Parameter-free superconvergent H(div)-conforming HDG methods for the Brinkman equations

Parameter-free superconvergent H(div)-conforming HDG methods for the Brinkman equations Abstract In this paper, we present new parameter-free superconvergent H(div)-conforming hybridizable discontinuous Galerkin (HDG) methods for the Brinkman equations on both simplicial and rectangular meshes. The methods are based on a velocity gradient–velocity–pressure formulation, which can be considered a natural extension of the H(div)-conforming HDG method (defined on simplicial meshes) for the Stokes flow (Cockburn, B. & Sayas, F.-J. (2014) Divergence-conforming HDG methods for Stokes flow. Math. Comp., 83, 1571–1598). We obtain an optimal L2-error estimate for the velocity in both the Stokes-dominated regime (high viscosity/permeability ratio) and Darcy-dominated regime (low viscosity/permeability ratio). We also obtain a superconvergent L2-estimate of one order higher for a suitable projection of the velocity error in the Stokes-dominated regime. Moreover, thanks to H(div)-conformity of the velocity, our velocity error estimates are independent of the pressure regularity. Furthermore, we provide a discrete H1-stability result for the velocity field, which is essential in the error analysis of the natural generalization of these new HDG methods to the incompressible Navier–Stokes equations. Preliminary numerical results on both triangular and rectangular meshes in two dimensions confirm our theoretical predictions. 1. Introduction In this paper, we devise a superconvergent H(div)-conforming hybridizable discontinuous Galerkin (HDG) method for the following Brinkman equations in a velocity gradient–velocity–pressure formulation: \begin{align} \mathrm{L}=\nabla\mathbf{u}\;\quad\, \textrm{in}\,\,\,\varOmega, \end{align} (1.1a) \begin{align} -\nu\nabla\cdot \mathrm{L}+\gamma\mathbf{u}+\nabla p=\mathbf{f}\;\qquad\,\textrm{in}\,\,\,\varOmega, \end{align} (1.1b) \begin{align} \nabla\cdot\mathbf{u}=g\;\qquad \textrm{in}\,\,\, \varOmega, \end{align} (1.1c) \begin{align} \mathbf{u}\cdot\mathbf{n}=0\;\qquad \textrm{on}\,\,\,\, \partial\varOmega, \end{align} (1.1d) \begin{align} \ \nu(I_{d}-\mathbf{n}\otimes\mathbf{n})\mathbf{u}=0\;\qquad \textrm{on}\quad \partial\varOmega, \end{align} (1.1e) \begin{align} \int_{\varOmega} p=0, \end{align} (1.1f) where L is the velocity gradient, u is the velocity, p is the pressure, ν is the effective viscosity constant, $$\gamma \in L^{\infty }(\varOmega )^{d\times d}$$ is the inverse of the permeability tensor, f ∈ L2$$(\varOmega)$$d is the external body force, $$g\in{L_{0}^{2}}(\varOmega ) := \{ q\in L^{2}(\varOmega ): (q, 1)_{\varOmega } = 0 \}$$ and n is the unit outward normal vector along ∂$$\varOmega$$. The domain $$\varOmega \subset \mathbb{R}^{d}$$ is a polygon (d = 2) or polyhedron (d = 3). Here (1.1e) indicates that we impose a homogeneous tangential trace of $$\boldsymbol{u} $$ on ∂$$\varOmega$$. We notice that when ν = 0, (1.1e) vanishes such that equations (1.1) become the Darcy equations. One challenging aspect of numerical discretization of the Brinkman equations is the construction of stable finite element methods in both Stokes-dominated and Darcy-dominated regimes. We refer to such methods as uniformly stable methods. Uniformly stable methods for the Brinkman equations have been extensively studied for the classical velocity–pressure formulation, including nonconforming methods with an H(div)-conforming velocity field (Mardal et al., 2002; Tai & Winther, 2006; Xie et al., 2008Guzmán & Neilan, 2012), conforming methods ( Xie et al., 2008Juntunen & Stenberg, 2010), stabilized methods ( Xie et al., 2008; Badia & Codina, 2009; Juntunen & Stenberg, 2010), the H(div)-conforming discontinuous Galerkin method (Könnö & Stenberg, 2011) and the hybridized H(div)-conforming discontinuous Galerkin method (Könnö & Stenberg, 2012), and other alternative formulations, including the vorticity–velocity–pressure formulation (Vassilevski & Villa, 2014,Anaya et al., 2015), the pseudostress-based formulation (Gatica et al., 2015) and a dual-mixed formulation (Howell & Neilan, 2016). In this paper, we propose and study a class of high-order, parameter-free, H(div)-conforming HDG methods for the Brinkman equations (1.1) on both simplicial and rectangular meshes. This is the first HDG method for the Brinkman equations based on a velocity gradient–velocity–pressure formulation. Our method can be considered a natural, stable extension to the Brinkman equations of the high-order, parameter-free, H(div)-conforming HDG method for the Stokes problem on simplicial meshes (Cockburn & Sayas, 2014). Three distinctive properties of the method make it attractive. Firstly, our method provides an optimal error estimate in L2-norms for the velocity that is robust with respect to viscosity/permeability ratio ν/γ (Theorem 2.4, Corollary 2.5), and a superconvergent error estimate in the L2-norm of one order higher for a suitable projection of the velocity error (under a regularity assumption on the dual problem). To the best of our knowledge, this is the first superconvergent velocity estimate for the Brinkman equations. Secondly, thanks to H(div)-conformity of the velocity, our velocity error estimates are independent of the pressure regularity (see Corollary 2.5 and Theorem 2.6). Such a pressure-robustness property is highly appreciated for incompressible flow problems (Linke, 2014; Linke & Merdon, 2016). Finally, our error analysis, which is quite different from and more straightforward than that in the study by Cockburn & Sayas (2014) for the Stokes flow, is based on a so-called discrete H1-stability result (see Theorem 2.1), which is the essential ingredient in the analysis of the velocity gradient–velocity–pressure HDG formulation of the incompressible Navier–Stokes equations. We specifically remark that no stabilization parameter enters our method, which has to be compared with the hybridized H(div)-conforming discontinuous Galerkin method (Könnö & Stenberg, 2012) in the classical velocity–pressure formulation, where Nitsche’s penalty method is used to impose tangential continuity of the velocity field and the stabilization parameter needs to be ‘sufficiently large’. The organization of the paper is as follows. In Section 2, we introduce the parameter-free H(div)-conforming HDG method and give the main results on a priori error estimates. In Section 3, we prove our main results in Section 2. In Section 4, we discuss the hybridization of the H(div)-conforming HDG method. In Section 5, we provide preliminary two-dimensional numerical experiments on triangular and rectangular meshes to validate our theoretical results. We end in Section 6 with some concluding remarks. 2. Main results: superconvergent H(div)-conforming HDG In this section, we first introduce the notation that will be used throughout the paper, and then present the finite element spaces that define the H(div)-conforming HDG methods. We conclude with an a priori error estimate along with a key inequality that we call discrete H1-stability. 2.1. Meshes and trace operators We denote by $${\mathcal{T}_{h}}:=\{K\}$$ (the mesh) a shape-regular conforming triangulation of the domain $$\varOmega \subset \mathbb{R}^{d}$$ into affine-mapped simplices (triangles if d = 2, tetrahedra if d = 3) or hypercubes (squares if d = 2, cubes if d = 3), and by $$\mathcal{E}_{h}$$ (the mesh skeleton) the set of facets F (edges if d = 2, faces if d = 3) of the elements $$K \in{\mathcal{T}_{h}}$$. Let $$\mathcal{F}(K)$$ denote the set of facets F of the element $$\textit{K} $$. We set hF := diam(F), hK := diam(K) and $$h := \max _{K\in{\mathcal{T}_{h}}}h_{K}$$. Let $$\underline{\textsf{K}}$$ be the reference element (d-dimensional simplex or hypercube), and $$\underline{\textsf{F}}$$ be the reference facet (d−1-dimensional simplex or hypercube). We denote by $$\varPhi _{K}: \underline{\textsf{K}}\rightarrow K$$ and $$\varPhi _{F}: \underline{\textsf{F}}\rightarrow F$$ the associated affine mappings. For a d-dimensional vector-valued function v on an element $$K\subset \mathbb{R}^{d}$$ with sufficient regularity, we denote by \begin{align} \textrm{tr}_{t}^{F}(\boldsymbol\upsilon):= \left.\left(\boldsymbol\upsilon-(\boldsymbol\upsilon\cdot\boldsymbol{n}_{F})\,\boldsymbol{n}_{F})\right)\right|{}_{F}\quad \textrm{and}\quad \textrm{tr}_{n}^{F}(\boldsymbol\upsilon):= \left.\left(\boldsymbol\upsilon\cdot\boldsymbol{n}_{F}\right)\boldsymbol{n}_{F} \right|{}_{F} \end{align} (2.1) the tangential and normal traces of v on the facet $$F\in \mathcal{F}(K)$$, where $$\boldsymbol{n} $$F is the unit normal vector to F. Note that the above trace operators are independent of the direction of the normal nF. Whenever there is no confusion, we suppress the superscript and denote by trt(v) and trn(v) the related tangential and normal traces, respectively. With an abuse of notation, we also denote $$ \textrm{tr}_{t}(\widehat{\mathbf{v}}):= \left.\left(\widehat{\mathbf{v}}-(\widehat{\mathbf{v}}\cdot\mathbf{n}_{F})\,\mathbf{n}_{F})\right)\right|{}_{F}\quad \textrm{and}\quad \textrm{tr}_{n}(\widehat{\mathbf{v}}):= \left.\left(\widehat{\mathbf{v}}\cdot\mathbf{n}_{F}\right)\mathbf{n}_{F} \right|{}_{F} $$ for a d-dimensional vector-valued function v on a facet $$F\subset \mathbb{R}^{d-1}$$ with sufficient regularity. 