 and the time step ratio $$\omega _{n}:=\frac{\tau _{n}}{\tau _{n-1}}$$. The finite difference operator $$\bar \partial $$ for variable time steps is defined recursively $$ \bar\partial U^{n}:=\bar\partial^{1} U^{n} := \frac{U^{n}-U^{n-1}}{\tau_{n}},\qquad \bar\partial^{k} U^{n}:= \frac{\bar\partial^{k-1}U^{n}-\bar\partial^{k-1}U^{n-1}}{\tau_{n}},\;k=2,\ldots,n.$$ The BDF2 operator is then given by $$ \bar\partial^{B} U^{n}:=\frac{1}{\tau_{n}}\left(\frac{1+2\omega_{n}}{1+\omega_{n}}U^{n}-(1+\omega_{n})U^{n-1}+\frac{{\omega_{n}^{2}}}{1+\omega_{n}}U^{n-2}\right) =\tau_{n}\frac{\omega_{n}}{1+\omega_{n}}\bar\partial^{2}U^{n}+\bar\partial U^{n}. $$ Let Xh, Yh be a mixed finite element approximation of X and Y. The discrete instationary Stokes system, cf. equations (2.9) and (2.10), with variable time steps reads: set U0 := u0, P0 := p0 and calculate U1 from \begin{align} \bar\partial U^{1}-\varDelta_{h} U^{1/2}+ B_{h}^{\top} P^{1/2}=F^{1/2},\qquad B_{h} U^{1} = 0,\qquad U^{1}|_{\varGamma}=0. \end{align} (7.1) Then for n ≥ 2 compute Un from \begin{align} \bar\partial^{B}U^{n} -\varDelta_{h} U^{n}+B_{h}^{\top} P^{n} =F^{n},\qquad B_{h} U^{n}=0. \end{align} (7.2) In the next theorem we present the estimators for variable time steps. Theorem 7.1 (L2–L2 a posteriori error estimates for variable time steps) The discrete solutions (Un, Pn) of equations (7.1) and (7.2) satisfy the estimates $$ \| U-u\|_{L^{2}(L^{2})}^{2}\ \lesssim\ \check{\mathcal{E}}_{1}+\check{\mathcal{E}}_{2} + \varLambda_{1} + {\mathcal{F}}_{1} + {\textit{I}}_{1}, $$ where $$\varLambda_{1} $$, $$\mathcal{F}_{1}$$ and $$ {\textit{I}}_{1} $$ are given in Definition 3.3 and \begin{align} \check{\mathcal{E}}_{1} =\sum_{n=2}^{N} {\tau_{n}^{5}}\|\bar\partial^{2} U^{n}\|^{2}\qquad \textrm{and}\qquad \check{\mathcal{E}}_{2} = \sum_{n=2}^{N} \frac{{\tau_{n}^{3}}}{{\omega_{n}^{2}}}\left\|\frac{\omega_{n}}{\omega_{n}+1}\bar\partial^{2} U^{n} -\frac{\omega_{n-1}}{\omega_{n-1}+1}\bar\partial^{2} U^{n-1}\right\|^{2}_{J^{\ast}}\!. \end{align} (7.3) Proof. For n ≥ 2, t ∈ In, define the three-point reconstruction for variable time steps $$ \hat{U}(t):= U(t)+\frac{\omega_{n}}{\omega_{n}+1}(t-t_{n})(t-t_{n-1})\bar\partial U^{n}.$$ Its time derivative is $$ \partial_{t} \hat{U}(t)=\bar\partial^{B} U^{n} + \frac{2\omega_{n}}{\omega_{n}+1}(t-t_{n})\bar\partial^{2} U^{n}.$$ With this definition, deduce the error equation like for equation (4.1). Now proceed exactly like in the proof of equation (3.7) of Theorem 3.4. Note that in case of constant time steps the above theorem reduces to Theorem 3.4 (in the case of constant meshes). 7.2 Adaptive strategy Based on the above result for non constant time steps we devise a simple adaptive algorithm and present some computational results. For n ≥ 1 define the local error indicator \begin{align} {\vartheta_{n}}^{2} :=\frac{{\tau_{n}^{2}}}{{\omega_{n}^{2}}} \left\|\frac{\omega_{n}}{\omega_{n}+1}\bar\partial^{2} U^{n}-\frac{\omega_{n-1}}{\omega_{n-1}+1}\bar\partial^{2} U^{n-1}\right\|^{2}_{J^{\ast}} +{\tau_{n}^{4}}\|\bar\partial^{2} U^{n}\|^{2}. \end{align} (7.4) Note that for n ≤ 3 the first term is dropped. From Theorem 7.1 we conclude $$ \|U-u\|_{L^{2}(L^{2})}^{2} \lesssim\ \sum_{n=1}^{N} \tau_{n}{\vartheta_{n}}^{2} \; + \varLambda_{1} + \mathcal{F}_{1} + {I}_{1}. $$ The optimal strategy to choose the time step sizes with respect to the L2–norm in time would consist in equilibrating the local contributions τnϑn2. This, however, is not possible within a time-marching scheme, since it would require to a priori know the total number N of time steps. Instead, one contents oneself with equilibrating the local contribution ϑn, which is more suited for the $$L^{\infty }$$–norm (compare also the discussion in the study by Kreuzer et al., 2012). Thus we propose the following simple adaptive time-stepping strategy, which is rather common for ordinary differential equations, see Deuflhard & Weiser (2012): in each time step a tentative size of the next time step is chosen according to the extrapolation formula $$ \tau_{n+1}:= \tau_{n} {\textrm{TOL}/{\vartheta_{n}}}^{\frac{1}{\gamma+1}}, $$ where γ is the order of the method, γ = 2 in our case. If the new time step generates too big an error, this time step is rejected and recomputed with a reduced time step. Otherwise the time step is accepted. With some ‘safety’ parameters ρ1, ρ2, ρ2 the algorithm reads as follows. Algorithm 