2.2. The finite element spaces Now, we define the finite element spaces associated with the mesh $${\mathcal{T}_{h}}$$ and mesh skeleton $$\mathcal{E}_{h}$$ via appropriate mappings (cf. Brenner & Scott, 2008) from (polynomial) spaces on the reference elements. We use the following mapped finite element spaces on the mapped element K and facet F: \begin{align} {\mathcal{G}}^{\mathrm{r}\mathrm{o}\mathrm{w}}(K):=\;\left\{\mathbf{v}\in{L}^{2}(K)^{d}:\;\;\mathbf{v}= \frac{1}{\mathrm{d}\mathrm{e}\mathrm{t}\,\varPhi_{K}^{\prime}}\varPhi_{K}^{\prime}\,\underline{\boldsymbol{\textsf{ v}}}\circ\varPhi_{K}^{-1},\;\;\underline{\boldsymbol{\textsf{ v}}}\in{\mathcal{G}}^{\mathrm{r}\mathrm{o}\mathrm{w}}(\underline{\textsf{K}})\right\}\!, \end{align} (2.2a) \begin{align} {\boldsymbol V}(K):=\;\left\{\mathbf{v}\in{L}^{2}(K)^{d}:\; \mathbf{v}= \frac{1}{\mathrm{d}\mathrm{e}\mathrm{t}\,\varPhi_{K}^{\prime}}\varPhi_{K}^{\prime}\,\underline{\boldsymbol{\textsf{ v}}}\circ\varPhi_{K}^{-1},\;\;\underline{\boldsymbol{\mathbf{ v}}}\in{\mathbf V}(\underline{\textsf{K}})\right\}\!, \end{align} (2.2b) \begin{align} Q(K):=\;\left\{q\in{L}^{2}(K):\;q=\underline{{\textsf{ q}}}\circ\varPhi_{K}^{-1},\;\;\underline{{\textsf{ q}}}\in{Q}(\underline{\textsf{K}})\right\}\!, \end{align} (2.2c) \begin{align} {\boldsymbol{M}}(F):=\;\left\{\widehat{\mathbf{v}}\in{L}^{2}(F)^{d}:\;\widehat{\mathbf{v}}=\underline{\widehat{\boldsymbol{\mathbf{v}}}}\circ\varPhi_{F}^{-1},\;\;\underline{\widehat{\boldsymbol{\mathbf{v}}}}\in{\boldsymbol{M}}(\underline{\textsf{F}})\right\}\!. \end{align} (2.2d) Here $$\varPhi$$K and $$\varPhi$$F are the affine mappings introduced above, and $$\varPhi _{K}^{\prime }$$ is the Jacobian matrix of the mapping $$\varPhi$$K. Note that the vector spaces in (2.2a) and (2.2b) are obtained from the well-known Piola transformation which preserves normal continuity (cf. Durán, 2008). The polynomial spaces on the reference elements are given in Table 1. Table 1 The reference finite element spaces Element $${\mathcal{G}}^{\mathrm{r}\mathrm{o}\mathrm{w}}(\underline{\textsf{K}})$$ $${\boldsymbol V}(\underline{\textsf{K}})$$ $${Q}(\underline{\textsf{K}})$$ $${\boldsymbol{M}}(\underline{\textsf{F}})$$ Simplex $$\mathcal{P}_{k}(\underline{\textsf{K}})^{d}$$ $$\boldsymbol{\mathrm{R}\mathrm{T}}_{k}(\underline{\textsf{K}})$$ $$\mathcal{P}_{k}(\underline{\textsf{K}})$$ $$\mathcal{P}_{k}(\underline{\textsf{F}})^{d}$$ Hypercube $$\boldsymbol{\mathrm{B}\mathrm{D}\mathrm{M}}_{k}(\underline{\textsf{K}})$$ $$\boldsymbol{\mathrm{B}\mathrm{D}\mathrm{F}\mathrm{M}}_{k}(\underline{\textsf{K}})$$ $$\mathcal{P}_{k}(\underline{\textsf{K}})$$ $$\mathcal{P}_{k}(\underline{\textsf{F}})^{d}$$ Element $${\mathcal{G}}^{\mathrm{r}\mathrm{o}\mathrm{w}}(\underline{\textsf{K}})$$ $${\boldsymbol V}(\underline{\textsf{K}})$$ $${Q}(\underline{\textsf{K}})$$ $${\boldsymbol{M}}(\underline{\textsf{F}})$$ Simplex $$\mathcal{P}_{k}(\underline{\textsf{K}})^{d}$$ $$\boldsymbol{\mathrm{R}\mathrm{T}}_{k}(\underline{\textsf{K}})$$ $$\mathcal{P}_{k}(\underline{\textsf{K}})$$ $$\mathcal{P}_{k}(\underline{\textsf{F}})^{d}$$ Hypercube $$\boldsymbol{\mathrm{B}\mathrm{D}\mathrm{M}}_{k}(\underline{\textsf{K}})$$ $$\boldsymbol{\mathrm{B}\mathrm{D}\mathrm{F}\mathrm{M}}_{k}(\underline{\textsf{K}})$$ $$\mathcal{P}_{k}(\underline{\textsf{K}})$$ $$\mathcal{P}_{k}(\underline{\textsf{F}})^{d}$$ Table 1 The reference finite element spaces Element $${\mathcal{G}}^{\mathrm{r}\mathrm{o}\mathrm{w}}(\underline{\textsf{K}})$$ $${\boldsymbol V}(\underline{\textsf{K}})$$ $${Q}(\underline{\textsf{K}})$$ $${\boldsymbol{M}}(\underline{\textsf{F}})$$ Simplex $$\mathcal{P}_{k}(\underline{\textsf{K}})^{d}$$ $$\boldsymbol{\mathrm{R}\mathrm{T}}_{k}(\underline{\textsf{K}})$$ $$\mathcal{P}_{k}(\underline{\textsf{K}})$$ $$\mathcal{P}_{k}(\underline{\textsf{F}})^{d}$$ Hypercube $$\boldsymbol{\mathrm{B}\mathrm{D}\mathrm{M}}_{k}(\underline{\textsf{K}})$$ $$\boldsymbol{\mathrm{B}\mathrm{D}\mathrm{F}\mathrm{M}}_{k}(\underline{\textsf{K}})$$ $$\mathcal{P}_{k}(\underline{\textsf{K}})$$ $$\mathcal{P}_{k}(\underline{\textsf{F}})^{d}$$ Element $${\mathcal{G}}^{\mathrm{r}\mathrm{o}\mathrm{w}}(\underline{\textsf{K}})$$ $${\boldsymbol V}(\underline{\textsf{K}})$$ $${Q}(\underline{\textsf{K}})$$ $${\boldsymbol{M}}(\underline{\textsf{F}})$$ Simplex $$\mathcal{P}_{k}(\underline{\textsf{K}})^{d}$$ $$\boldsymbol{\mathrm{R}\mathrm{T}}_{k}(\underline{\textsf{K}})$$ $$\mathcal{P}_{k}(\underline{\textsf{K}})$$ $$\mathcal{P}_{k}(\underline{\textsf{F}})^{d}$$ Hypercube $$\boldsymbol{\mathrm{B}\mathrm{D}\mathrm{M}}_{k}(\underline{\textsf{K}})$$ $$\boldsymbol{\mathrm{B}\mathrm{D}\mathrm{F}\mathrm{M}}_{k}(\underline{\textsf{K}})$$ $$\mathcal{P}_{k}(\underline{\textsf{K}})$$ $$\mathcal{P}_{k}(\underline{\textsf{F}})^{d}$$ Here we denote by $$\mathcal{P}_{k}(D)$$ and $$\widetilde{\mathcal{P}}_{k}(D)$$ the polynomials of degree no greater than k, and homogeneous polynomials of degree k, respectively, on the domain D. The vector space $$\boldsymbol{\mathrm{R}\mathrm{T}}_{k}(\underline{\textsf{K}})$$ on the reference simplex is the Raviart–Thomas–Nedéléc space (see Raviart & Thomas 1977; Nédélec 1980) $$ \boldsymbol{\mathrm{R}\mathrm{T}}_{k}(\underline{\textsf{K}}):=\mathcal{P}_{k}(\underline{\textsf{K}})^{d}\oplus \mathbf{x}\,\widetilde{\mathcal{P}}_{k}(\underline{\textsf{K}}), $$ the vector space $$\boldsymbol{\mathrm{B}\mathrm{D}\mathrm{M}}_{k}(\underline{\textsf{K}})$$ on the reference hypercube is the Brezzi–Douglas–Marini space, (see Brezzi et al., 1985; Brezzi et al., 1987a;Arnold & Awanou 2014) $$ \boldsymbol{\mathrm{B}\mathrm{D}\mathrm{M}}_{k}(\underline{\textsf{K}}):=\begin{cases} \mathcal{P}_{k}(\underline{\textsf{K}})^{d}\oplus{{\nabla\times}}\{x\,y^{k+1},y\,x^{k+1}\} & \textrm{if}\ d=2, \\ \mathcal{P}_{k}(\underline{\textsf{K}})^{d}\oplus{{\nabla\times}}\left\{ \begin{array}{c} x\,\widetilde{\mathcal{P}}_{k}(y,z)(y{\nabla} z-z{\nabla} y),\\ y\,\widetilde{\mathcal{P}}_{k}(z,x)(z{\nabla} x-x{\nabla} z),\\ z\,\widetilde{\mathcal{P}}_{k}(x,y)(x{\nabla} y-y{\nabla} x)\\ \end{array} \right\}& \textrm{if}\ d=3, \end{cases} $$ and the vector space $$\boldsymbol{\mathrm{B}\mathrm{D}\mathrm{F}\mathrm{M}}_{k}(\underline{\textsf{K}})$$ on the reference hypercube is the Brezzi–Douglas–Fortin–Marini space, (see Brezzi et al., 1987b) $$ \boldsymbol{\mathrm{B}\mathrm{D}\mathrm{F}\mathrm{M}}_{k}(\underline{\textsf{K}}):=\begin{cases} \mathcal{P}_{k}(\underline{\textsf{K}})^{d}\oplus \left[ \begin{array}{c} x\,\widetilde{\mathcal{P}}_{k}(\underline{\textsf{K}}) \\ y\,\widetilde{\mathcal{P}}_{k}(\underline{\textsf{K}}) \end{array} \right] & \textrm{if}\ d=2, \\ \mathcal{P}_{k}(\underline{\textsf{K}})^{d}\oplus \left[ \begin{array}{c} x\,\widetilde{\mathcal{P}}_{k}(\underline{\textsf{K}}) \\ y\,\widetilde{\mathcal{P}}_{k}(\underline{\textsf{K}}) \\ z\,\widetilde{\mathcal{P}}_{k}(\underline{\textsf{K}}) \end{array} \right] & \textrm{if}\ d=3. \end{cases} $$ Next, for the vector-valued finite element space $${\mathcal{G}}^{\mathrm{r}\mathrm{o}\mathrm{w}}(K)$$ given in (2.2a), we denote by \begin{align} {\mathcal{G}}(K):= \left[{\mathcal{G}}^{\mathrm{r}\mathrm{o}\mathrm{w}}(K)\right]^{d} \end{align} (2.3) the tensor-valued space such that each row is the space $${\mathcal{G}}^{\mathrm{r}\mathrm{o}\mathrm{w}}(K)$$. We use the following finite element spaces on the mesh $${\mathcal{T}_{h}}$$ and mesh skeleton $$\mathcal{E}_{h}$$ to define the H(div)-conforming HDG method in the next section: \begin{align} {\mathcal{G}}_{h}:=\ \left\{\mathrm{g}\in L^{2}({\mathcal{T}_{h}})^{d\times d}:\quad\mathrm{g}|_{K}\in{\mathcal{G}}(K),\;\; K\in{\mathcal{T}_{h}}\right\}\!, \end{align} (2.4a) \begin{align}{\boldsymbol{V}}_{h}:=\, \left\{\mathbf{v}\in L^{2}({\mathcal{T}_{h}})^{d}:\quad\quad\mathbf{v}|_{K} \in{\textbf V}(K),\;\; K\in{\mathcal{T}_{h}}\right\}\!,\end{align} (2.4b) \begin{align} {\textbf V}_{h}^{\textrm{div}}:=\; \left\{\mathbf{v}\in{\textbf V}_{h}:\,\,\quad\quad\quad\quad\mathbf{v}\in H(\textrm{div};\varOmega)\right\}\!, \end{align} (2.4c) \begin{align}{\boldsymbol V}_{h}^{\textrm{div}}(0):=\; \left\{\mathbf{v}\in{\textbf V}_{h}^{\textrm{div}}:\,\,\, \quad\quad\quad\textrm{tr}_{n}(\mathbf{v})|_{\partial \varOmega}=0 \right\}\!, \end{align} (2.4d) \begin{align} Q_{h}:=\;\left\{q\in L^{2}({\mathcal{T}_{h}}): q|_{K}\in{Q}(K),\;\; K\in{\mathcal{T}_{h}}\right\}\!, \end{align} (2.4e) \begin{align} \mathring{Q_{h}}:=\;\left\{q\in Q_{h}:\qquad\qquad ( q\,,\,1 )_{{\mathcal{T}_{h}}} = 0 \right\}\!,\end{align} (2.4f) \begin{align}{\boldsymbol{M}}_{h}:=\ \left\{\widehat{\mathbf{v}}\in L^{2}(\mathcal{E}_{h})^{d}:\qquad\;\widehat{\mathbf{v}}|_{F}\in{\boldsymbol{M}}(F),\quad F\in\mathcal{E}_{h}\right\}\!, \end{align} (2.4g) \begin{align} \boldsymbol{M}_{h}(0):=\;\left\{\widehat{\mathbf{v}}\in{\boldsymbol{M}}_{h}:\qquad\qquad\,\widehat{\mathbf{v}}|_{\partial\varOmega} = \mathbf{0} \right\}\!, \end{align} (2.4h) \begin{align} {\boldsymbol{M}}^{t}_{h}:=\;\left\{\widehat{\mathbf{v}}\in \boldsymbol{M}_{h}:\textrm{tr}_{n}(\widehat{\mathbf{v}})|_{F}=0,\;\; F\in\mathcal{E}_{h}\right\}\!, \end{align} (2.4i) \begin{align} {\boldsymbol{M}}^{t}_{h}(0):=\, \left\{\widehat{\mathbf{v}}\in{\boldsymbol{M}}^{t}_{h}:\qquad\;\;\;\;\;\;\;\textrm{tr}_{t}(\widehat{\mathbf{v}})|_{\partial \varOmega}=\mathbf{0}\right\}\!.