1 Let ρ1, ρ2, ρ3 > 1 and TOL > 0 be given. Initialize all unknowns. Let τ1 be given. do n ← n + 1 loop forever [Un, Pn] = solve (Un−1, Un−2) compute ϑnfrom Eq. (83) if ϑnρ1* TOL *     time step rejected τn ← τn/ρ2 else *     time step accepted break end if end loop forever τn+1 := min{τn(TOL/ϑn)1/3, ρ3τn} tn+1 := tn + τn+1 whiletn+1 < T Note that solve returns the unknowns for time tn by solving one time step of equation (2.9). The above adaptive method is tested with the same example as in Section 6, except for taking \begin{align*} u(t,x) & = g(t)\pi\,\left[\sin(2\pi x_{2})\sin(\pi x_{1})^{2},\;-\sin(2\pi x_{1})\sin(\pi x_{2})^{2}\right]^{\top}\!,\\ p(t,x) & = g(t)\cos(\pi x_{1})\sin(\pi x_{2}) \end{align*} with $$g(t) = \sin (t)\left (1-\exp \left (-5.e2(t-0.25)^{2}\right )\right )$$ exhibiting a ‘bump’ around t = 0.25, see Fig. 4. This behavior of g is well suited to study the performance of a time adaptive strategy (compare also Kreuzer et al., 2012). Fig. 4. View largeDownload slide Graph of function g. Fig. 4. View largeDownload slide Graph of function g. Fig. 5. View largeDownload slide Instantaneous L2(Ω) error versus time with uniform time step and adaptive strategy (left) and local error estimator θn versus time (right). Fig. 5. View largeDownload slide Instantaneous L2(Ω) error versus time with uniform time step and adaptive strategy (left) and local error estimator θn versus time (right). Fig. 6. View largeDownload slide Time step sizes τn versus time. Fig. 6. View largeDownload slide Time step sizes τn versus time. Figure 5 (left) shows the result for a computation with uniform time step size and with adaptive strategy. For a fair comparison the uniform time step is chosen such that the number of time steps is the same as for the adaptive run. As can be seen, the local error of the computation with uniform time step is rather big around t = 0.25, otherwise they are quite small, thus reflecting the character of the function g. The run with the adaptive strategy is done with TOL = 5.e − 3, ρ1 := 1.2, ρ2 = 1.43 and ρ3 = 1.8. As one can see the error is fairly well controlled. Figure 5 (right) shows the error estimator θn, equilibrated around the value 5.e − 3, and Fig. 6 reports the time step size as a function of time for the adaptive strategy. We conclude that in principle the derived error estimator is capable of effectively monitoring the time step size. References Ainsworth , M. & Oden , J. T. ( 2011 ) A Posteriori Error Estimation in Finite element Analysis , vol. 37. New York : John Wiley . Akrivis , G. & Chatzipantelidis , P. ( 2010 ) A posteriori error estimates for the two-step backward differentiation formula method for parabolic equations . SIAM J. Numer. Anal. , 48 , 109 – 132 . Google Scholar CrossRef Search ADS Akrivis , G. , Makridakis , Ch. & Nochetto , R. H. ( 2006 ) A posteriori error estimates for the Crank–Nicolson method for parabolic equations . Math. Comp. , 75 , 511 – 531 . Google Scholar CrossRef Search ADS Akrivis , G. , Makridakis , Ch. & Nochetto , R. H. ( 2009 ) Optimal order a posteriori error estimates for a class of Runge–Kutta and Galerkin methods . Numer. Math. , 114 , 133 – 160 . Google Scholar CrossRef Search ADS Alnæs , M. , Blechta , J. , Hake , J. , Johansson , A. , Kehlet , B. , Logg , A. , Richardson , C. , Ring , J. , Rognes , M. & Wells , G. ( 2015 ) The FEniCS project version 1.5 . Archive of Num. Software , 3 , 9 – 23 . Bank , R. E. & Welfert , B. D. ( 1990 ) A posteriori error estimates for the Stokes equations: a comparison . Comput. Methods Appl. Mech. Eng. , 82 , 323 – 340 . Google Scholar CrossRef Search ADS Bänsch , E. ( 1991a ) An adaptive finite-element strategy for the three-dimensional time-dependent Navier–Stokes equations . J. Comput. Appl. Math. , 36 , 3 – 28 . Google Scholar CrossRef Search ADS Bänsch , E. ( 1991b ) Local mesh refinement in 2 and 3 dimensions . IMPACT Comput. Sci. Engrg. , 3 , 181 – 191 . Google Scholar CrossRef Search ADS Bänsch , E. ( 1998 ) Simulation of instationary, incompressible flows . Acta Math. Univ. Comenianae , 67 , 101 – 114 . Bänsch , E. & Brenner , A. ( 2016 ) A posteriori error estimates for pressure-correction schemes . SIAM J. Numer. Anal. , 54 , 2323 – 2358 . Google Scholar CrossRef Search ADS Bänsch , E. , Karakatsani , F. & Makridakis , Ch. ( 2012 ) A posteriori error control for fully discrete Crank–Nicolson schemes . SIAM J. Numer. Anal. , 50 , 2845 – 2872 . Google Scholar CrossRef Search ADS Besier , M. & Wollner , W. ( 2012 ) On the pressure approximation in nonstationary incompressible flow simulations on dynamically varying spatial meshes . Int. J. Numer. Methods Fluids , 69 , 1045 – 1064 . Google Scholar CrossRef Search ADS Brenner , A. , Bänsch , E. & Bause , M. ( 2014 ) A priori error analysis for finite element approximations of the stokes problem on dynamic meshes . IMA J. Numer. Anal. , 34 , 123 – 146 . Google Scholar CrossRef Search ADS Crouzeix , M. & Raviart , P.