\end{align} (2.4j) 2.3. The H(div)-conforming HDG method Now, we are ready to present the H(div)-conforming HDG method for the Brinkman equations (1.1). It is defined as the unique element $$(\mathrm{L}^{h},\mathbf \;{u}^{h},\;p^{h},\;{\widehat{\mathbf{u}}_{t}}^{h})\in{\mathcal{G}}_{h}\times{\boldsymbol V}_{h}^{\textrm{div}}(0)\times \mathring{{Q}_{h}}\times{\boldsymbol{M}}^{t}_{h}(0)$$ such that the following weak formulation holds: \begin{align} ( \mathrm{L}^{h},\,\nu\,\mathrm{g}^{h} )_{{\mathcal{T}_{h}}}-( {\boldsymbol{\nabla}} \mathbf{u}^{h},\,\nu\,\mathrm{g}^{h} )_{{\mathcal{T}_{h}}} + \left\langle \textrm{tr}_{t}(\mathbf{u}^{h})-\widehat{\mathbf{u}}_{t}^{h},\, \textrm{tr}_{t}(\nu\,\mathrm{g}^{h}\, \mathbf{n}) \right\rangle_{\partial{{\mathcal{T}_{h}}}} = 0, \end{align} (2.5a) \begin{align} ( \nu\,\mathrm{L}^{h},\,{\boldsymbol{\nabla}} \mathbf{v}^{h} )_{{\mathcal{T}_{h}}} - \left\langle \textrm{tr}_{t}(\nu\,\mathrm{L}^{h}\,\mathbf{n}),\,\textrm{tr}_{t}(\mathbf{v}^{h})-\widehat{\mathbf{v}}_{t}^{h} \right\rangle_{\partial{{\mathcal{T}_{h}}}} \end{align} (2.5b) \begin{align} - ( p^{h},\,{\nabla\cdot} \mathbf{v}^{h} )_{{\mathcal{T}_{h}}}+( \gamma\, \mathbf{u}^{h},\,\mathbf{v}^{h} )_{{\mathcal{T}_{h}}}&=(\mathbf{f},\mathbf{v}^{h})_{\mathcal{T}_{h}},\nonumber \\ ( {\nabla\cdot}\mathbf{u}^{h},\,q^{h} )_{{\mathcal{T}_{h}}} & = (g, q^{h})_{\mathcal{T}_{h}}, \end{align} (2.5c) for all $$(\mathrm{g}^{h},\mathbf{v}^{h},q^{h},{\widehat{\mathbf{v}}_{t}}^{h})\in{\mathcal{G}}_{h}\times{\boldsymbol V}_{h}^{\textrm{div}}(0)\times \mathring{{Q}_{h}}\times{\boldsymbol{M}}^{t}_{h}(0)$$. Here we write $$(\eta, \, \zeta )_{{\mathcal{T}_{h}}} := \sum _{K \in{\mathcal{T}_{h}}} (\eta , \zeta )_{K},$$ where (η, ζ)K denotes the integral of ηζ over the domain $$K \subset \mathbb{R}^{n}$$. We also write $$\langle \eta,\,\zeta \rangle _{\partial{{\mathcal{T}_{h}}}}:= \sum _{K \in{\mathcal{T}_{h}}}\langle \eta,\,\zeta \rangle _{{\partial K}}$$, where $$\langle \eta,\,\zeta \rangle _{{\partial K}}:=\sum _{F \in \mathcal{F}(K)} \langle \eta,\,\zeta \rangle _{F},$$ and ⟨η, ζ⟩F denotes the integral of ηζ over the facet $$F \subset \mathbb{R}^{n-1}$$ and where $$\partial{\mathcal{T}_{h}} := \{ \partial K: K \subset{\mathcal{T}_{h}} \}$$. When vector-valued or tensor-valued functions are involved, we use similar notation. We specifically remark that, when γ = 0 and g = 0 in $$\varOmega$$, our method on simplicial meshes is identical to the one for the Stokes equations introduced in the study by Cockburn & Sayas (2014). As mentioned in the introduction, we postpone to Section 4 discussing the efficient implementation of the above method via hybridization. Here we focus on the presentation of its (superconvergent) a priori error estimates. 2.3.1. Discrete H1-stability We first obtain a key result, which will be used to prove the error estimates presented in the Section 2.3.3, on the control of a discrete H1-norm of the pair $$(\mathbf{u}^{h}, \widehat{\mathbf{u}}_{t}^{h})\in{\boldsymbol V}_{h}^{\textrm{div}}\times{\boldsymbol{M}}^{t}_{h}$$ by the L2-norm of a tensor field. For a pair $$(\mathbf{v}^{h}, \widehat{\mathbf{v}}_{t}^{h})\in{\boldsymbol V}_{h}^{\textrm{div}}\times{\boldsymbol{M}}^{t}_{h}$$, we denote its discrete H1-norm as: \begin{align} {\left\vert\kern-0.25ex\left\vert\kern-0.25ex\left\vert (\mathbf{u}^{h}, \widehat{\mathbf{u}}_{t}^{h}) \right\vert\kern-0.25ex\right\vert\kern-0.25ex\right\vert}_{1,{\mathcal{T}_{h}}} :=\left( \sum_{K\in{\mathcal{T}_{h}}} \|{\boldsymbol{\nabla}} \mathbf{u}^{h}\|_{K}^{2} + \sum_{F\in\mathcal{E}_{h}}h_{F}^{-1}\| \textrm{tr}_{t}(\mathbf{u}^{h})-\widehat{\mathbf{u}}_{t}^{h}\|_{F}^{2} \right)^{1/2} .\end{align} (2.6) Theorem 2.1 (Discrete H1-stability). Let $$(\mathrm{r}, \mathbf{z}^{h},\; \widehat{\mathbf{z}}_{t}^{h})\in L^{2}({\mathcal{T}_{h}})^{d\times d}\times{\boldsymbol V}_{h}^{\textrm{div}}\times{\boldsymbol{M}}^{t}_{h}\ $$ satisfy the following equation \begin{align} ( \mathrm{r},\,\mathrm{g}^{h} )_{{\mathcal{T}_{h}}}-( {\boldsymbol{\nabla}} \mathbf{z}^{h},\,\mathrm{g}^{h} )_{{\mathcal{T}_{h}}} + \left\langle \textrm{tr}_{t}(\mathbf{z}^{h})-\widehat{\mathbf{z}}_{t}^{h},\, \textrm{tr}_{t}(\mathrm{g}^{h}\, \mathbf{n}) \right\rangle_{\partial{{\mathcal{T}_{h}}}} = 0 \end{align} (2.7) for all $$\mathrm{g}^{h}\in{\mathcal{G}}_{h}$$; then we have \begin{align} {\left\vert\kern-0.25ex\left\vert\kern-0.25ex\left\vert \left(\mathbf{z}^{h},\; \widehat{\mathbf{z}}_{t}^{h}\right) \right\vert\kern-0.25ex\right\vert\kern-0.25ex\right\vert}_{1,{\mathcal{T}_{h}}}\le C\,\|\mathrm{r}\|_{\mathcal{T}_{h}}, \end{align} (2.8) with a constant C depending only on the polynomial degree k and the shape regularity of the elements $$K\in{\mathcal{T}_{h}}$$. Here $$\Vert \cdot \Vert _{{\mathcal{T}_{h}}}$$ is the standard L2-norm on $$\varOmega$$. 2.3.2. Well-posedness of the HDG method Theorem 2.2 shows the well-posedness of the HDG method (2.5), which is a direct consequence of Theorem 2.4. Theorem 2.2 For any $$(\mathbf{f}, g) \in L^{2}(\varOmega )^{d} \times{L_{0}^{2}}(\varOmega )$$, the HDG method (2.5) has a unique solution $$(\mathrm{L}^{h},\mathbf{u}^{h},p^{h},{\widehat{\mathbf{u}}_{t}}^{h})\in{\mathcal{G}}_{h}\times{\boldsymbol V}_{h}^{\textrm{div}}(0)\times \mathring{{Q}_{h}}\times{\boldsymbol{M}}^{t}_{h}(0)$$. 2.3.3. A priori error estimates We are now ready to present the a priori error estimates for the method (2.5). We compare the numerical solution against suitably chosen projections. The projections In the following, we denote by $$P_{\mathcal{G}}$$, PV, PQ and $$P_{\boldsymbol{M}^{t}}$$ the L2-projections onto $${\mathcal{G}}_{h}$$, Vh, $$\mathring{{Q}_{h}} $$ and $${\boldsymbol{M}}^{t}_{h}$$, respectively. Moreover, we set \begin{align*} \mathrm{e}_{L} &= P_{\mathcal{G}} \mathrm{L} - \mathrm{L}^{h},\;\; \mathbf{e}_{u} = \Pi_{{\boldsymbol V}} \mathbf{u} - \mathbf{u}^{h}, \;\; {e}_{p} = P_{Q} p - p^{h}, \;\;\mathbf{e}_{\widehat u_{t}} = P_{\boldsymbol{M}^{t}} \mathbf{u} - \widehat{\mathbf{u}}_{t}^{h},\\{\delta}_{L} &= \mathrm{L} -P_{\mathcal{G}} \mathrm{L}, \;\; \boldsymbol{\delta}_{u} \;= \mathbf{u} - \Pi_{{\boldsymbol V}} \mathbf{u}, \;\; {\delta}_{p} = p -P_{Q} p, \;\;\boldsymbol{\delta}_{\widehat u_{t}} \;=\textrm{tr}_{t}(\mathbf{u}) - P_{\boldsymbol{M}^{t}} \mathbf{u}. \end{align*} Here the projection ΠVu ∈ Vh whose restriction to an element K is the unique function in V(K) such that \begin{align} (\Pi_{{\boldsymbol V}} \mathbf{u}, \mathbf{v})_{K} = \; (\mathbf{u},\mathbf{v})_{K} \quad \forall\,\mathbf{v}\in{\boldsymbol{\nabla\cdot}} {\mathcal{G}}(K), \end{align} (2.9a) \begin{align} \left\langle \textrm{tr}_{n}(\Pi_{{\boldsymbol V}} \mathbf{u}),\,\textrm{tr}_{n}(\widehat{\mathbf{v}}) \right\rangle_{F} = \; \left\langle \textrm{tr}_{n}(\mathbf{u}),\,\textrm{tr}_{n}(\widehat{\mathbf{v}}) \right\rangle_{F} \quad \forall\, \widehat{\mathbf{v}}\in{\boldsymbol{M}}(F),\quad \forall\, F\in \mathcal{F}(K). \end{align} (2.9b) Recall that the spaces V(K), M(F) and $${\mathcal{G}}(K)$$ are defined in (2.2) and (2.3), respectively. When K is a simplex, the above projection is nothing but the Raviart–Thomas projection (see Raviart & Thomas, 1977: Nédélec, 1980); when K is a hypercube, the above projection is nothing but the Brezzi–Douglas–Fortin–Marini projection (see Brezzi et al., 1987b). The following approximation property of the above projection is well known; see Boffi et al. (2013, Chapter 2). Lemma 2.3 There exists a unique function $$\Pi _{{\boldsymbol V}} \mathbf{u}\in{\boldsymbol V}_{h}^{\textrm{div}}$$ defined elementwise by equations (2.9). Moreover, there exists a constant C depending only on the polynomial degree and shape regularity of the elements $$K\in{\mathcal{T}_{h}}$$ such that \begin{align} \|\Pi_{{\boldsymbol V}} \mathbf{u} - \mathbf{u}\|_{\mathcal{T}_{h}}\le \;C\, \left(\|P_{{\boldsymbol V}} \mathbf{u} - \mathbf{u}\|_{\mathcal{T}_{h}} +\sum_{K\in{\mathcal{T}_{h}}}h_{K}^{1/2}\|P_{{\boldsymbol V}} \mathbf{u} -\mathbf{u}\|_{\partial K} \right). \end{align} (2.10) The projection errors Now, we state our main results on the superconvergent error estimates. Theorem 2.4 Let $$(\mathrm{L}^{h},\mathbf{u}^{h},p^{h}, \widehat{\mathbf{u}}_{t}^{h})\in{\mathcal{G}}_{h}\times{\boldsymbol V}_{h}^{\textrm{div}}(0)\times \mathring{{Q}_{h}}\times{\boldsymbol{M}}^{t}_{h}(0)$$ be the numerical solution of (2.5); then there exists a constant C, depending only on the polynomial degree k, the shape regularity of the mesh $${\mathcal{T}_{h}}$$ and the domain Ω, such that \begin{align}{2} \|\mathbf{e}_{u}\|_{{\mathcal{T}_{h}}} \le \;C\, {\left\vert\kern-0.25ex\left\vert\kern-0.25ex\left\vert (\mathbf{e}_{u},\mathbf{e}_{\widehat u_{t}}) \right\vert\kern-0.25ex\right\vert\kern-0.25ex\right\vert}_{1,{\mathcal{T}_{h}}}, \end{align} (2.11a) \begin{align} {\left\vert\kern-0.25ex\left\vert\kern-0.25ex\left\vert (\mathbf{e}_{u},\mathbf{e}_{\widehat u_{t}}) \right\vert\kern-0.25ex\right\vert\kern-0.25ex\right\vert}_{1,{\mathcal{T}_{h}}} \le \; C\,\|\mathrm{e}_{L}\|_{{\mathcal{T}_{h}}}, \end{align} (2.11b) \begin{align} \nu\|\mathrm{e}_{L}\|_{{\mathcal{T}_{h}}}^{2} +\|\gamma^{1/2}\,\mathbf{e}_{u}\|_{\mathcal{T}_{h}}^{2} \le \;C\,\left(\sum_{F\in\mathcal{E}_{h}}\nu\,h_{F}\|{\delta}_{L}\,\mathbf{n}\|_{F}^{2}+\|\gamma^{1/2}\,\boldsymbol{\delta}_{u}\|_{\mathcal{T}_{h}}^{2} \right)\!\!. \end{align} (2.11c) Combining this result with Lemma 2.3, we immediately obtain optimal convergence of the L2-error for Lh and uh, and superconvergent discrete H1-error for the pair $$(\mathbf{u}^{h},\widehat{\mathbf{u}}_{t}^{h})$$ comparing with the projection $$(\Pi _{{\boldsymbol V}} \mathbf{u}, P_{\mathbf{M}^{t}} \mathbf{u})$$; see the following corollary. We omit the proof due to its simplicity. We specifically remark that the errors below are independent of the regularity of the pressure. Corollary 2.5 Let $$(\mathrm{L}^{h},\mathbf{u}^{h},p^{h}, \widehat{\mathbf{u}}_{t}^{h})\in{\mathcal{G}}_{h}\times{\boldsymbol V}_{h}^{\textrm{div}}(0)\times \mathring{{Q}_{h}}\times{\boldsymbol{M}}^{t}_{h}(0)$$ be the numerical solution of (2.5); then there exists a constant C, depending only on the polynomial degree k, the shape regularity of the mesh $${\mathcal{T}_{h}}$$ and the domain $$\varOmega$$, such that $$ \nu^{1/2}\left(\|\mathrm{e}_{L}\|_{{\mathcal{T}_{h}}} +{\left\vert\kern-0.25ex\left\vert\kern-0.25ex\left\vert (\mathbf{e}_{u},\mathbf{e}_{\widehat u_{t}}) \right\vert\kern-0.25ex\right\vert\kern-0.25ex\right\vert}_{1,{\mathcal{T}_{h}}}\right) + \max\{ \nu^{1/2}\|\mathbf{e}_{u}\|_{\mathcal{T}_{h}}, \|\gamma^{1/2}\,\mathbf{e}_{u}\|_{\mathcal{T}_{h}}\} \le \;C\,\varTheta\, h^{k+1}, $$ where $$ \varTheta:= \nu^{1/2}\,\|\mathrm{L}\|_{k+1,\varOmega}+ \gamma_{\max}^{1/2}\,\|\mathbf{u}\|_{k+1,\varOmega}, $$ and $$\gamma _{\max }$$ is the maximum eigenvalue of the inverse permeability tensor γ, and ∥⋅∥m, denotes the Hm-norm on $$\varOmega$$. Next we obtain optimal L2-estimates for pressure for k ≥ 0 and superconvergent L2-estimates for the projection error eu for k ≥ 1 (with an H2-regularity assumption for the dual problem). We assume that the regularity estimate \begin{align} \|\varPhi\|_{1,\varOmega}+\|\phi\|_{2,\varOmega}+\|\varphi\|_{1,\varOmega}\le C_{r}\|\boldsymbol{\theta}\|_{\varOmega} \end{align} (2.12) holds for the dual problem \begin{align} \varPhi-{\boldsymbol{\nabla}}\phi=0\;\qquad \textrm{in}\;\varOmega, \end{align} (2.13a) \begin{align} -\nu{\boldsymbol{\nabla\cdot}} \varPhi+\gamma\phi-\nabla \varphi=\boldsymbol{\theta}\;\qquad\textrm{in}\;\varOmega, \end{align} (2.13b) \begin{align} {\boldsymbol{\nabla\cdot}}\phi=0\;\qquad \textrm{in}\;\varOmega, \end{align} (2.13c) \begin{align} \phi=0\;\qquad \textrm{on}\; \partial\varOmega. \end{align} (2.13d) We notice that it is easy to see that the dual problem (2.13) is well posed. Obviously, ($$\varPhi$$, ϕ, φ) is the solution of the Stokes problem with the source term θ − γϕ. So, the regularity estimate (2.12) comes from that of the Stokes problem (see Girault & Raviart, 1986). Theorem 2.6 Let $$(\mathrm{L}^{h},\mathbf{u}^{h},p^{h}, \widehat{\mathbf{u}}_{t}^{h})\in{\mathcal{G}}_{h}\times{\boldsymbol V}_{h}^{\textrm{div}}(0)\times{\mathring{Q}_{h}}\times{\boldsymbol{M}}^{t}_{h}(0)$$ be the numerical solution of (2.5); then there exists a constant C, depending only on the polynomial degree k, the shape regularity of the mesh $${\mathcal{T}_{h}}$$ and the domain $$\varOmega$$, such that \begin{align} \|{e}_{p}\|_{{\mathcal{T}_{h}}}\le C\left(\nu^{1/2}+\gamma_{\max}^{1/2}\right)\,\varTheta\,h^{k+1}. \end{align} (2.14) Here $$\gamma _{\max }$$ and $$\varTheta $$ are defined in Corollary 2.5. In addition, if k ≥ 1, the regularity assumption (2.12) holds and $$\gamma \in W^{1,\infty }(\varOmega )^{d\times d}$$, then we have \begin{align} \|\mathbf{e}_{u}\|_{{\mathcal{T}_{h}}}\le C\,C_{r}\,\left(\left(\nu^{1/2}+\gamma_{\max}^{1/2}\right)\,\varTheta+ \|\gamma\|_{1,\infty}\|\mathbf{u}\|_{k+1}\right)h^{k+2}. \end{align} (2.15) 3. Proofs of Theorems 2.1, 2.4 and 2.6 In this section, we prove the main results in Section 2, namely, Theorems 2.1, 2.4 and 2.6. The following results, Lemmas 3.1 and 3.2, are key ingredients to prove Theorem 2.1. The proof of Lemma 3.1 comes directly from Lemma 2.3 and the usual scaling argument. Lemma 3.1 Given $$(\mathrm{r}^{h},\; \widehat{\mathbf{z}}^{h})\in{\mathcal{G}}(K)\times{\boldsymbol{M}}({\partial K})$$ where $$ {\boldsymbol{M}}({\partial K}):=\left\{\widehat{\mathbf{v}}\in L^{2}({\partial K})^{d}:\;\widehat{\mathbf{v}}|_{F}\in{\boldsymbol{M}}(F)\quad \forall \, F\in\mathcal{F}(K)\right\}\!, $$ there exists a unique function wh ∈ V(K) such that \begin{align*} ({\mathbf{w}^{h}},{\mathbf{v}^{h}})_{K} = &\; ({{\boldsymbol{\nabla\cdot}} \mathrm{r}^{h}},{\mathbf{v}^{h}})_{K}\quad \forall \, \mathbf{v}^{h}\in{\boldsymbol{\nabla\cdot}}{\mathcal{G}}(K),\\ \left\langle \textrm{tr}_{n}(\mathbf{w}^{h}),\,\mathrm{t}\mathrm{r}_{n}(\widehat{\mathbf{v}}) \right\rangle_{\partial{K}} = &\; \left\langle \textrm{tr}_{n}(\widehat{\mathbf{z}}^{h}),\,\textrm{tr}_{n}(\widehat{\mathbf{v}}) \right\rangle_{\partial{K}}\quad \forall \, \widehat{\mathbf{v}}^{h}\in{\boldsymbol{M}}({\partial K}). \end{align*} Moreover, there exists a constant C depending only on the shape regularity of the element K such that \begin{align} \|\mathbf{w}^{h}\|_{K}\le C\left(\|{\boldsymbol{\nabla\cdot}} \mathrm{r}^{h}\|_{K}^{2} + \sum_{F\in\mathcal{F}(K)}h_{F}\|\textrm{tr}_{n}(\,\widehat{\mathbf{z}}^{h})\|_{F}^{2}\right)^{1/2} .\end{align} (3.1) Lemma 3.2 Given $$(\mathbf{z}^{h},\; \widehat{\mathbf{z}}^{h})\in{\boldsymbol V}(K)\times{\boldsymbol{M}}({\partial K})$$ where $$ {\boldsymbol{M}}({\partial K}):=\left\{\widehat{\mathbf{v}}\in L^{2}({\partial K})^{d}:\;\widehat{\mathbf{v}}|_{F}\in{\boldsymbol{M}}(F)\quad \forall \, F\in\mathcal{F}(K)\right\}\!, $$ there exists a unique function $$\mathrm{r}^{h}\in{\mathcal{G}}(K)$$ such that \begin{align} ({\mathrm{r}^{h}},{\mathrm{g}^{h}})_{K} = \; ({{\boldsymbol{\nabla}} \mathbf{z}^{h}},{\mathrm{g}^{h}})_{K}\,\quad \forall \, \mathrm{g}^{h}\in{\boldsymbol{\nabla}}{\boldsymbol V}(K)\oplus{\mathcal{G}}_{\mathrm{s}\mathrm{b}\mathrm{b}}(K), \end{align} (3.2a) \begin{align} \left\langle \textrm{tr}_{t}(\mathrm{r}^{h}\,\mathbf{n}),\,\textrm{tr}_{t}(\widehat{\mathbf{v}}) \right\rangle_{\partial{K}} = \; \left\langle \textrm{tr}_{t}(\widehat{\mathbf{z}}^{h}),\,\textrm{tr}_{t}(\widehat{\mathbf{v}}) \right\rangle_{\partial{K}}\quad \forall \, \widehat{\mathbf{v}}^{h}\in{\textbf{M}}({\partial K}), \end{align} (3.2b) where $$ {\mathcal{G}}_{\mathrm{s}\mathrm{b}\mathrm{b}}(K):=\left\{\mathrm{g}\in{\mathcal{G}}(K):\;\;{\boldsymbol{\nabla\cdot}} \mathrm{g} = 0,\;\;\textrm{tr}_{n}^{F}(\mathrm{g}\,\mathbf{n})=0 \quad \forall \, F\in\mathcal{F}(K)\right\}\!. $$ Moreover, there exists a constant C depending only on the shape regularity of the element K such that \begin{align} \|\mathrm{r}^{h}\|_{K}\le C\left(\|{\boldsymbol{\nabla}} \mathbf{z}^{h}\|_{K}^{2} + \sum_{F\in\mathcal{F}(K)}h_{F}\|\textrm{tr}_{t}(\,\widehat{\mathbf{z}}^{h})\|_{F}^{2}\right)^{1/2}. \end{align} (3.3) Proof. We prove only the existence and uniqueness of the function $$\mathrm{r}^{h}\in{\mathcal{G}}(K)$$ satisfying equations (3.2) on the reference element $$K=\underline{\textsf{K}}$$; the result on an affine-mapped element K can be easily obtained from that on the reference element (cf. Boffi et al., 2013, Chapter 2), and the estimate (3.3) is a direct consequence of the usual scaling argument and equivalence of norms on finite-dimensional spaces. We first show that (3.2) defines a square system. We use the concept of an M-decomposition (Cockburn & Fu, 2017a,b; Cockburn et al., 2017) to prove it. By the choice of $${\mathcal{G}}^{\mathrm{r}\mathrm{o}\mathrm{w}}(K)$$ in Table 1, we obtain that the pair $$ {\mathcal{G}}^{\mathrm{r}\mathrm{o}\mathrm{w}}(K)\times \mathcal{P}_{k}(K)$$ admits an M-decomposition with the trace space $$ M({\partial K}):=\left\{\widehat{w}\in L^{2}({\partial K}):\;\;\widehat{w}|_{F}\in\mathcal{P}_{k}(F)\quad \forall \, F\in\mathcal{F}(K)\right\}\!. $$ Hence \begin{align*} \dim{\mathcal{G}}^{\mathrm{r}\mathrm{o}\mathrm{w}}(K)+ \dim\mathcal{P}_{k}(K) = &\; \dim{\mathcal{G}}^{\mathrm{r}\mathrm{o}\mathrm{w}}_{\mathrm{s}\mathrm{b}\mathrm{b}}(K) + \dim{\nabla\cdot}{\mathcal{G}}^{\mathrm{r}\mathrm{o}\mathrm{w}}(K)\\ & +\dim{\nabla} \mathcal{P}_{k}(K) +\dim M({\partial K}). \end{align*} Here $$ {\mathcal{G}}^{\mathrm{r}\mathrm{o}\mathrm{w}}_{\mathrm{s}\mathrm{b}\mathrm{b}}(K):= \left \{ \mathbf{v}\in{\mathcal{G}}^{\mathrm{r}\mathrm{o}\mathrm{w}}(K):\;{\nabla \cdot } \mathbf{v}=0,\; \textrm{tr}_{n}(\mathbf{v})=0\ \textrm{on}\ {\partial K} \right \}. $$ This immediately implies that \begin{align} \dim{\mathcal{G}}(K)+ \dim\mathcal{P}_{k}(K)^{d} = & \dim{\mathcal{G}}_{\mathrm{s}\mathrm{b}\mathrm{b}}(K) + \dim{\boldsymbol{\nabla\cdot}}{\mathcal{G}}(K)\\ & +\dim{\boldsymbol{\nabla}} \mathcal{P}_{k}(K)^{d} +\dim{\boldsymbol{M}}({\partial K}).\nonumber \end{align} (3.4) By Lemma 2.3, we have $$ \dim{\boldsymbol V}(K) = \dim{\boldsymbol{\nabla\cdot}}{\mathcal{G}}(K)+\dim \textrm{tr}_{n}({\boldsymbol{M}}({\partial K})). $$ Combining the above equality with (3.4) and reordering the terms, we get \begin{align} \dim{\mathcal{G}}(K) = & \dim{\mathcal{G}}_{\mathrm{s}\mathrm{b}\mathrm{b}}(K) +\dim \textrm{tr}_{t}({\boldsymbol{M}}({\partial K})) \\ & +\dim{\boldsymbol V}(K)- \dim\mathcal{P}_{k}(K)^{d} +\dim{\boldsymbol{\nabla}} \mathcal{P}_{k}(K)^{d}.\nonumber \end{align} (3.5) Since it is trivial to prove that $$ \dim{\boldsymbol V}(K)- \dim\mathcal{P}_{k}(K)^{d} +\dim{\boldsymbol{\nabla}} \mathcal{P}_{k}(K)^{d} = \dim{\boldsymbol{\nabla}} {\boldsymbol V}(K) $$ for the vector space V(K) in Table 1, we conclude that equation (3.2) is indeed a square system. Hence, we are left to prove the uniqueness. To this end, we take $$\mathbf{z}^{h}=0, \widehat{\mathbf{z}}^{h}=0$$ in (3.2). By (3.2b), we have \begin{align} \textrm{tr}_{t}(\mathrm{r}^{h}\mathbf{n})=0. \end{align} (3.6) By (3.2a), we have, for all v ∈ V(K), \begin{align*} 0= (\mathrm{r}^{h}, {\boldsymbol{\nabla}} \mathbf{v})_{K} &= -({\boldsymbol{\nabla\cdot}}\mathrm{r}^{h}, \mathbf{v})_{K} +\left\langle \textrm{tr}_{n}(\mathrm{r}^{h}\mathbf{n}),\,\textrm{tr}_{n}(\mathbf{v}) \right\rangle_{\partial{K}} +\left\langle \textrm{tr}_{t}(\mathrm{r}^{h}\mathbf{n}),\,\textrm{tr}_{t}(\mathbf{v}) \right\rangle_{\partial{K}}\\ &= -({\boldsymbol{\nabla\cdot}}\mathrm{r}^{h}, \mathbf{v})_{K} +\left\langle \textrm{tr}_{n}(\mathrm{r}^{h}\mathbf{n}),\,\textrm{tr}_{n}(\mathbf{v}) \right\rangle_{\partial{K}}. \end{align*} Then, by Lemma 3.1, there exists a function v ∈ V(K) such that $$ -({\boldsymbol{\nabla\cdot}}\mathrm{r}^{h}, \mathbf{v})_{K} +\left\langle \textrm{tr}_{n}(\mathrm{r}^{h}\mathbf{n}),\,\textrm{tr}_{n}(\mathbf{v}) \right\rangle_{\partial{K}} =({\boldsymbol{\nabla\cdot}}\mathrm{r}^{h}, {\boldsymbol{\nabla\cdot}}\mathrm{r}^{h})_{K} +\left\langle \textrm{tr}_{n}(\mathrm{r}^{h}\mathbf{n}),\,\textrm{tr}_{n}(\mathrm{r}^{h}\mathbf{n}) \right\rangle_{\partial{K}}.$$ Hence ∇⋅rh = 0 and trn(rhn) = 0. This implies that $$\mathrm{r}^{h}\in{\mathcal{G}}_{\mathrm{s}\mathrm{b}\mathrm{b}}(K)$$. Then taking $$\mathrm{g}^{h} :=\mathrm{r}^{h}\in{\mathcal{G}}_{\mathrm{s}\mathrm{b}\mathrm{b}}(K)$$ in (3.2a), we conclude that rh = 0. This concludes the proof of Lemma 3.2. Now, we are ready to prove Theorem 2.1. 