-A. ( 1973 ) Conforming and nonconforming finite element methods for solving the stationary stokes equations i . Esaim. Math. Model. and Numer. Anal. , 7 , 33 – 75 . Demlow , A. , Lakkis , O. & Makridakis , Ch. ( 2009 ) A posteriori error estimates in the maximum norm for parabolic problems . SIAM J. Numer. Anal. , 47 , 2157 – 2176 . Google Scholar CrossRef Search ADS Deuflhard , P. & Weiser , M. ( 2012 ) Adaptive numerical solution of PDEs . Berlin : de Gruyter . Google Scholar CrossRef Search ADS Dupont , T. ( 1982 ) Mesh modification for evolution equations . Math. Comp. , 39 , 85 – 107 . Google Scholar CrossRef Search ADS Emmrich , E. ( 2005 ) Stability and error of the variable two-step BDF for semilinear parabolic problems . JAMS , 19 , 33 – 55 . Ern , A. , Smears , I. & Vohralík , M. ( 2016 ) Guaranteed, locally space-time efficient, and polynomial-degree robust a posteriori error estimates for high-order discretizations of parabolic problems . Technical report . arXiv:1610.01804 . Evans , L. C. ( 2010 ) Partial Differential Equations , 2nd edn. Graduate Studies in Mathematics , vol. 33 . Providence, RI : American Mathematical Society . Georgoulis , E. , Lakkis , O. & Virtanen , J. ( 2011 ) A posteriori error control for discontinuous Galerkin methods for parabolic problems . SIAM J. Numer. Anal. , 49 , 427 – 458 . Google Scholar CrossRef Search ADS Heywood , J. G. ( 1980 ) The Navier–Stokes equations: on the existence, regularity and decay of solutions . Indiana Univ. Math. J. , 29 , 639 – 681 . Google Scholar CrossRef Search ADS Heywood , J. G. & Rannacher , R. ( 1982 ) Finite-element approximation of the nonstationary Navier–Stokes problem. I. Regularity of solutions and second-order error estimates for spatial discretization. SIAM J. Numer. Anal. , 19 , 275 – 311 . Heywood , J. G. & Rannacher , R. ( 1988 ) Finite-element approximation of the nonstationary Navier–Stokes problem. III. Smoothing property and higher order error estimates for spatial discretization . SIAM J. Numer. Anal. , 25 , 489 – 512 . Google Scholar CrossRef Search ADS Karakatsani , F. & Makridakis , Ch. ( 2007 ) A posteriori estimates for approximations of time-dependent Stokes equations . IMA J. Numer. Anal. , 27 , 741 – 764 . Google Scholar CrossRef Search ADS Kreuzer , C. , Schmidt , A. , Möller , C. A. & Siebert , K. G. ( 2012 ) Design and convergence analysis for an adaptive discretization of the heat equation . IMA J. Numer. Anal. , 32 , 1375 – 1403 . Google Scholar CrossRef Search ADS Lakkis , O. & Makridakis , Ch. ( 2006 ) Elliptic reconstruction and a posteriori error estimates for fully discrete linear parabolic problems . Math. Comp. , 75 , 1627 – 1658 . Google Scholar CrossRef Search ADS Lozinski , A. , Picasso , M. & Prachittham , V. ( 2009 ) An anisotropic error estimator for the Crank–Nicolson method: application to a parabolic problem . SIAM J. Sci. Comput. , 31 , 1757 – 1783 . Google Scholar CrossRef Search ADS Makridakis , Ch. & Nochetto , R. H. ( 2003 ) Elliptic reconstruction and a posteriori error estimates for parabolic problems . SIAM J. Numer. Anal. , 41 , 1585 – 1594 . Google Scholar CrossRef Search ADS Sohr , H. ( 2001 ) The Navier–Stokes Equations . Basel : Birkhäuser . Temam , R. ( 1977 ) Navier–Stokes Equations , 3rd edn . Amsterdam : North-Holland . Verfürth , R. ( 1996 ) A Review of A Posteriori Error Estimation and Adaptive Mesh-refinement Techniques . Wiley-Teubner Series . Chichester-Stuttgart . Verfürth , R. ( 2003 ) A posteriori error estimates for finite element discretizations of the heat equation . Calcolo , 40 , 195 – 212 . Google Scholar CrossRef Search ADS Verführt , R. ( 2013 ) A Posteriori Error Estimation Techniques for Finite Element Methods . Oxford Science Publications . Google Scholar CrossRef Search ADS Appendix A. Proof of Theorem 2.5 In the first step a