3.1. Proof of Theorem 2.1 Proof. By Lemma 3.2, for any zh ∈ V(K) and $$\widehat{\mathbf{z}}_{t}^{h}\in \{\widehat{\mathbf{v}}\in{\boldsymbol{M}}({\partial K}):\; \textrm{tr}_{n}(\widehat{\mathbf{v}})=0\}$$, there exists $$\mathrm{g}^{h}\in{\mathcal{G}}(K)$$ such that \begin{align*} ({\boldsymbol{\nabla}} \mathbf{z}^{h}, \mathrm{g}^{h})_{K} - \left\langle \textrm{tr}_{t}(\mathbf{z}^{h})-\widehat{\mathbf{z}}_{t}^{h},\,\textrm{tr}_{t}(\mathrm{g}^{h}\,\mathbf{n}) \right\rangle_{\partial{K}} &=\|{\boldsymbol{\nabla}} \mathbf{z}^{h}\|_{K}^{2}\\ &\quad+\sum_{F\in\mathcal{F}(K)}h_{F}^{-1}\|P_{\boldsymbol{M}^{t}}\left(\textrm{tr}_{t}(\mathbf{z}^{h})\right)-\widehat{\mathbf{z}}_{t}^{h}\|_{F}^{2} \end{align*} and $$\|\mathrm{g}^{h}\|_{K}\le C\, \left (\|{\boldsymbol{\nabla }} \mathbf{z}^{h}\|_{K}^{2}+\sum _{F\in \mathcal{F}(K)}h_{F}^{-1}\|P_{\boldsymbol{M}^{t}}(\textrm{tr}_{t}(\mathbf{z}^{h}))-\widehat{\mathbf{z}}_{t}^{h}\|_{F}^{2}\right )^{1/2}$$. Taking such gh in (2.7), we get \begin{align*} \|{\boldsymbol{\nabla}} \mathbf{z}^{h}\|_{K}^{2}+\sum_{F\in\mathcal{F}(K)}h_{F}^{-1}\|P_{\boldsymbol{M}^{t}}(\textrm{tr}_{t}(\mathbf{z}^{h}))-\widehat{\mathbf{z}}_{t}^{h}\|_{F}^{2} &=(\mathrm{r}, \mathrm{g}^{h})_{K}\\ &\le C\, \left(\|{\boldsymbol{\nabla}} \mathbf{z}^{h}\|_{K}^{2}+\sum_{F\in\mathcal{F}(K)}h_{F}^{-1}\|P_{\boldsymbol{M}^{t}}(\textrm{tr}_{t}(\mathbf{z}^{h}))-\widehat{\mathbf{z}}_{t}^{h}\|_{F}^{2}\right)^{1/2}\,\|\mathrm{r}\|_{K}. \end{align*} Hence \begin{align} \left(\|{\boldsymbol{\nabla}} \mathbf{z}^{h}\|_{K}^{2}+\sum_{F\in\mathcal{F}(K)}h_{F}^{-1}\|P_{\boldsymbol{M}^{t}}(\textrm{tr}_{t}(\mathbf{z}^{h}))-\widehat{\mathbf{z}}_{t}^{h}\|_{F}^{2}\right)^{1/2}\le C\,\|\mathrm{r}\|_{K}. \end{align} (3.7) Moreover, on each facet $$F\in \mathcal{F}(K)$$, we have $$ \| \textrm{tr}_{t}(\mathbf{z}^{h}) -P_{\boldsymbol{M}^{t}}(\textrm{tr}_{t}(\mathbf{z}^{h}))\|_{F} = \| \mathbf{z}^{h} -P_{\boldsymbol{M}}(\mathbf{z}^{h})\|_{F} \le \,\| \mathbf{z}^{h} -\overline{\mathbf{z}^{h}}\,\|_{F} \le C\,h_{K}^{1/2}\| {\boldsymbol{\nabla}} \mathbf{z}^{h}\,\|_{K}, $$ where $$\overline{\mathbf{z}^{h}}$$ is the average of zh in the element K and the last inequality is the inverse inequality. Combining the above result with (3.7), we obtain $$ {\left\vert\kern-0.25ex\left\vert\kern-0.25ex\left\vert \left(\mathbf{z}^{h}, \widehat{\mathbf{z}}_{t}^{h}\right) \right\vert\kern-0.25ex\right\vert\kern-0.25ex\right\vert}_{1,K}\le C\,\|\mathrm{r}\|_{K}. $$ The proof of Theorem 2.1 is completed by summing the above estimate over all the elements $$K\in{\mathcal{T}_{h}}$$. We use the following error equation to prove Theorem 2.4. To simplify notation, we denote \begin{align} B_{h}\left(\mathrm{L}, \mathbf{u}, \;p,\;\widehat{\mathbf{u}}_{t};\; \mathrm{g}, \mathbf{v},\; q,\;\widehat{\mathbf{v}}_{t}\right)&:= ( \mathrm{L}\,,\,\nu\,\mathrm{g} )_{{\mathcal{T}_{h}}}-( {\boldsymbol{\nabla}} \mathbf{u}\,,\,\nu\,\mathrm{g} )_{{\mathcal{T}_{h}}} \\ &\quad + \left\langle \textrm{tr}_{t}(\mathbf{u})-\widehat{\mathbf{u}}_{t},\, \textrm{tr}_{t}(\nu\,\mathrm{g}\, \mathbf{n}) \right\rangle_{\partial{{\mathcal{T}_{h}}}} \nonumber\\ &\quad + ( \nu\,\mathrm{L}\,,\,{\boldsymbol{\nabla}} \mathbf{v} )_{{\mathcal{T}_{h}}} - \left\langle \textrm{tr}_{t}(\nu\,\mathrm{L}\,\mathbf{n}),\,\textrm{tr}_{t}(\mathbf{v})-\widehat{\mathbf{v}}_{t} \right\rangle_{\partial{{\mathcal{T}_{h}}}}\nonumber\\ &\quad- ( p,\,{\nabla\cdot} \mathbf{v} )_{{\mathcal{T}_{h}}}+( \gamma\, \mathbf{u},\,\mathbf{v} )_{{\mathcal{T}_{h}}}\nonumber\\ &\quad + ( {\nabla\cdot}\mathbf{u},\,q )_{{\mathcal{T}_{h}}}.\nonumber \end{align} (3.8) Lemma 3.3 Let (L, u, p) be the solution to (1.1), and $$(\mathrm{L}^{h},\; \mathbf{u}^{h}, \;p^{h},\;\widehat{\mathbf{u}}_{t}^{h})$$ be the numerical solution to (2.5). Then we have \begin{align} B_{h}\left(\mathrm{e}_{L},\; \mathbf{e}_{u},\; {e}_{p},\;\mathbf{e}_{\widehat u_{t}}; \mathrm{g}^{h},\; \mathbf{v}^{h}, q^{h},\;\widehat{\mathbf{v}}_{t}^{h}\right) &= \left\langle \textrm{tr}_{t}(\nu\,{\delta}_{L}\,\mathbf{n}),\,\textrm{tr}_{t}(\mathbf{v}^{h})-\widehat{\mathbf{v}}_{t}^{h} \right\rangle_{\partial{{\mathcal{T}_{h}}}}\\[-5pt] & \quad-( \gamma\,\boldsymbol{\delta}_{u},\,\mathbf{v}^{h} )_{{\mathcal{T}_{h}}}\nonumber \end{align} (3.9) for all $$(\mathrm{g}^{h},\mathbf{v}^{h},q^{h},{\widehat{\mathbf{v}}_{t}}^{h})\in{\mathcal{G}}_{h}\times{\boldsymbol V}_{h}^{\textrm{div}}(0)\times \mathring{{Q}_{h}}\times{\boldsymbol{M}}^{t}_{h}(0)$$. Proof. By (1.1), (2.5) and (3.8), we have \begin{align*} B_{h}\left(\mathrm{L}^{h},\; \mathbf{u}^{h},\; p^{h},\widehat{\mathbf{u}}_{t}^{h};\; \mathrm{g}^{h}, \mathbf{v}^{h},\; q^{h},\;\widehat{\mathbf{v}}_{t}^{h}\right) &=(\mathbf{f},\mathbf{v}^{h})_{\mathcal{T}_{h}} + (g,q^{h})_{\mathcal{T}_{h}},\\ B_{h}\left(\mathrm{L},\; \mathbf{u},\; p,\;\textrm{tr}_{t}(\mathbf{u});\; \mathrm{g}^{h}, \mathbf{v}^{h},\; q^{h},\;\widehat{\mathbf{v}}_{t}^{h}\right) &=(\mathbf{f},\mathbf{v}^{h})_{\mathcal{T}_{h}} + (g,q^{h})_{\mathcal{T}_{h}}, \end{align*} for all $$(\mathrm{g}^{h},\;\mathbf{v}^{h},\;q^{h},\;{\widehat{\mathbf{v}}_{t}}^{h})\in{\mathcal{G}}_{h}\times{\boldsymbol V}_{h}^{\textrm{div}}(0)\times \mathring{{Q}_{h}}\times{\boldsymbol{M}}^{t}_{h}(0)$$. Hence $$ B_{h}\left(\mathrm{e}_{L},\; \mathbf{e}_{u},\; {e}_{p},\;\mathbf{e}_{\widehat u_{t}};\; \mathrm{g}^{h}, \mathbf{v}^{h},\; q^{h},\;\widehat{\mathbf{v}}_{t}^{h}\right) = -{B}_{h}\left({\delta}_{L},\; \boldsymbol{\delta}_{u},\; {\delta}_{p},\;\boldsymbol{\delta}_{\widehat u_{t}};\; \mathrm{g}^{h},\; \mathbf{v}^{h},\; q^{h},\;\widehat{\mathbf{v}}_{t}^{h}\right). $$ Using orthogonality properties of the projections, we easily obtain $$ B_{h}\left({\delta}_{L},\; \boldsymbol{\delta}_{u},\; {\delta}_{p},\;\boldsymbol{\delta}_{\widehat u_{t}};\; \mathrm{g}^{h},\; \mathbf{v}^{h},\; q^{h},\;\widehat{\mathbf{v}}_{t}^{h}\right) = -\left\langle \textrm{tr}_{t}(\nu\,{\delta}_{L}\,\mathbf{n}),\,\textrm{tr}_{t}(\mathbf{v}^{h})-\widehat{\mathbf{v}}_{t}^{h} \right\rangle_{\partial{{\mathcal{T}_{h}}}}+( \gamma\,\boldsymbol{\delta}_{u},\,\mathbf{v}^{h} )_{{\mathcal{T}_{h}}}. $$ This completes the proof. Now we are ready to prove Theorem 2.4. 3.2. Proof of Theorem 2.4 Proof. By Di Pietro & Ern (2010, Theorem 2.1), we have $$ \|\mathbf{e}_{u}\|_{\mathcal{T}_{h}} \le C\, \left( \|{\boldsymbol{\nabla}} \mathbf{e}_{u}\|_{\mathcal{T}_{h}} + \sum_{F\in\mathcal{F}(K)}h_{F}^{-1} \left\|\,[\![\mathbf{e}_{u}]\!]\right\|_{F}^{2} \right)^{1/2}. $$ Here $$\,[\![\mathbf{e}_{u}]\!]:=\mathbf{e}_{u}^{+}-\mathbf{e}_{u}^{-}$$ denotes the jump of $$\mathbf{e}_{u}\in{\boldsymbol V}_{h}^{\textrm{div}}(0)$$ on an interior facet F := K+ ∩ K−, and $$[\![\mathbf{e}_u]\!] :=\mathbf{e}_u $$ on a boundary facet F ⊂ ∂$$\varOmega $$, where $$\mathbf{e}_{u}^{\pm } = \mathbf{e}_{u}|_{K^{\pm }}$$. Since eu is H(div)-conforming and has vanishing normal trace on the boundary, we have trn$$([\![\mathbf{e}]\!]) = 0 $$ for all facets $$F\in \mathcal{E}_{h}$$. Hence $$ \,[\![\mathbf{e}_{u}]\!]=\textrm{tr}_{t}(\,[\![\mathbf{e}_{u}]\!]).$$ By the triangle inequality, we have $$ \|\mathrm{t}\mathrm{r}_{t}(\,[\![\mathbf{e}_{u}]\!])\|_{F}\le \|\mathrm{t}\mathrm{r}_{t}(\mathbf{e}_{u}^{+})-\mathbf{e}_{\widehat u_{t}}\|_{F} +\|\mathrm{t}\mathrm{r}_{t}(\mathbf{e}_{u}^{-})-\mathbf{e}_{\widehat u_{t}}\|_{F}. $$ Combining the above estimates, we finish the proof of the first error estimate (2.11a). The second error estimate (2.11b) comes directly from Theorem 2.1. Now, let us prove the last error estimate (2.11c). Taking $$(\mathrm{g}^{h}, \mathbf{v}^{h}, q^{h},\widehat{\mathbf{v}}_{t}^{h}):=(\mathrm{e}_{L}, \mathbf{e}_{u}, {e}_{p},\mathbf{e}_{\widehat u_{t}})$$, we obtain \begin{align*} \nu\|\mathrm{e}_{L}\|_{\mathcal{T}_{h}}^{2}+ \|\gamma^{1/2}\mathbf{e}_{u}\|_{\mathcal{T}_{h}}^{2} &= -\left\langle \mathrm{t}\mathrm{r}_{t}(\nu\,{\delta}_{L}\,\mathbf{n}),\,\mathrm{t}\mathrm{r}_{t}(\mathbf{e}_{u})-\mathbf{e}_{\widehat u_{t}} \right\rangle_{\partial{{\mathcal{T}_{h}}}}+( \gamma\,\boldsymbol{\delta}_{u},\,\mathbf{e}_{u} )_{{\mathcal{T}_{h}}}\\ &\le \sum_{F\in\mathcal{E}_{h}}\left(h_{F}^{1/2}\|\mathrm{t}\mathrm{r}_{t}(\nu\,{\delta}_{L}\,\mathbf{n})\|_{F}\,h_{F}^{-1/2}\|{\mathrm{t}\mathrm{r}_{t}(\mathbf{e}_{u})-\mathbf{e}_{\widehat u_{t}}}\|_{F} \right) \\ &\quad +\|\gamma^{1/2}\boldsymbol{\delta}_{u}\|_{\mathcal{T}_{h}}\|\gamma^{1/2}\mathbf{e}_{u}\|_{\mathcal{T}_{h}}\\ &\le \;C\, \left(\sum_{F\in\mathcal{E}_{h}}\nu\,h_{F}\|{\delta}_{L}\,\mathbf{n}\|_{F}^{2}+\|\gamma^{1/2}\,\boldsymbol{\delta}_{u}\|_{\mathcal{T}_{h}}^{2} \right)^{{1/2}} \,\left( \nu\|\mathrm{e}_{L}\|_{\mathcal{T}_{h}}^{2}+ \|\gamma^{1/2}\mathbf{e}_{u}\|_{\mathcal{T}_{h}}^{2} \right)^{1/2} \end{align*} by the Cauchy–Schwarz inequality. This completes the proof of Theorem 2.4. The following result is used to prove the velocity estimate in Theorem 2.6. Lemma 3.4 Let ($$\varPhi$$, ϕ, φ) be the solution to the dual problem (2.13) for $$\boldsymbol{\theta }\in L^{2}({\mathcal{T}_{h}})^{d}$$. We have \begin{align} ( \mathbf{e}_{u},\,\boldsymbol{\theta} )_{{\mathcal{T}_{h}}}&=\left\langle \nu\,\mathrm{e}_{L}\,\mathbf{n},\,\boldsymbol{\delta}_{\phi} \right\rangle_{\partial{{\mathcal{T}_{h}}}}+\left\langle \mathrm{t}\mathrm{r}_{t}(\nu\,{\delta}_{L}\,\mathbf{n})+\mathrm{t}\mathrm{r}_{t}(\nu\,\mathrm{e}_{L}\,\mathbf{n}),\,\Pi_{{\boldsymbol V}}\phi-P_{M}\phi \right\rangle_{\partial{{\mathcal{T}_{h}}}}\nonumber\\ &\quad+\left\langle \mathrm{t}\mathrm{r}_{t}(\mathbf{e}_{u})-\mathbf{e}_{\widehat u_{t}},\,\nu\,\delta_{\varPhi}\mathbf{n} \right\rangle_{\partial{{\mathcal{T}_{h}}}}+( \gamma\,\mathbf{e}_{u},\,\boldsymbol{\delta}_{\phi} )_{{\mathcal{T}_{h}}}-( \gamma\,\boldsymbol{\delta}_{u},\,\Pi_{{\boldsymbol V}}\phi )_{{\mathcal{T}_{h}}}\nonumber\\ &=:T_{1}+T_{2}+T_{3}+T_{4}+T_{5}, \end{align} (3.10) where $$\delta _{\varPhi }=\varPhi -P_{\mathcal{G}}\varPhi ,\boldsymbol{\delta }_{\phi }=\phi -\Pi _{{\boldsymbol V}}\phi ,\delta _{\varphi }=\varphi -P_{Q}\varphi $$. Proof. By (2.13a)–(2.13c), we have \begin{align*} ( \mathbf{e}_{u},\,\boldsymbol{\theta} )_{{\mathcal{T}_{h}}}=&-( \mathbf{e}_{u},\,\nu\,{\boldsymbol{\nabla\cdot}}\varPhi )_{{\mathcal{T}_{h}}}+( \mathbf{e}_{u},\,\nu\phi )_{{\mathcal{T}_{h}}}-( \mathbf{e}_{u},\,{\nabla}\varphi )_{{\mathcal{T}_{h}}}\\ &-( \nu\mathrm{e}_{L},\,\varPhi )_{{\mathcal{T}_{h}}}+( \nu\,\mathrm{e}_{L},\,{\boldsymbol{\nabla}}\phi )_{{\mathcal{T}_{h}}}-( {e}_{p},\,{\boldsymbol{\nabla\cdot}}\phi )_{{\mathcal{T}_{h}}}\\ =&-( \mathbf{e}_{u},\,\nu\,{\boldsymbol{\nabla\cdot}}P_{\mathcal{G}}\varPhi )_{{\mathcal{T}_{h}}}-( \mathbf{e}_{u},\,\nu\,{\boldsymbol{\nabla\cdot}}\delta_{\varPhi} )_{{\mathcal{T}_{h}}}-( \mathbf{e}_{u},\,{\nabla}P_{Q}\varphi )_{{\mathcal{T}_{h}}}-( \mathbf{e}_{u},\,{\nabla}\delta_{\varphi} )_{{\mathcal{T}_{h}}}\\ &\quad+( \mathbf{e}_{u},\,\gamma\phi )_{{\mathcal{T}_{h}}}-( \nu\,\mathrm{e}_{L},\,P_{\mathcal{G}}\varPhi )_{{\mathcal{T}_{h}}}+( \nu\,\mathrm{e}_{L},\,{\boldsymbol{\nabla}}\phi )_{{\mathcal{T}_{h}}}-( {e}_{p},\,{\boldsymbol{\nabla\cdot}}\phi )_{{\mathcal{T}_{h}}}. \end{align*} Taking $$(\mathrm{g}^{h}, \mathbf{v}^{h}, q^{h},\widehat{\mathbf{v}}_{t}^{h}):=(P_{\mathcal{G}}\varPhi, \boldsymbol{0}, -P_{Q}\varphi ,0)$$ in the error equation (3.9), putting the resulting identity into the above expression and simplifying, we have \begin{align*} ( \mathbf{e}_{u},\,\boldsymbol{\theta} )_{{\mathcal{T}_{h}}}=&-\left\langle \mathbf{e}_{u},\,\nu\,P_{\mathcal{G}}\varPhi\mathbf{n} \right\rangle_{\partial{{\mathcal{T}_{h}}}}-\left\langle \mathbf{e}_{u},\,P_{Q}\varphi\mathbf{n} \right\rangle_{\partial{{\mathcal{T}_{h}}}}\\ &+\left\langle \mathrm{t}\mathrm{r}_{t}(\mathbf{e}_{u})-\mathbf{e}_{\widehat u_{t}},\,\mathrm{t}\mathrm{r}_{t}(\nu\,P_{\mathcal{G}}\varPhi\mathbf{n}) \right\rangle_{\partial{{\mathcal{T}_{h}}}}-( \mathbf{e}_{u},\,\nu\,{\boldsymbol{\nabla\cdot}}\delta_{\varPhi} )_{{\mathcal{T}_{h}}}-( \mathbf{e}_{u},\,{\nabla}\delta_{\varphi} )_{{\mathcal{T}_{h}}}\\ &+( \mathbf{e}_{u},\,\gamma\phi )_{{\mathcal{T}_{h}}}+( \nu\,\mathrm{e}_{L},\,{\boldsymbol{\nabla}}\phi )_{{\mathcal{T}_{h}}}-( {e}_{p},\,{\boldsymbol{\nabla\cdot}}\phi )_{{\mathcal{T}_{h}}}\\ =&-\langle \mathbf{e}_{u},\,\nu\,P_{\mathcal{G}}\varPhi\mathbf{n} \rangle_{\partial{{\mathcal{T}_{h}}}}-\langle \mathbf{e}_{u},\,P_{Q}\varphi\mathbf{n} \rangle_{\partial{{\mathcal{T}_{h}}}}\\ &+\left\langle \mathrm{t}\mathrm{r}_{t}(\mathbf{e}_{u})-\mathbf{e}_{\widehat u_{t}},\,\mathrm{t}\mathrm{r}_{t}(\nu\,P_{\mathcal{G}}\varPhi\mathbf{n}) \right\rangle_{\partial{{\mathcal{T}_{h}}}}-\langle \mathbf{e}_{u},\,\nu\,\delta_{\varPhi}\mathbf{n} \rangle_{\partial{{\mathcal{T}_{h}}}}-\langle \mathbf{e}_{u},\,\delta_{\varphi}\mathbf{n} \rangle_{\partial{{\mathcal{T}_{h}}}}\\ &+( \mathbf{e}_{u},\,\gamma\phi )_{{\mathcal{T}_{h}}}+( \nu\,\mathrm{e}_{L},\,{\boldsymbol{\nabla}}\phi )_{{\mathcal{T}_{h}}}-( {e}_{p},\,{\boldsymbol{\nabla\cdot}}\phi )_{{\mathcal{T}_{h}}}\\ =&-\langle \mathbf{e}_{u},\,\nu\,\varPhi\mathbf{n} \rangle_{\partial{{\mathcal{T}_{h}}}}+\langle \mathrm{t}\mathrm{r}_{t}(\mathbf{e}_{u})-\mathbf{e}_{\widehat u_{t}},\,\nu\,P_{\mathcal{G}}\varPhi\mathbf{n} \rangle_{\partial{{\mathcal{T}_{h}}}}\\ &+( \mathbf{e}_{u},\,\gamma\phi )_{{\mathcal{T}_{h}}}+( \nu\,\mathrm{e}_{L},\,{\boldsymbol{\nabla}}\phi )_{{\mathcal{T}_{h}}}-( {e}_{p},\,{\boldsymbol{\nabla\cdot}}\phi )_{{\mathcal{T}_{h}}}\\ =&-\left\langle \mathrm{t}\mathrm{r}_{t}(\mathbf{e}_{u})-\mathbf{e}_{\widehat u_{t}},\,\nu\,\delta_{\varPhi}\mathbf{n} \right\rangle_{\partial{{\mathcal{T}_{h}}}}\\ &+( \mathbf{e}_{u},\,\gamma\phi )_{{\mathcal{T}_{h}}}+( \nu\,\mathrm{e}_{L},\,{\boldsymbol{\nabla}}\phi )_{{\mathcal{T}_{h}}}-( {e}_{p},\,{\boldsymbol{\nabla\cdot}}\phi )_{{\mathcal{T}_{h}}}, \end{align*} by inserting the zero term $$\langle \mathbf{e}_{\widehat u_{t}},\,\nu \,\varPhi \mathbf{n} \rangle _{\partial{{\mathcal{T}_{h}}}}$$ and using the fact that $$\langle \mathbf{e}_{u},\,\nu \,\varPhi \mathbf{n} \rangle _{\partial{{\mathcal{T}_{h}}}}=\langle \mathrm{t}\mathrm{r}_{t}(\mathbf{e}_{u}),\,\nu \,\varPhi \mathbf{n} \rangle _{\partial{{\mathcal{T}_{h}}}}$$ and $$\langle \mathbf{e}_{u},\,\varphi \mathbf{n} \rangle _{\partial{{\mathcal{T}_{h}}}}=0$$. Take $$(\mathrm{g}^{h}, \mathbf{v}^{h}, q^{h},\widehat{\mathbf{v}}_{t}^{h}):=(0, \Pi _{{\boldsymbol V}}\phi , 0, P_{\textbf{M}^{t}}\phi )$$ in the error equation (3.9). Denoting by $$I:=( \mathbf{e}_{u},\,\gamma \phi )_{{\mathcal{T}_{h}}}+( \nu \,\mathrm{e}_{L},\,{\boldsymbol{\nabla }}\phi )_{{\mathcal{T}_{h}}}-( {e}_{p},\,{\boldsymbol{\nabla \cdot }}\phi )_{{\mathcal{T}_{h}}}$$, we obtain \begin{align*} I&=( \mathbf{e}_{u},\,\gamma\boldsymbol{\delta}_{\phi} )_{{\mathcal{T}_{h}}}+( \nu\,\mathrm{e}_{L},\,{\boldsymbol{\nabla}}\boldsymbol{\delta}_{\phi} )_{{\mathcal{T}_{h}}}-( {e}_{p},\,{\boldsymbol{\nabla\cdot}}\boldsymbol{\delta}_{\phi} )_{{\mathcal{T}_{h}}}\\ &\quad+( \mathbf{e}_{u},\,\gamma\Pi_{{\boldsymbol V}}\phi )_{{\mathcal{T}_{h}}}+( \nu\,\mathrm{e}_{L},\,{\boldsymbol{\nabla}}\Pi_{{\boldsymbol V}}\phi )_{{\mathcal{T}_{h}}}-( {e}_{p},\,{\boldsymbol{\nabla\cdot}}\Pi_{{\boldsymbol V}}\phi )_{{\mathcal{T}_{h}}}\\ &=( \mathbf{e}_{u},\,\gamma\boldsymbol{\delta}_{\phi} )_{{\mathcal{T}_{h}}}+( \nu\,\mathrm{e}_{L},\,{\boldsymbol{\nabla}}\boldsymbol{\delta}_{\phi} )_{{\mathcal{T}_{h}}}-( {e}_{p},\,{\boldsymbol{\nabla\cdot}}\boldsymbol{\delta}_{\phi} )_{{\mathcal{T}_{h}}}\\ &\quad\left\langle \mathrm{t}\mathrm{r}_{t}(\nu\,{\delta}_{L}\mathbf{n})+\mathrm{t}\mathrm{r}_{t}(\nu\,\mathrm{e}_{L}\mathbf{n}),\,\mathrm{t}\mathrm{r}_{t}(\Pi_{{\boldsymbol V}}\phi)-P_{\textbf{M}^{t}}\phi \right\rangle_{\partial{{\mathcal{T}_{h}}}}-( \gamma\boldsymbol{\delta}_{u},\,\Pi_{{\boldsymbol V}}\phi )_{{\mathcal{T}_{h}}}\\ &=( \mathbf{e}_{u},\,\gamma\boldsymbol{\delta}_{\phi} )_{{\mathcal{T}_{h}}}+\langle \nu\,\mathrm{e}_{L}\mathbf{n},\,\boldsymbol{\delta}_{\phi} \rangle_{\partial{{\mathcal{T}_{h}}}}\\ &\quad\left\langle \mathrm{t}\mathrm{r}_{t}(\nu\,{\delta}_{L}\mathbf{n})+\mathrm{t}\mathrm{r}_{t}(\nu\,\mathrm{e}_{L}\mathbf{n}),\,\Pi_{{\boldsymbol V}}\phi-P_{M}\phi \right\rangle_{\partial{{\mathcal{T}_{h}}}}-( \gamma\boldsymbol{\delta}_{u},\,\Pi_{{\boldsymbol V}}\phi )_{{\mathcal{T}_{h}}}. \end{align*} This completes the proof of Lemma 3.4. Now we are ready to prove Theorem 2.6. 3.3. Proof of Theorem 2.6 Proof. We first present the optimal error estimate for ep by applying an inf–sup argument. It is well known that the following inf–sup condition holds for a positive constant κ (cf. Girault & Raviart, 1986, Chapter 1, Corollary 2.4): \begin{align} \sup_{\boldsymbol{\omega}\in{H^{1}_{0}}(\varOmega)^{d}\backslash\{0\}}\frac{({\boldsymbol{\nabla\cdot}}\boldsymbol{\omega},q)_{\varOmega}}{\|\boldsymbol{\omega}\|_{1,\varOmega}}\ge\kappa\|q\|_{\varOmega}. \end{align} (3.11) Here ∥⋅∥$$1{,\varOmega}$$ is the standard H1-norm on $$\varOmega$$. Since $${e}_{p}\in{L^{2}_{0}}(\varOmega )$$, we have by (3.11), \begin{align} \|e_{p}\|_{\varOmega}\le\frac{1}{\kappa}\sup_{\boldsymbol{\omega}\in{H^{1}_{0}}(\varOmega)^{d}\backslash\{0\}}\frac{({\boldsymbol{\nabla\cdot}}\boldsymbol{\omega},e_{p})_{\varOmega}}{\|\boldsymbol{\omega}\|_{1,\varOmega}}. \end{align} (3.12) Taking $$(\mathrm{g}^{h},\; \mathbf{v}^{h},\; q^{h},\;\widehat{\mathbf{v}}_{t}^{h}):=(0,\; \Pi _{{\boldsymbol V}}\boldsymbol{\omega },\; 0,\; P_{\boldsymbol{M}^{t}}\boldsymbol{\omega })$$ in the error equation (3.9) and applying integration by parts, we can rewrite the numerator as: \begin{align*} ( {\boldsymbol{\nabla\cdot}} \boldsymbol{\omega},\,{e}_{p} )_{{\mathcal{T}_{h}}}&=( {\boldsymbol{\nabla\cdot}}\Pi_{{\boldsymbol V}}\boldsymbol{\omega},\,{e}_{p} )_{{\mathcal{T}_{h}}}+( {\boldsymbol{\nabla\cdot}}(\boldsymbol{\omega}-\Pi_{{\boldsymbol V}}\boldsymbol{\omega}),\,{e}_{p} )_{{\mathcal{T}_{h}}}=( {\boldsymbol{\nabla\cdot}}\Pi_{{\boldsymbol V}}\boldsymbol{\omega},\,{e}_{p} )_{{\mathcal{T}_{h}}}\\ &=( \nu\,\mathrm{e}_{L},\,{\boldsymbol{\nabla}}\Pi_{{\boldsymbol V}}\boldsymbol{\omega} )_{{\mathcal{T}_{h}}}-\langle \mathrm{t}\mathrm{r}_{t}(\nu\,\mathrm{e}_{L}\mathbf{n})+\mathrm{t}\mathrm{r}_{t}(\nu\,{\delta}_{L}\mathbf{n}),\,\mathrm{t}\mathrm{r}_{t}(\Pi_{{\boldsymbol V}}\boldsymbol{\omega})-P_{\boldsymbol{M}^{t}}\boldsymbol{\omega} \rangle_{\partial{{\mathcal{T}_{h}}}}\\ &\quad+( \gamma\mathbf{e}_{u},\,\Pi_{{\boldsymbol V}}\boldsymbol{\omega} )_{{\mathcal{T}_{h}}}+( \gamma\boldsymbol{\delta}_{u},\,\Pi_{{\boldsymbol V}}\boldsymbol{\omega} )_{{\mathcal{T}_{h}}}\\ &=( \nu\,\mathrm{e}_{L},\,{\boldsymbol{\nabla}}\Pi_{{\boldsymbol V}}\boldsymbol{\omega} )_{{\mathcal{T}_{h}}}-\langle \mathrm{t}\mathrm{r}_{t}(\nu\,\mathrm{e}_{L}\mathbf{n})+\mathrm{t}\mathrm{r}_{t}(\nu\,{\delta}_{L}\mathbf{n}),\,\Pi_{{\boldsymbol V}}\boldsymbol{\omega}-P_{M}\boldsymbol{\omega} \rangle_{\partial{{\mathcal{T}_{h}}}}\\ &\quad+( \gamma\mathbf{e}_{u},\,\Pi_{{\boldsymbol V}}\boldsymbol{\omega} )_{{\mathcal{T}_{h}}}+( \gamma\boldsymbol{\delta}_{u},\,\Pi_{{\boldsymbol V}}\boldsymbol{\omega} )_{{\mathcal{T}_{h}}}\\ &=:\,I_{1}+I_{2}+I_{3}+I_{4}. \end{align*} Then we will bound I1 to I4 by Corollary 2.5 as: \begin{align*} I_{1}\le&\;\nu\|\mathrm{e}_{L}\|_{{\mathcal{T}_{h}}}\|{\boldsymbol{\nabla}}\Pi_{{\boldsymbol V}}\boldsymbol{\omega}\|_{{\mathcal{T}_{h}}}\le C\nu^{1/2}\varTheta h^{k+1}\|\boldsymbol{\omega}\|_{1,\varOmega},\\ I_{2}\le&\;\nu\left(\|\mathrm{e}_{L}\mathbf{n}\|_{\partial{\mathcal{T}_{h}}}+\|{\delta}_{L}\mathbf{n}\|_{\partial{\mathcal{T}_{h}}}\right)\|\Pi_{{\boldsymbol V}}\boldsymbol{\omega}-P_{M}\boldsymbol{\omega}\|_{\partial{\mathcal{T}_{h}}}\\ \le&\;C(\nu^{1/2}\varTheta h^{k+1/2}+\nu\|\mathrm{L}\|_{k+1}h^{k+1/2})h^{1/2}\|\boldsymbol{\omega}\|_{1,\varOmega}\le C\nu^{1/2}\varTheta h^{k+1}\|\boldsymbol{\omega}\|_{1,\varOmega},\\ I_{3}\le&\;C\gamma^{1/2}_{\max}\|\gamma^{1/2}\mathbf{e}_{u}\|_{{\mathcal{T}_{h}}}\|\Pi_{{\boldsymbol V}}\boldsymbol{\omega}\|_{{\mathcal{T}_{h}}}\le C\gamma^{1/2}_{\max}\varTheta h^{k+1}\|\boldsymbol{\omega}\|_{1,\varOmega},\\ I_{4}\le&\;C\gamma_{\max}\|\boldsymbol{\delta}_{u}\|_{{\mathcal{T}_{h}}}\|\Pi_{{\boldsymbol V}}\boldsymbol{\omega}\|_{{\mathcal{T}_{h}}}\le C\gamma_{\max}\|\mathbf{u}\|_{k+1} h^{k+1}\|\boldsymbol{\omega}\|_{1,\varOmega}\\ \le&\;C\gamma^{1/2}_{\max}\varTheta h^{k+1}\|\boldsymbol{\omega}\|_{1,\varOmega}. \end{align*} Then we have $$ ( {\boldsymbol{\nabla\cdot}} \boldsymbol{\omega},\,{e}_{p} )_{{\mathcal{T}_{h}}}\le C\left(\nu^{1/2}+\gamma^{1/2}_{\max}\right)\varTheta h^{k+1}\|\boldsymbol{\omega}\|_{1,\varOmega}.