priori estimates of the initial errors are presented: Lemma A.1 Assume the exact solution of the Stokes system (2.1) is sufficiently regular. Determine p0 from u0 and equation (2.1). Set $$U^{0}=\mathcal{S}_{0}^{u}(u_{0},p_{0})$$ and $$P^{0}:=\mathcal{S}^{p}_{0}(u_{0},p_{0})$$ of Lemma 2.3. Calculate U1 ∈ Xh and P1 ∈ Yh by equation (2.10). Then the approximation estimate of the Stokes operator holds \begin{align*} \|U^{1}-\mathcal{S}_{1}^{u}(u(t_{1}),p(t_{1}))\|^{2}+\tau\|\nabla(U^{1}-\mathcal{S}_{1}^{u}(u(t_{1}),p(t_{1})))\|^{2}+\tau^{2}\|P^{1}-\mathcal{S}_{1}^{p}(u(t_{1}),p(t_{1}))\|^{2} \end{align*} \begin{align} \lesssim \tau^{2} h^{2(k+1)}A^{1} + \tau^{6}\|{\partial^{3}_{t}} u\|^{2}_{L^{\infty}((0,t_{1});L^{2})} \end{align} (A.1) and the fully discrete estimates \begin{align} \|U^{1}-u(t_{1})\|^{2} \lesssim h^{2(k+1)} A^{0} + \tau^{6}\|{\partial^{3}_{t}} u\|^{2}_{L^{\infty}((0,t_{1});L^{2})}, \end{align} (A.2) \begin{align} \|\nabla(U^{1}-u(t_{1}))\|^{2} \lesssim h^{2k} A^{0}+\tau h^{2(k+1)} A^{1} + \tau^{5}\|{\partial^{3}_{t}} u\|^{2}_{L^{\infty}((0,t_{1});L^{2})}, \end{align} (A.3) \begin{align} \|P^{1}-p(t_{1})\|^{2} \lesssim h^{2k} A^{0}+ h^{2(k+1)} A^{1} + \tau^{4}\|{\partial^{3}_{t}} u\|^{2}_{L^{\infty}((0,t_{1});L^{2})}, \end{align} (A.4) where $$ A^{i}:=\|{\partial^{i}_{t}} u\|^{2}_{L^{\infty}((0,t_{1});H^{k+1})}+ \|{\partial^{i}_{t}} p\|^{2}_{L^{\infty}((0,t_{1});H^{k})}.$$ Proof. Set $$\theta ^{i}:=U^{i}-\mathcal{S}_{i}^{u}(u_{i},p_{i})$$. Subtract the half of equation (2.1) at times t = 0 and t = t1 from equation (2.10), use the Stokes projection defined in Lemma 2.3 and note that θ0 = 0 and $$P^{0}-~\mathcal{S}_{0}^{p}(u_{0},p_{0})=0$$ to deduce the error equation \begin{align} \frac{1}{\tau}\theta^{1}-\frac{1}{2}\varDelta_{h}\theta^{1}+\frac{1}{2}B_{h}^{\top}(P^{1}-\mathcal{S}_{1}^{p}(u(t_{1}),p(t_{1}))) = \bar\partial(u(t_{1})-\mathcal{S}^{u}_{n}(u(t_{1}),p(t_{1})))+(\partial_{t} u(t_{1/2})-\bar\partial u(t_{1})). \end{align} (A.5) Test this equation by τθ1 ∈ Jh to obtain \begin{align} \|\theta^{1}\|^{2}+\tau\|\nabla\theta^{1}\|^{2} &\lesssim \tau^{2}\|\bar\partial(u(t_{1})-\mathcal{S}_{1}^{u} (u(t_{1}),p(t_{1})))\|^{2}+\tau^{2}\|(\partial_{t}u(t_{1/2})-\bar\partial u(t_{1}))\|^{2}\nonumber\\ &\lesssim \tau^{2}h^{2(k+1)}A^{1} + \tau^{6}\|{\partial^{3}_{t}} u\|^{2}_{L^{\infty}((0,t_{1});L^{2})}, \end{align} (A.6) where Lemma 2.2 and Taylor expansion has been used. This proves the estimate for the first and second term in equation (A.1). Equations (A.2) and (A.3) are then easily deduced by the triangle inequality and Lemma 2.2. To show the pressure estimate (A.4) let us first show the estimate for $$\|P^{1}-\mathcal{S}_{1}^{p}(u(t_{1}),p(t_{1}))\|$$ in equation (A.1). Use the inf–sup condition and insert the error equation (A.5) to get \begin{align*} \|P^{1}-\mathcal{S}_{1}^{p}(u(t_{1}),p(t_{1}))\| &\lesssim \sup_{v_{h}\in X_{h}}\frac{|\langle P^{1}-\mathcal{S}_{1}^{p}(u(t_{1}),p(t_{1})),\operatorname{div} v_{h}\rangle|^{2}}{\|\nabla v_{h}\|}\\ &\leq \frac{1}{\tau}\|\theta^{1}\|_{X_{h}^{\ast}}\!+\!\|\nabla\theta^{1}\|\!+\!\|\bar\partial(u(t_{1})-\mathcal{S}_{1}^{u}(u(t_{1}),p(t_{1})))\|_{X_{h}^{\ast}}+\!\|\partial_{t} u(t_{1/2})\!-\!\bar\partial u(t_{1})\|_{X^{\ast}}\\ &\leq h^{2(k+1)}A^{1} + \tau^{4}\|{\partial^{3}_{t}} u\|^{2}_{L^{\infty}((0,t_{1});L^{2})}, \end{align*} where we have bounded $$\|\theta ^{1}\|_{X_{h}^{\ast }}\leq \|\theta ^{1}\|$$ and used equation (A.6) in the last step. The pressure estimate (A.4) is then obtained by the triangle inequality and Lemma 2.3: $$ \|P^{1}-p(t_{1})\|\;\leq \|P^{1}-\mathcal{S}_{1}^{p}(u(t_{1}),p(t_{1}))\|+\|p(t_{1})-\mathcal{S}_{1}^{p}(u(t_{1}),p(t_{1}))\|.$$ Now we are able to prove Theorem 2.5: Proof. (Theorem 2.5) Set $$\theta ^{n}:= U^{n}-\mathcal{S}^{u}_{n}(u(t_{n}),p(t_{n}))$$. Subtract equation (2.1) from equation (2.9) and combine with the Stokes projector $$\mathcal{S}^{u}_{n}$$ to get the error equation \begin{align} \bar\partial^{B}\theta^{n}-\varDelta_{h}\theta^{n} + B_{h}^{\top}(P^{n}-\mathcal{S}^{p}_{n}(u(t_{n}),p(t_{n})))\; =\; \bar\partial^{B}(u(t_{n})-\mathcal{S}^{u}_{n}(u(t_{n}),p(t_{n})))+(\partial_{t}-\bar\partial^{B})u(t_{n}),\quad n\geq2. \end{align} (A.7) Test this equation by 4τθn ∈ Jh. By the identity \begin{align} 2\langle 3a^{n}-4a^{n-1}+a^{n-2},a^{n}\rangle = \|a^{n}\|^{2}-\|a^{n-1}\|^{2}+\tau^{4}\|\bar\partial^{2} a^{n}\|^{2}+\|2a^{n}-a^{n-1}\|^{2}-\|2a^{n-1}-a^{n-2}\|^{2} \end{align} (A.8) the