$$ By (3.12), we obtain the estimate for ep. Now we give a superconvergent estimate for eu. By (3.10), it suffices to estimate the terms T1 to T5. We apply Corollary 2.5, the regularity assumption (2.12) and the inverse inequality to bound these terms: \begin{align*} T_{1}\le&\;\nu\|\mathrm{e}_{L}\mathbf{n}\|_{\partial{\mathcal{T}_{h}}}\|\boldsymbol{\delta}_{\phi}\|_{\partial{\mathcal{T}_{h}}}\le C\nu h^{-1/2}\|\mathrm{e}_{L}\|_{{\mathcal{T}_{h}}}h^{3/2}\|\phi\|_{2}\\ \le&\;C\nu^{1/2}\varTheta h^{k+2}\|\boldsymbol{\theta}\|_{{\mathcal{T}_{h}}},\\ T_{2}\le&\;\nu\left(\|{\delta}_{L}\mathbf{n}\|_{\partial{\mathcal{T}_{h}}}+\|\mathrm{e}_{L}\mathbf{n}\|_{\partial{\mathcal{T}_{h}}}\right)\|\Pi_{{\boldsymbol V}}\phi-P_{M}\phi\|_{\partial{\mathcal{T}_{h}}}\\ \le&\;C\left(\nu\|\mathrm{L}\|_{k+1}h^{k+1/2}+\nu^{1/2}\varTheta h^{k+1/2}\right)h^{3/2}\|\phi\|_{2}\le C\nu^{1/2}\varTheta h^{k+2}\|\boldsymbol{\theta}\|_{{\mathcal{T}_{h}}},\\ T_{3}\le&\;\nu h^{-1/2}\|\mathrm{t}\mathrm{r}_{t}(\mathbf{e}_{u})-\mathbf{e}_{\widehat u_{t}}\|_{\partial{\mathcal{T}_{h}}}h^{1/2}\|\delta_{\varPhi}\mathbf{n}\|_{\partial{\mathcal{T}_{h}}}\\ \le&\;C\nu^{1/2}{\left\vert\kern-0.25ex\left\vert\kern-0.25ex\left\vert (\mathbf{e}_{u},\mathbf{e}_{\widehat u_{t}}) \right\vert\kern-0.25ex\right\vert\kern-0.25ex\right\vert}_{1,{\mathcal{T}_{h}}}h\|\varPhi\|_{1,\varOmega}\le C\nu^{1/2}\varTheta h^{k+2}\|\boldsymbol{\theta}\|_{{\mathcal{T}_{h}}},\\ T_{4}\le&\;\gamma^{1/2}_{\max}\|\gamma^{1/2}\mathbf{e}_{u}\|_{{\mathcal{T}_{h}}}\|\boldsymbol{\delta}_{\phi}\|_{{\mathcal{T}_{h}}}\le C\gamma^{1/2}_{\max}\varTheta h^{k+2}\|\boldsymbol{\theta}\|_{{\mathcal{T}_{h}}},\\ T_{5}=&\;\left( (\gamma-P_{0,h}\gamma)\boldsymbol{\delta}_{u},\,\Pi_{{\boldsymbol V}}\phi \right)_{{\mathcal{T}_{h}}}+( P_{0,h}\gamma\boldsymbol{\delta}_{u},\,\Pi_{{\boldsymbol V}}\phi- \bar{\phi} )_{{\mathcal{T}_{h}}}\\ \le&\;\|\gamma-P_{0,h}\gamma\|_{\infty}\|\boldsymbol{\delta}_{u}\|_{{\mathcal{T}_{h}}}\|\Pi_{{\boldsymbol V}}\phi\|_{{\mathcal{T}_{h}}}+|P_{0,h}\gamma|\|\boldsymbol{\delta}_{u}\|_{{\mathcal{T}_{h}}}\|\Pi_{{\boldsymbol V}}(\phi-\bar{\phi})\|_{{\mathcal{T}_{h}}}\\ \le&\;Ch\|\gamma\|_{1,\infty}h^{k+1}\|\mathbf{u}\|_{k+1}\|\phi\|_{2}+C\|\gamma\|_{0,\infty}\,h^{k+1}\|\mathbf{u}\|_{k+1}h\|{\boldsymbol{\nabla}}\phi\|_{{\mathcal{T}_{h}}}\\ \le&\;C \|\gamma\|_{1,\infty}\|\mathbf{u}\|_{k+1}h^{k+2}\|\boldsymbol{\theta}\|_{{\mathcal{T}_{h}}}, \end{align*} where P0, h is the L2- orthogonal projection onto $$\mathcal{P}_{0}({\mathcal{T}_{h}})^{d\times d}$$ and $$\bar{\phi }$$ is defined as $$ \bar{\phi}=\frac{1}{|K|}( \phi,\,1 )_{K}\quad \forall \, K\in{\mathcal{T}_{h}}.$$ Combining all the above estimates, we have $$ \|\mathbf{e}_{u}\|_{{\mathcal{T}_{h}}}\le C\left(\nu^{1/2}\varTheta+\gamma_{\max}^{1/2}\varTheta+\|\gamma\|_{1,\infty}\|\mathbf{u}\|_{k+1}\right)h^{k+2}.$$ This completes the proof of Theorem 2.6. 4. Hybridization In this section, we hybridize the H(div)-conforming HDG method (2.5) by relaxing the H(div)-conformity of the velocity field via Lagrange multipliers; similar treatment was used in the study by Cockburn & Sayas (2014). The resulting global linear system is a saddle point system for $$(\widehat{\mathbf{u}}_{t}^{h},\; \widehat{\mathbf{u}}_{n}^{h},\; \bar{p}^{h}) \in{\boldsymbol{M}}^{t}_{h}(0)\times{\boldsymbol{M}}^{n}_{h}(0) \times \overline{Q}_{h}$$, where \begin{align} {\boldsymbol{M}}^{n}_{h}(0):= \left\{ \widehat{\mathbf{v}} \in \boldsymbol{M}_{h}(0) :\;\; \mathrm{t}\mathrm{r}_{t} (\widehat{\mathbf{v}})|_{F} = \boldsymbol{0}\quad \forall \, F\in\mathcal{E}_{h}\right\}\!, \end{align} (4.1a) \begin{align} \overline{Q}_{h} := \left\{ q\in L^{2}({\mathcal{T}_{h}}):\; q|_{K} \in \mathcal{P}_{0}(K)\quad \forall \, K\in{\mathcal{T}_{h}}\right\}\!. \end{align} (4.1b) We show that $$\widehat{\mathbf{u}}_{t}^{h}$$ here is the same as that in (2.5), $$\widehat{\mathbf{u}}_{n}^{h} = \mathrm{t}\mathrm{r}_{n} (\mathbf{u}^{h})$$ on $$\mathcal{E}_{h}$$, $$\bar{p}^{h}$$ is equal to the average of ph on each element of $${\mathcal{T}_{h}}$$. Here we first relax H(div)-conformity of the velocity field in (2.5) to obtain the following result. Theorem 4.1 There exists a unique element $$(\mathrm{L}^{h},\,\mathbf{u}^{h},\,p_{\perp }^{h},\, \bar{p}^{h},\, {\widehat{\mathbf{u}}_{t}}^{h},\,{\widehat{\mathbf{u}}_{n}}^{h},\, \lambda ^{h})\in{\mathcal{G}}_{h}\times \textbf{V}_{h}\times Q^{\perp }_{h} \times \overline{Q}_{h} \times{\boldsymbol{M}}^{t}_{h}(0)\times{\boldsymbol{M}}^{n}_{h}(0) \times M_{h}^{\partial }$$ such that the following weak formulation holds: \begin{align} ( \mathrm{L}^{h},\,\nu\,\mathrm{g}^{h} )_{{\mathcal{T}_{h}}}+\left( \mathbf{u}^{h},\,{\boldsymbol{\nabla}} \cdot (\nu\,\mathrm{g}^{h}) \right)_{{\mathcal{T}_{h}}} - \left\langle \widehat{\mathbf{u}}_{t}^{h} + \widehat{\mathbf{u}}_{n}^{h},\, \nu\,\mathrm{g}^{h}\, \mathbf{n} \right\rangle_{\partial{{\mathcal{T}_{h}}}} = 0, \end{align} (4.2a) \begin{align} \left( \nu\,\mathrm{L}^{h} - (p_{\perp}^{h}+\bar{p}^{h})I_{d},\,{\boldsymbol{\nabla}} \mathbf{v}^{h} \right)_{{\mathcal{T}_{h}}} +( \gamma\, \mathbf{u}^{h},\,\mathbf{v}^{h} )_{{\mathcal{T}_{h}}} \end{align} (4.2b) \begin{align} - \left\langle \nu\,\mathrm{L}^{h} \mathbf{n} - (p_{\perp}^{h}+\bar{p}^{h}) \mathbf{n} +\lambda^{h}\mathbf{n},\,\mathbf{v}^{h} \right\rangle_{\partial{{\mathcal{T}_{h}}}} &=(\mathbf{f},\mathbf{v}^{h} )_{\mathcal{T}_{h}},\nonumber \\ \left( {\nabla\cdot}\mathbf{u}^{h},\,q_{\perp}^{h} + \bar{q}^{h} \right)_{{\mathcal{T}_{h}}} & = \left(g, q_{\perp}^{h} + \bar{q}^{h}\right)_{\mathcal{T}_{h}}, \end{align} (4.2c) \begin{align} \left\langle \nu\,\mathrm{L}^{h} \mathbf{n} - (p_{\perp}^{h}+\bar{p}^{h}) \mathbf{n} +\lambda^{h}\mathbf{n},\,\widehat{\mathbf{v}}_{t}^{h}+\widehat{\mathbf{v}}_{n}^{h} \right\rangle_{\partial{{\mathcal{T}_{h}}}} = 0, \end{align} (4.2d) \begin{align} \left\langle (\mathbf{u}^{h} - \widehat{\mathbf{u}}_{n}^{h})\cdot \mathbf{n},\,\mu^{h} \right\rangle_{\partial{{\mathcal{T}_{h}}}}=0, \end{align} (4.2e) \begin{equation} (\bar{p}^{h}, 1)_{\mathcal{T}_{h}} =0, \end{equation} (4.2f) for all $$(\mathrm{g}^{h},\mathbf{v}^{h},\, q_{\perp }^{h},\,\bar{q}^{h},\,{\widehat{\mathbf{v}}_{t}}^{h},\,\widehat{\mathbf{v}}_{n}^{h},\,\mu ^{h}) \in{\mathcal{G}}_{h}\times \boldsymbol{V}_{h}\times Q^{\perp }_{h} \times \overline{Q}_{h} \times{\boldsymbol{M}}^{t}_{h}(0)\times{\boldsymbol{M}}^{n}_{h}(0) \times M_{h}^{\partial }$$, where \begin{align*} Q_{h}^{\perp} := & \left\{ q\in L^{2}({\mathcal{T}_{h}}):\; (q,1)_{K}=0\quad \forall \, K\in{\mathcal{T}_{h}}\right\}\!, \\ M_{h}^{\partial} := & \left\{ \mu \in L^{2}(\partial{\mathcal{T}_{h}}): \mu|_{\partial K}\in \mathcal{P}_{k}(\partial K)\quad \forall \, K\in{\mathcal{T}_{h}}\right\}\!, \\ \mathcal{P}_{k}(\partial K) := & \left\{\mu \in L^{2}(\partial K): \mu |_{F} \in \mathcal{P}_{k}(F)\quad \forall \, F\in \mathcal{F}(K)\right\}\!. \end{align*} Moreover, if $$(\mathrm{L}^{h},\,\mathbf{u}^{h},\, p_{\perp }^{h}, \bar{p}^{h},\, {\widehat{\mathbf{u}}_{t}}^{h},\,{\widehat{\mathbf{u}}_{n}}^{h},\, \lambda ^{h})\in{\mathcal{G}}_{h}\times \boldsymbol{V}_{h}\times Q^{\perp }_{h} \times \overline{Q}_{h} \times{\boldsymbol{M}}^{t}_{h}(0)\times{\boldsymbol{M}}^{n}_{h}(0) \times M_{h}^{\partial }$$ is the numerical solution to the above equations, then $$(\mathrm{L}^{h},\mathbf{u}^{h},p_{\perp }^{h}+\bar{p}^{h},\,{\widehat{\mathbf{u}}_{t}}^{h})\in{\mathcal{G}}_{h}\times{\boldsymbol V}_{h}^{\textrm{div}}(0)\times \mathring{{Q}_{h}} \times{\boldsymbol{M}}^{t}_{h}(0)$$ is the only solution to (2.5). Note that $$\lambda ^{h}\in M_{h}^{\partial }$$ is a quantity that approximates $$0|_{\partial{\mathcal{T}_{h}}}$$. Proof. Let $$(\mathrm{L}^{h},\,\mathbf{u}^{h},\,p_{\perp }^{h},\, \bar{p}^{h},\,{\widehat{\mathbf{u}}_{t}}^{h},\,{\widehat{\mathbf{u}}_{n}}^{h},\, \lambda ^{h}) \in{\mathcal{G}}_{h}\times \boldsymbol{V}_{h}\times Q^{\perp }_{h} \times \overline{Q}_{h} \times{\boldsymbol{M}}^{t}_{h}(0)\times{\boldsymbol{M}}^{n}_{h}(0) \times M_{h}^{\partial }$$ be a numerical solution to equations (4.2). We prove such a numerical solution is unique and $$(\mathrm{L}^{h},\,\mathbf{u}^{h},\,p_{\perp }^{h} + \bar{p}^{h},\,\widehat{\mathbf{u}}_{t}^{h})$$ is the unique solution to equations (2.5). Since $$ \left(\mathbf{u}^{h} - \widehat{\mathbf{u}}_{n}^{h}\right)\cdot \mathbf{n} |_{\partial K} \in \mathcal{P}_{k}(\partial K)= M_{h}^{\partial}(K) \quad \forall \, K \in{\mathcal{T}_{h}}, $$ we have $$ \mathrm{t}\mathrm{r}_{n}^{F} (\mathbf{u}^{h}) = {\widehat{\mathbf{u}}_{n}}^{h}$$ on any facet $$F\in \mathcal{E}_{h}$$ by equations (4.2e). Hence, $$\mathbf{u}^{h}\in{\boldsymbol V}_{h}^{\textrm{div}}(0)$$. By equation (4.2f), we have $$p_{\perp }^{h} + \bar{p}^{h} \in \mathring{{Q}_{h}}$$. Then, taking $$\mathbf{v}^{h}\in{\boldsymbol V}_{h}^{\textrm{div}}(0)$$ in (4.2b), $$\widehat{\mathbf{v}}_{n}^{h} |_{F} = \mathrm{t}\mathrm{r}_{n}^{F} (\mathbf{v}^{h})$$ on any facet $$F\in \mathcal{E}_{h}$$ in (4.2d) and qh ∈ $$\mathring{{Q}_{h}}$$ in (4.2c), we have $$ \left(\mathrm{L}^{h},\mathbf{u}^{h},p_{\perp}^{h} + \bar{p}^{h},{\widehat{\mathbf{u}}_{t}}^{h}\right)\in{\mathcal{G}}_{h}\times{\boldsymbol V}_{h}^{\textrm{div}}(0)\times \mathring{{Q}_{h}} \times{\mathbf{M}}^{t}_{h}(0)$$ is the unique solution to equations (2.5). Now, we only need to show the uniqueness of λh. If there are two λh, then by equation (4.2b), their difference which we still call λh, satisfies $$ \langle \lambda^{h},\,\mathbf{v}^{h}\cdot \mathbf{n} \rangle_{\partial{{\mathcal{T}_{h}}}} = 