first term can be transformed into \begin{align*} 4\tau\langle\bar\partial^{B}\theta^{n},\theta^{n}\rangle &= 2\langle 3\theta^{n}-4\theta^{n-1}+\theta^{n-2},\theta^{n}\rangle\\ &=\|\theta^{n}\|^{2}-\|\theta^{n-1}\|^{2}+\tau^{4}\|\bar\partial^{2} \theta^{n}\|^{2}+\|2\theta^{n}-\theta^{n-1}\|^{2}-\|2\theta^{n-1}-\theta^{n-2}\|^{2}. \end{align*} Since θn ∈ Jh the pressure terms drop out and one obtains the error inequality \begin{align*} \|\theta^{n}\|^{2}-\|\theta^{n-1}\|^{2}+\tau^{4}\|\bar\partial^{2} \theta^{n}\|^{2}+\|2\theta^{n}-\theta^{n-1}\|^{2}-\|2\theta^{n-1}-\theta^{n-2}\|^{2} + \tau\|\nabla\theta^{n}\|^{2}\\ \lesssim \tau\|\bar\partial^{B}(u(t_{n})-\mathcal{S}^{u}_{n}(u(t_{n}),p(t_{n})))\|^{2}_{J_{h}^{\ast}}+\tau\|(\partial_{t}-\bar\partial^{B})u(t_{n})\|^{2}_{J_{h}^{\ast}}. \end{align*} Summing up from n = 2 to N yields \begin{align*} \max_{1\leq n\leq N}\|\theta^{n}\|^{2}+\sum_{n=2}^{N}\tau\|\nabla\theta^{n}\|^{2} &\lesssim \sum_{n=2}^{N} \tau\|\bar\partial^{B}(u(t_{n})-\mathcal{S}^{u}_{n}(u(t_{n}),p(t_{n})))\|^{2}_{J_{h}^{\ast}}\\ &\quad+\sum_{n=2}^{N}\tau\|(\partial_{t}-\bar\partial^{B})u(t_{n})\|^{2}_{J_{h}^{\ast}}+\|\theta^{1}\|^{2}+\|\theta^{1}-\theta^{0}\|^{2}.\end{align*} For the first term on the right-hand side we have by using Lemma 2.2 \begin{align} \sum_{n=2}^{N}\tau\|\bar\partial^{B}(u(t_{n})-\mathcal{S}^{u}_{n}(u(t_{n}),p(t_{n})))\|^{2}_{J_{h}^{\ast}} \lesssim \sum_{n=1}^{N} \tau\|\bar\partial u(t_{n})-\mathcal{S}^{u}_{n}(\bar\partial u(t_{n}),\bar\partial p(t_{n}))\|^{2} \lesssim h^{2(k+1)} A^{1}. \end{align} (A.9) For the second term on the right-hand side use standard estimates for the BDF2 operator $$\bar \partial ^{B}$$ and get \begin{align} \max_{1\leq n\leq N}\|\theta^{n}\|^{2}+\sum_{n=2}^{N}\tau\|\nabla\theta^{n}\|^{2} \lesssim \|\theta^{1}\|^{2} +h^{2(k+1)} A^{1} + \tau^{4}\|{\partial_{t}^{3}} u\|^{2}_{L^{\infty}(H^{-1})}. \end{align} (A.10) The velocity estimate (2.11) is finally obtained using the triangle inequality: $$ \max_{1\leq n\leq N}\|U^{n}-u(t_{n})\|\lesssim\max_{1\leq n\leq N}\|\theta^{n}\|+\max_{1\leq n\leq N} \|u(t_{n})-\mathcal{S}^{u}_{n}(u(t_{n}),p(t_{n}))\|,$$ the initial estimate (A.1), equation (A.10) and Lemma 2.2. In the same way equation (2.12) is proved. To derive the pressure estimate (2.13) we first bound the term $$\|\bar \partial ^{B} U^{n}-\partial _{t} u\|$$. To this end test the error equation (A.7) by $$4\tau \bar \partial ^{B}\theta ^{n}$$ and use the identity (A.8) to get \begin{align*} \tau\|\bar\partial^{B}\theta^{n}\|^{2}+\|\nabla\theta^{n}\|^{2}-\|\nabla\theta^{n-1}\|^{2}+\tau^{4}\|\nabla\bar\partial\theta^{n}\|^{2}+ \|\nabla(2\theta^{n}-\theta^{n-1})\|^{2}-\|\nabla(2\theta^{n-1}-\theta^{n-2})\|^{2}\\ \lesssim\tau\|\bar\partial^{B}(u(t_{n})-\mathcal{S}^{u}_{n}(u(t_{n}),p(t_{n})))\|^{2}+\tau\|(\partial_{t}-\bar\partial^{B})u(t_{n})\|^{2}. \end{align*} Sum up from n = 2 to N, observe that θ0 = 0 to obtain \begin{align} \sum_{n=2}^{N}\tau\|\bar\partial^{B}\theta^{n}\|^{2}+\max_{1\leq n\leq N}\|\nabla\theta^{n}\|^{2} &\lesssim\|\nabla\theta^{1}\|^{2}+\sum_{n=2}^{N}\tau\|\bar\partial(u(t_{n})-\mathcal{S}^{u}_{n}(u(t_{n}),p(t_{n})))\|^{2}\nonumber\\&\quad+\sum_{n=2}^{N}\tau\|(\partial_{t}-\bar\partial^{B})u(t_{n})\|^{2}. \end{align} (A.11) Now we are able to prove the pressure estimate (2.13). The inf–sup stability yields \begin{align*} \|P^{n}-p(t_{n})\| &\leq \|P^{n}-\mathcal{S}^{p}_{n}(u(t_{n}),p(t_{n}))\|+\|\mathcal{S}^{p}_{n}(u(t_{n}),p(t_{n}))- p(t_{n})\|\\ &\lesssim \sup_{v_{h}\in X_{h}}\frac{\langle P^{n}-\mathcal{S}^{p}_{n}(u(t_{n}),p(t_{n})),\operatorname{div} v_{h}\rangle}{\|\nabla v_{h}\|}+\|\mathcal{S}^{p}_{n}(u(t_{n}),p(t_{n}))- p(t_{n})\|\\ &= \sup_{v_{h}\in X_{h}}\frac{\langle \bar\partial^{B} U^{n}-\partial_{t} u(t_{n}),v_{n}\rangle+\langle\nabla\theta^{n},\nabla v_{h}\rangle}{\|\nabla v_{h}\|}+\|\mathcal{S}^{p}_{n}(u(t_{n}),p(t_{n}))- p(t_{n})\|\\ &\leq \|\bar\partial^{B}\theta^{n}\| + \|\bar\partial^{B}(u(t_{n})-\mathcal{S}^{u}_{n}(u(t_{n}),p(t_{n})))\|+\|(\bar\partial^{B}-\partial_{t})u(t_{n})\|\\ &\quad+ \|\nabla\theta^{n}\|+\|\mathcal{S}^{p}_{n}(u(t_{n}),p(t_{n}))- p(t_{n})\|. \end{align*} Square this inequality, multiply by τ and use equation (A.11) to get \begin{align*} \sum_{n=1}^{N}\tau\|P^{n}-p(t_{n})\|^{2} &\lesssim \|\nabla\theta^{1}\|^{2}+\tau\|P^{1}-p(t_{1})\|^{2}\\ &\quad+ \sum_{n=2}^{N}\tau\left(\|\bar\partial^{B}(u(t_{n})-\mathcal{S}^{u}_{n}(u(t_{n}),p(t_{n})))\|^{2}+\|(\partial_{t}-\bar\partial^{B})u(t_{n})\|^{2}\right.\\ &\qquad\qquad\quad\left.+\|\nabla\theta^{n}\|^{2}+\|\mathcal{S}^{p}_{n}(u(t_{n}),p(t_{n}))- p(t_{n})\|^{2}\right)\!. \end{align*} Then by using the initial estimate equations (A.1)–(A.4), the velocity estimate (A.10) and Lemma 2.3 the pressure estimate in equation (2.13) is proven. The $$\ell ^{\infty }(L^{2})$$ estimate for the gradients of the velocity is easily derived by equation (A.11) and Lemma 2.2. Appendix B. Discrete regularity In this section we derive regularity estimates for the solution of the semidiscrete instationary Stokes problem \begin{align} \partial_{t} Z(t) - \varDelta_{h} Z(t) + B_{h}^{\top} Q(t) = \tilde G (t),\qquad B_{h} Z(t)=0,\qquad Z(t)=V_{0}, \end{align} (B.1) for a sufficiently regular (in time) right-hand side $$\tilde G(t)\in X_{h}$$ and initial value V0 or, equivalently, by multiplying the above equation by $$\varPi _{J_{h}}$$, \begin{align} \partial_{t} Z(t) + \mathcal{A}_{h} Z(t) = F(t):=\varPi_{J_{h}}\tilde G(t),\qquad B_{h} Z(t)=0,\qquad Z(0)=V_{0}. \end{align} (B.2) Here, $$\varPi _{J_{h}}:X\to J_{h}$$ denotes the L2 projection onto the discrete solenoidal space Jh and \begin{align} \mathcal{A}_{h}:=\varPi_{J_{h}}(-\varDelta_{h}) \end{align} (B.3) the discrete Stokes operator. Regularity results for the full continuous system equation (2.1) can be found for instance in the study by Temam (1977, Chapter III), Sohr (2001, Chapter IV.2.7) and Heywood (1980). The following remark seems to be in order. The crucial difference between the continuous and the semidiscrete case is that in the latter case no compatibility conditions between the initial data and right-hand side are needed for getting higher regularity, since the domain of the discrete operator $$\mathcal{A}_{h}$$ is the whole of Jh. In contrast, in the continuous case higher regularity in time is always linked to such compatibility conditions, even if data are arbitrarily smooth. This can readily be seen for the heat equation: assume that u, △u, f and ∂tu are continuous on $$[0,T]\times \bar{\varOmega }$$, satisfying $$ \partial_{t}u -\varDelta u = f,\qquad u_{|\varGamma}=0,\qquad u(0)=u_{0}.$$ Since u(t, x)|Γ = 0 for t ∈ [0, T], we see that ∂tu(0)|Γ = 0 and therefore the additional condition for given data f, u0 must be satisfied: $$ \varDelta u_{0|\varGamma} = f(0)_{|\varGamma}.$$ Further compatibility conditions are required for higher order time derivatives $$\partial _{t}^{(m)} u$$, see for instance Evans (2010). For the Stokes problem the situation is even more involved because of the presence of the pressure gradient. Only if the pressure fulfills a Poisson equation with both, a Dirichlet and a Neumann boundary condition (and is thus overdetermined), higher time regularity can hold. This kind of compatibility condition cannot be checked in practice, see also Heywood & Rannacher (1982). To avoid these complicated conditions Sohr presents the following local results, cf. Sohr (2001, Theorem 2.7.2): Let $$m\in \mathbb{N}$$, let $$\varOmega \subset \mathbb{R}^{d}$$, d ≥ 2, be a uniform C2-domain, let $$0<T\leq \infty $$, $$1<s<\infty $$, $$u_{0}\in D\left (\mathcal{A}^{1-\frac{1}{s}}\right )\cap D\left (\mathcal{A}^{\frac{1}{2}}\right )$$, $$f\in L^{2}_{loc}\left ([0,T);L^{2}(\varOmega )^{d}\right )$$, and suppose u is a solution of the Stokes system (2.1) with data f, u0 together with an associated pressure p. Then $$ \partial_{t} f,\partial_{tt}f,\ldots,\partial_{t}^{(m)}f\in L^{s}_{loc}\left((0,T);L^{2}(\varOmega)^{d}\right)$$ implies $$ \partial_{t} u,\partial_{tt} u,\ldots,\partial_{t}^{(m+1)}u,\, \mathcal{A}\partial_{t} u,\mathcal{A}\partial_{tt} u,\ldots, \mathcal{A}\partial_{t}^{(m+1)}u\in L^{s}_{loc}((0,T);H),$$ and $$ \partial_{t} u,\partial_{tt} u,\ldots,\partial_{t}^{(m+1)}u \in L^{s}_{loc}\left((0,T);H^{2}(\varOmega)^{d}\right)\!.$$ Let us return to the problem of regularity for the semidiscrete problem. Since (B.2) is a linear, finite-dimensional evolution equation, it admits a unique solution for all times t > 0. Moreover, differentiability of u with respect to time directly follows from regularity of G(t). The following lemma gives an equivalent expression for the dual norm of Jh in terms of the discrete Stokes operator. Lemma B.1 For yh ∈ Jh there is a unique xh ∈ Jh, solution of the discrete Stokes problem $$\mathcal{A}_{h}x_{h}=y_{h}$$. Moreover, \begin{align} \|y_{h}\|_{J_{h}^{\ast}} = \|\nabla x_{h}\|. \end{align} (B.4) Proof. One has \begin{align} \langle \mathcal{A}_{h} v_{h},w_{h}\rangle = \langle\varPi_{J_{h}}(-\varDelta_{h} v_{h}),w_{h}\rangle = \langle -\varDelta_{h} v_{h}, w_{h}\rangle = \langle \nabla v_{h},\nabla w_{h}\rangle,\qquad \forall\ v_{h},\ w_{h}\in J_{h}. \end{align} (B.5) This shows that $$\mathcal{A}_{h}$$ is symmetric and coercive on Jh and in particular bijective. Using the above representation for $$\mathcal{A}_{h}$$, the lemma can be easily proven. First, $$ \|y_{h}\|_{J^{\ast}_{h}}=\sup_{z_{h}\in J_{h}}\frac{\langle y_{h},z_{h}\rangle}{\|\nabla z_{h}\|} =\sup_{z_{h}\in J_{h}}\frac{\langle \mathcal{A}_{h}x_{h},z_{h}\rangle}{\|\nabla z_{h}\|} =\sup_{z_{h}\in J_{h}}\frac{\langle -\varDelta_{h}x_{h},z_{h}\rangle}{\|\nabla z_{h}\|}\leq \|\nabla x_{h}\|. $$ Second, $$ \|\nabla x_{h}\| = \frac{\|\nabla x_{h}\|^{2}}{\|\nabla x_{h}\|}=\frac{\langle-\varDelta_{h}x_{h},x_{h}\rangle}{\|\nabla x_{h}\|} =\frac{\langle \mathcal{A}_{h}x_{h},x_{h}\rangle}{\|\nabla x_{h}\|}=\frac{\langle y_{h},x_{h}\rangle}{\|\nabla x_{h}\|} \leq\sup_{x_{h}\in J_{h}}\frac{\langle y_{h},x_{h}\rangle}{\|\nabla x_{h}\|}=\|y_{h}\|_{J^{\ast}_{h}}. $$ In the following theorem, regularity estimates for the solution of the semidiscrete system (B.2) in terms of the regularity of the right-hand side G and the initial data V0 are presented. Theorem B.2 Suppose that $$\partial _{t}^{(m)}G\in L^{2}((0,T),J_{h})$$ for all 1 ≤ m ≤ M. Then the solution Z(t) of equation (B.2) satisfies for all 1 ≤ m ≤ M \begin{align} \|\partial_{t}^{(m)} Z\|^{2}_{L^{\infty}(L^{2})} + \|\partial_{t}^{(m)} \nabla Z\|^{2}_{L^{2}(L^{2})} \lesssim \|\partial_{t}^{(m)} G\|^{2}_{L^{2}(J_{h}^{\ast})} + \|{Y^{m}_{0}}\|^{2}, \end{align} (B.6) \begin{align} \|\partial_{t}^{(m+1)}Z\|^{2}_{L^{2}(L^{2})} + \|\partial_{t}^{(m)} \nabla Z\|^{2}_{L^{\infty}(L^{2})} \lesssim \|\partial_{t}^{(m)} G\|^{2}_{L^{2}(L^{2})} + \|{Y^{m}_{0}}\|_{1}^{2}, \end{align} (B.7) \begin{align} \|\partial_{t}^{(m)}\nabla Z\|^{2}_{L^{\infty}(L^{2})} + \|\partial_{t}^{(m)}\mathcal{A}_{h} Z\|^{2}_{L^{2}(L^{2})} \lesssim \|\partial_{t}^{(m)} G\|^{2}_{L^{2}(L^{2})} + \|{Y^{m}_{0}}\|_{1}^{2}, \end{align} (B.8) \begin{align} \|\partial_{t}^{(m+1)}\nabla Z\|^{2}_{L^{2}(L^{2})} + \|\partial_{t}^{(m)} \mathcal{A}_{h} Z\|^{2}_{L^{\infty}(L^{2})} \lesssim \|\nabla\partial_{t}^{(m)} G\|^{2}_{L^{2}(L^{2})} + \|{Y^{m}_{0}}\|_{\mathcal{A}}^{2}, \end{align} (B.9) \begin{align} \|\partial_{t}^{(m)}Z\|^{2}_{L^{\infty}(J_{h}^{\ast})} + \|\partial_{t}^{(m)} Z\|^{2}_{L^{2}(L^{2})} \lesssim \|\partial_{t}^{(m)} G\|^{2}_{L^{1}(J_{h}^{\ast})} + \|{Y^{m}_{0}}\|_{J_{h}^{\ast}}^{2}, \end{align} (B.10) \begin{align} \|\partial_{t}^{(m+1)}Z\|^{2}_{L^{2}(J_{h}^{\ast})} + \|\partial_{t}^{(m)} Z\|^{2}_{L^{\infty}(L^{2})} \lesssim \|\partial_{t}^{(m)} G\|^{2}_{L^{2}(J_{h}^{\ast})} + \|{Y^{m}_{0}}\|^{2}, \end{align} (B.11) where \begin{align*} \|{Y^{m}_{0}}\|_{*}^{2} &:= \|\mathcal{A}_{h}^{m}Z(0)\|_{*}^{2}+\sum_{i=0}^{m-1}\|\partial_{t}^{(i)}\mathcal{A}_{h}^{m-1-i}G(0)\|_{*}^{2},\qquad *\in\{1,0,J_{h}^{\ast}\}, \\[-2pt] \|{Y^{m}_{0}}\|_{\mathcal{A}}^{2} &:= \|\mathcal{A}_{h}^{m+1}Z(0)\|^{2}+\sum_{i=0}^{m-1}\|\partial_{t}^{(i)}\mathcal{A}_{h}^{m-i}G(0)\|^{2}. \end{align*} Proof. Let 1 ≤ m ≤ M. Differentiating equation (B.2) m-times with respect to time leads to \begin{align} \partial_{t}^{(m+1)} Z(t) + \partial_{t}^{(m)} \mathcal{A}_{h} Z(t) = \partial_{t}^{(m)} G(t). \end{align} (B.12) First, to prove equation (B.6), test equation (B.12) with $$2\partial _{t}^{(m)} Z(t)\in J_{h}$$ to get $$ \frac{\mathrm{d}}{\mathrm{d}t}\|\partial_{t}^{(m)} Z(t)\|^{2}+\|\nabla \partial_{t}^{(m)} Z(t)\|^{2} \lesssim \|\partial_{t}^{(m)} G(t)\|^{2}_{J_{h}^{\ast}},$$ where we have used equation (B.5) for the second term on the left-hand side. Now, integrating the above equation from t = 0 to T yields $$ \|\partial_{t}^{(m)} Z\|^{2}_{L^{\infty}(L^{2})} + \|\partial_{t}^{(m)} \nabla Z\|^{2}_{L^{2}(L^{2})} \lesssim \|\partial_{t}^{(m)} G\|^{2}_{L^{2}(J_{h}^{\ast})} + \|\partial_{t}^{(m)} Z(0)\|^{2}.$$ Next, to prove equations (B.7)–(B.9), test equation (B.12) with $$v_{h} = \partial _{t}^{(m+1)} Z(t)$$, $$\partial _{t}^{(m)} \mathcal{A}_{h} Z(t)$$ and $$\partial _{t}^{(m+1)} \mathcal{A}_{h} Z(t)$$, and proceed exactly as above to get $$v_{h} = \partial _{t}^{(m+1)} Z(t)$$: $$\|\partial _{t}^{(m+1)}Z\|^{2}_{L^{2}(L^{2})} + \|\partial _{t}^{(m)}\nabla Z\|^{2}_{L^{\infty }(L^{2})} \lesssim \|\partial _{t}^{(m)} G\|^{2}_{L^{2}(L^{2})} + \|\partial _{t}^{(m)} \nabla Z(0)\|^{2}$$, $$v_{h}= \partial _{t}^{(m)} \mathcal{A}_{h} Z(t)$$: $$\|\partial _{t}^{(m)} \nabla Z\|^{2}_{L^{\infty }(L^{2})} + \|\partial _{t}^{(m)} \mathcal{A}_{h} Z\|^{2}_{L^{2}(L^{2})} \lesssim \|\partial _{t}^{(m)} G\|^{2}_{L^{2}(L^{2})} + \|\partial _{t}^{(m)} \nabla Z(0)\|^{2}$$, $$v_{h} = \partial _{t}^{(m+1)} \mathcal{A}_{h} Z(t)$$: $$\|\partial _{t}^{(m+1)}\nabla Z\|^{2}_{L^{2}(L^{2})} + \|\partial _{t}^{(m)} \mathcal{A}_{h} Z\|^{2}_{L^{\infty }(L^{2})} \lesssim \|\nabla \partial _{t}^{(m)} G\|^{2}_{L^{2}(L^{2})} + \|\partial _{t}^{(m)} \mathcal{A}_{h} Z(0)\|^{2}$$, respectively. For proving equations (B.10) and (B.11) we use a duality argument. We test equation (B.12) with vh, where vh is solution of the Stokes problem $$\mathcal{A}_{h} v_{h} = \partial _{t}^{(m)} Z(t)$$. For the left-hand side, one gets \begin{align*} \langle \partial_{t}^{(m+1)} Z(t),v_{h}(t)\rangle &= \langle \partial_{t} \mathcal{A}_{h} v_{h}(t),v_{h}(t)\rangle = \frac{1}{2}\frac{\mathrm{d}}{\mathrm{d}t}\|\nabla v_{h}(t)\|^{2},\\ \langle \partial_{t}^{(m)}\mathcal{A}_{h} Z(t),v_{h}\rangle &= \langle \partial_{t}^{(m)}-\varDelta_{h} Z(t),v_{h}\rangle = \langle \partial_{t}^{(m)} Z(t),\mathcal{A}_{h} v_{h}\rangle = \|\partial_{t}^{(m)} Z(t)\|^{2}, \end{align*} where for the second term equation (B.5) was used. For the right-hand side it follows $$\langle \partial _{t}^{(m)} G(t),v_{h}\rangle \leq \|\partial _{t}^{(m)} G(t)\|_{J^{\ast }_{h}}\|\nabla v_{h}\|$$ and then $$ \frac{1}{2}\frac{\mathrm{d}}{\mathrm{d}t}\|\nabla v_{h}(t)\|^{2}+ \|\partial_{t}^{(m)} Z(t)\|^{2} \leq \|\partial_{t}^{(m)} G(t)\|_{J^{\ast}_{h}}\|\nabla v_{h}\|.$$ Integrate from t = 0 to T and use the norm equivalence in Lemma B.1 to get $$ \|\partial_{t}^{(m)} Z\|^{2}_{L^{\infty}(J_{h}^{\ast})} + \|\partial_{t}^{(m)} Z\|^{2}_{L^{2}(L^{2})} \lesssim \|\partial_{t}^{(m)} G\|^{2}_{L^{1}(J_{h}^{\ast})} + \|\partial_{t}^{(m)} Z(0)\|^{2}_{J_{h}^{\ast}}. $$ Now let vh be solution of the Stokes problem $$\mathcal{A}_{h} v_{h} = \partial _{t}^{(m+1)} Z(t)$$. The same procedure as above yields $$ \|\partial_{t}^{(m+1)}Z\|^{2}_{L^{2}(J_{h}^{\ast})} + \|\partial_{t}^{(m)} Z\|^{2}_{L^{\infty}(L^{2})} \lesssim \|\partial_{t}^{(m)} G\|^{2}_{L^{2}(J_{h}^{\ast})} + \|\partial_{t}^{(m)} Z(0)\|^{2}.$$ It remains to bound the initial values $$\| \partial _{t}^{(m)}Z(0)\|^{2}$$, $$\| \partial _{t}^{(m)}\nabla Z(0)\|^{2}$$, $$\| \partial _{t}^{(m)} \mathcal{A}_{h} Z(0)\|^{2}$$ and $$\| \partial _{t}^{(m)} Z(0)\|^{2}_{J_{h}^{\ast }}$$. To this end we prove the following identity for m ≥ 1 \begin{align} \partial_{t}^{(m)}Z(0) = \displaystyle\sum_{i=0}^{m-1}(-1)^{i-1+m}\partial_{t}^{(i)}\mathcal{A}_{h}^{m-1-i}G(0) + (-1)^{m}\mathcal{A}_{h}^{m}Z(0). \end{align} (B.13) The cases m = 1 and m = 2 are easily shown by using equation (B.12) and taking the limit t ↘ 0. \begin{align*} \partial_{t} Z(0) &= G(0) - \mathcal{A}_{h}Z(0),\\ \partial_{tt} Z(0) &= \partial_{t} G(0) - \mathcal{A}_{h}\partial_{t}Z(0) = \partial_{t} G(0) - \mathcal{A}_{h}G(0) + \mathcal{A}_{h}^{2} Z(0). \end{align*} Now assume equation (B.13) holds for m ≥ 2. Then we have by another use of equation (B.12) \begin{align*} \partial_{t}^{(m+1)}Z(0) &= \partial_{t}^{(m)}G(0)-\mathcal{A}_{h}\partial_{t}^{(m)}Z(0)\\& = \partial_{t}^{(m)} G(0) - \mathcal{A}_{h}\sum_{i=0}^{m-1}(-1)^{i-1+m}\partial_{t}^{(i)}\mathcal{A}_{h}^{m-1-i}G(0) + (-1)^{m+1}\mathcal{A}_{h}^{m+1}Z(0)\\ &= \sum_{i=0}^{m}(-1)^{i+m}\partial_{t}^{(i)}\mathcal{A}_{h}^{m-i}G(0) + (-1)^{m+1}\mathcal{A}_{h}^{m+1}Z(0) \end{align*} and equation (B.13) is shown. Hence, the initial values can be easily bounded by equation (B.13) using the triangle inequality. © The Author(s) 2018. Published by Oxford University Press on behalf of the Institute of Mathematics and its Applications. All rights reserved. This article is published and distributed under the terms of the Oxford University Press, Standard Journals Publication Model (https://academic.oup.com/journals/pages/about_us/legal/notices) For permissions, please e-mail: journals. permissions@oup.com

Journal

IMA Journal of Numerical AnalysisOxford University Press

Published: Apr 6, 2018

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