0\quad \forall \, \mathbf{v}^{h} \in \mathbf{V}_{h}. $$ Since $$M_{h}^{\partial }(K) = \mathrm{t}\mathrm{r}_{n}(\mathbf{V}_{h}(K))$$ for any $$K \in{\mathcal{T}_{h}}$$, we have $$\lambda ^{h} = 0|_{\partial{\mathcal{T}_{h}}}$$. So, λh is also unique. This completes the proof. Then we identify local and global solvers. Because of the lack of uniqueness of pressure in the Brinkman equations, we will keep $$\bar{p}_{h}\in \overline{Q}_{h}$$ as a separate unknown. Given $$(\widehat{\mathbf{u}}_{t}, \widehat{\mathbf{u}}_{n})\in{\boldsymbol{M}}^{t}_{h}(0)\times{\boldsymbol{M}}^{n}_{h}(0)$$, $$\mathbf{f}\in L^{2}({\mathcal{T}_{h}})^{d}$$ and $$g\in L^{2}({\mathcal{T}_{h}})$$, we consider the solution to the set of local problems in each element $$K\in{\mathcal{T}_{h}}$$: find $$ \left(\mathrm{L}^{h},\mathbf{u}^{h},p_{\perp}^{h}, \lambda^{h}\right)\in{\mathcal{G}}(K)\times{\boldsymbol V}(K)\times{Q}^{\perp}(K) \times M_{h}^{\partial}(K) $$ such that \begin{align} ( \mathrm{L}^{h},\,\nu\,\mathrm{g}^{h} )_{K}+\left( \mathbf{u}^{h},\,{\boldsymbol{\nabla}} \cdot (\nu\,\mathrm{g}^{h}) \right)_{K} = \left\langle \widehat{\mathbf{u}}_{t}+ \widehat{\mathbf{u}}_{n},\, \nu\,\mathrm{g}^{h}\, \mathbf{n} \right\rangle_{\partial{K}}, \end{align} (4.3a) \begin{align} -\left( {\boldsymbol{\nabla}} \cdot (\nu\,\mathrm{L}^{h})-\nabla p_{\perp}^{h} - \gamma\, \mathbf{u}^{h},\,\mathbf{v}^{h} \right)_{K} - \langle \lambda^{h}\,\mathbf{n},\,\mathbf{v}^{h} \rangle_{\partial{K}} =(\mathbf{f},\mathbf{v}^{h})_{\mathcal{T}_{h}}, \end{align} (4.3b) \begin{align} \left( {\nabla\cdot}\mathbf{u}^{h},\,q_{\perp}^{h} \right)_{K} = (g, q_{\perp}^{h})_{\mathcal{T}_{h}}, \end{align} (4.3c) \begin{align} \left\langle (\mathbf{u}^{h} - \widehat{\mathbf{u}}_{n})\cdot \mathbf{n},\,\mu^{h} \right\rangle_{\partial{K}} = 0, \end{align} (4.3d) for all $$(\mathrm{g}^{h},\mathbf{v}^{h},q_{\perp }^{h}, \mu ^{h})\in{\mathcal{G}}(K)\times{\boldsymbol V}(K) \times{Q}_{\perp }(K) \times M_{h}^{\partial }(K)$$. The unique solvability of this problem is a simple consequence of the unique solvability of equations (4.2). The solution to (4.3) can be written $$ \left(\mathrm{L}^{h},\mathbf{u}^{h},p_{\perp}^{h}, \lambda^{h}\right) = \left(\mathrm{L}^{h}_{(\widehat{\mathbf{u}}_{t},\widehat{\mathbf{u}}_{n})},\,\mathbf{u}^{h}_{(\widehat{\mathbf{u}}_{t},\widehat{\mathbf{u}}_{n})},\, p^{h}_{\perp,(\widehat{\mathbf{u}}_{t},\widehat{\mathbf{u}}_{n})},\, \lambda^{h}_{(\widehat{\mathbf{u}}_{t},\widehat{\mathbf{u}}_{n})}\right) +\left(\mathrm{L}^{h}_{(\mathbf{f},g)},\,\mathbf{u}^{h}_{(\mathbf{f},g)},\,p^{h}_{\perp,(\mathbf{f},g)},\,\lambda^{h}_{(\mathbf{f}, g)}\right) $$ by considering separately the influence of $$(\widehat{\mathbf{u}}_{t},\widehat{\mathbf{u}}_{t})$$ and (f, g) in the solution. For example, $$\big(\mathrm{L}^{h}_{(\widehat{\mathbf{u}}_{t},\widehat{\mathbf{u}}_{n})},\,\mathbf{u}^{h}_{(\widehat{\mathbf{u}}_{t},\widehat{\mathbf{u}}_{n})}, \,p^{h}_{\perp ,(\widehat{\mathbf{u}}_{t},\widehat{\mathbf{u}}_{n})},\, \lambda ^{h}_{(\widehat{\mathbf{u}}_{t},\widehat{\mathbf{u}}_{n})}\big )$$ is the solution of (4.3) when (f, g) = (0, 0). According to equations (4.2c) (4.2d) (4.2f), the global (hybrid) problem is to find $$(\widehat{\mathbf{u}}_{t}^{h}, \widehat{\mathbf{u}}_{n}^{h}, \bar{p}^{h}) \in{\boldsymbol{M}}^{t}_{h}(0)\times{\boldsymbol{M}}^{n}_{h}(0) \times \overline{Q}_{h}$$ such that \begin{align} \left\langle \nu\,\mathrm{L}^{h}_{\left(\widehat{\mathbf{u}}_{t}^{h},\widehat{\mathbf{u}}_{n}^{h}\right)} \mathbf{n} - \left(p_{\perp, (\widehat{\mathbf{u}}_{t}^{h},\widehat{\mathbf{u}}_{n}^{h})}^{h}+\bar{p}^{h}\right) \mathbf{n} +\lambda^{h}_{\left(\widehat{\mathbf{u}}_{t}^{h},\widehat{\mathbf{u}}_{n}^{h}\right)}\mathbf{n},\,\widehat{\mathbf{v}}_{t}^{h}+\widehat{\mathbf{v}}_{n}^{h} \right\rangle_{\partial{{\mathcal{T}_{h}}}} \end{align} (4.4a) \begin{align} \nonumber &\qquad = \left\langle \nu\,\mathrm{L}^{h}_{(\mathbf{f}, g)} \mathbf{n} - p_{\perp, (\mathbf{f}, g)}^{h} \mathbf{n} +\lambda^{h}_{(\mathbf{f}, g)}\mathbf{n},\,\widehat{\mathbf{v}}_{t}^{h}+\widehat{\mathbf{v}}_{n}^{h} \right\rangle_{\partial{{\mathcal{T}_{h}}}}\!, \\ & \left( {\nabla\cdot}\left(\mathbf{u}^{h}_{(\widehat{\mathbf{u}}_{t}^{h},\widehat{\mathbf{u}}_{n}^{h})} + \mathbf{u}^{h}_{(\mathbf{f}, g)}\right)\!,\,\bar{q}^{h} \right)_{{\mathcal{T}_{h}}} = (g, \bar{q}^{h})_{\mathcal{T}_{h}}, \end{align} (4.4b) \begin{align} (\bar{p}^{h}, 1)_{\mathcal{T}_{h}} =0, \end{align} (4.4c) for all $$(\widehat{\mathbf{v}}_{t}^{h},\, \widehat{\mathbf{v}}_{n}^{h},\, \bar{q}^{h}) \in{\mathbf{M}}^{t}_{h}(0)\times{\mathbf{M}}^{n}_{h}(0) \times \overline{Q}_{h}$$. Again, the unique solvability of this problem is a simple consequence of that for equations (4.2). Moreover, we have the following characterization of equations (4.4). Its proof is trivial; see, e.g., Cockburn & Sayas (2014). Proposition 4.2 Equations (4.4) can be rewritten \begin{align*} A_{h}\left(\widehat{\mathbf{u}}_{t}^{h},\,\widehat{\mathbf{u}}_{n}^{h};\,\widehat{\mathbf{v}}_{t}^{h},\,\widehat{\mathbf{v}}_{n}^{h}\right) + B_{h}\left(\widehat{\mathbf{v}}_{n}^{h};\, \bar{p}^{h}\right) &= F_{h} \left(\widehat{\mathbf{v}}_{t}^{h},\,\widehat{\mathbf{v}}_{n}^{h}\right)\!,\\ B_{h}\left(\widehat{\mathbf{u}}_{n}^{h}; \bar{q}^{h}\right) &= 0,\\ (\bar{p}^{h}, 1)_{\mathcal{T}_{h}} &= 0, \end{align*} where \begin{align} A_{h}\left(\widehat{\mathbf{u}}_{t}^{h},\,\widehat{\mathbf{u}}_{n}^{h};\,\widehat{\mathbf{v}}_{t}^{h},\,\widehat{\mathbf{v}}_{n}^{h}\right) := \left( \nu \mathrm{L}^{h}_{\left(\widehat{\mathbf{u}}_{t}^{h},\widehat{\mathbf{u}}_{n}^{h}\right)},\,\mathrm{L}^{h}_{\left(\widehat{\mathbf{v}}_{t},\widehat{\mathbf{v}}_{n}\right)} \right)_{{\mathcal{T}_{h}}} + \left( \gamma \mathbf{u}^{h}_{\left(\widehat{\mathbf{u}}_{t}^{h},\widehat{\mathbf{u}}_{n}^{h}\right)} ,\,\mathbf{u}^{h}_{\left(\widehat{\mathbf{v}}_{t}^{h},\widehat{\mathbf{v}}_{n}^{h}\right)} \right)_{{\mathcal{T}_{h}}}\!, \end{align} (4.5a) \begin{align} B_{h}\left(\widehat{\mathbf{v}}_{n}^{h};\, \bar{p}^{h}\right) := - \left\langle \bar{p}^{h},\,\widehat{\mathbf{v}}_{n}^{h}\cdot \mathbf{n} \right\rangle_{\partial{{\mathcal{T}_{h}}}}\!, \end{align} (4.5b) \begin{align}\!\!\!\!\!\!\!\!\!\!\!\!\! F_{h} \left(\widehat{\mathbf{v}}_{t}^{h},\,\widehat{\mathbf{v}}_{n}^{h}\right) := & \left(\mathbf{f}, \mathbf{u}^{h}_{\left(\widehat{\mathbf{v}}_{t}^{h},\widehat{\mathbf{v}}_{n}^{h}\right)}\right)_{{\mathcal{T}_{h}}} - \left( \nu \mathrm{L}^{h}_{(\mathbf{f},g)},\,\mathrm{L}^{h}_{\left(\widehat{\mathbf{v}}_{t},\widehat{\mathbf{v}}_{n}\right)} \right)_{{\mathcal{T}_{h}}} \\ \nonumber & - \left( \gamma \mathbf{u}^{h}_{(\mathbf{f},g)} ,\,\mathbf{u}^{h}_{\left(\widehat{\mathbf{v}}_{t}^{h},\widehat{\mathbf{v}}_{n}^{h}\right)} \right)_{{\mathcal{T}_{h}}}\!. \end{align} (4.5c) 5. Numerical results In this section, we present two-dimensional numerical studies on both rectangular and triangular meshes to validate the theoretical results in Section 2. We use the deal.II (Bangerth et al., 2016) software to implement the HDG method (2.5) on rectangular meshes, and NGSolve (Schöberl, 1997; 2016) on triangular meshes. Recall that our approximation spaces are given in Table 1. The implementation on rectangular meshes uses the hybridization discussed in Section 4, while the implementation on triangular meshes uses NGSolve’s built-in static condensation approach; see Schöberl (2016). We present four numerical tests with a manufactured solution to validate our theoretical results in Section 2. For all the tests, the body forces f and g are chosen such that the exact solution (u, p) takes the form \begin{align*} \mathbf{u} =&\; \left(\sin(2\,\pi x)\sin(2\,\pi y), \sin(2\,\pi x)\sin(2\,\pi y)\right)^{\textrm{T}}\!,\\ p =&\; \sin(m\,\pi x)\sin(m\,\pi y),\ \textrm{where}\ m\ \textrm{is a fixed number.} \end{align*} We take ν = 1, γ = 1 and m = 2 for the first test, ν = 1, γ = 1 and m = 20 for the second test, ν = 0.0001, γ = 1 and m = 2 for the third test and ν = 1 e−8, γ = 1 and m = 2 for the last test. The first two tests are in the Stokes-dominated regime, while the last two tests are in the Darcy-dominated regime. The second test examines the effect of pressure regularity on the convergence of the velocity field. Table 2 History of convergence for the H(div)-conforming HDG method on square meshes Mesh D.O.F. $$\|\mathrm{L} - \mathrm{L}^{h} \|_{{\mathcal{T}_{h}}}$$ $$\|\mathbf{u}-\mathbf{u}^{h}\|_{{\mathcal{T}_{h}}}$$ $$\|p - p^{h} \|_{{\mathcal{T}_{h}}}$$ $$\|\mathbf{u}-\mathbf{u}^{*,h}\|_{{\mathcal{T}_{h}}}$$ k Nele Nglobal Nlocal Error Order Error Order Error Order Error Order First test: ν = 1, γ = 1, m = 2. 0 64 288 704 2.393e+00 — 1.622e-01 — 4.133e-01 — 5.398e-02 — 256 1088 2816 1.224e+00 0.97 8.043e-02 1.01 1.300e-01 1.67 1.337e-02 2.01 1024 4224 11264 6.157e-01 0.99 4.011e-02 1.00 4.782e-02 1.44 3.335e-03 2.00 4096 16640 45056 3.083e-01 1.00 2.004e-02 1.00 2.108e-02 1.18 8.331e-04 2.00 1 64 576 1856 4.951e-01 — 1.829e-02 — 1.178e-01 — 6.955e-03 — 256 2176 7424 1.286e-01 1.94 4.211e-03 2.12 1.559e-02 2.92 7.790e-04 3.16 1024 8448 29696 3.245e-02 1.99 1.026e-03 2.04 2.518e-03 2.63 9.367e-05 3.06 4096 33280 118784 8.131e-03 2.00 2.546e-04 2.01 5.171e-04 2.28 1.159e-05 3.02 2 64 864 3328 5.810e-02 — 1.399e-03 — 1.281e-02 — 7.069e-04 — 256 3264 13312 7.352e-03 2.98 1.481e-04 3.24 9.173e-04 3.80 4.129e-05 4.10 1024 12672 53248 9.223e-04 2.99 1.731e-05 3.10 7.743e-05 3.57 2.533e-06 4.03 4096 49920 212992 1.154e-04 3.00 2.122e-06 3.03 8.097e-06 3.26 1.575e-07 4.01 3 64 1152 5248 5.598e-03 — 9.147e-05 — 1.740e-03 — 6.264e-05 — 256 4352 20992 3.600e-04 3.96 4.127e-06 4.47 9.163e-05 4.25 2.049e-06 4.93 1024 16896 83968 2.272e-05 3.99 2.222e-07 4.21 5.203e-06 4.14 6.492e-08 4.98 4096 66560 335872 1.424e-06 4.00 1.325e-08 4.07 3.112e-07 4.06 2.036e-09 5.00 Second test: ν = 1, γ = 1, m = 20. 0 64 288 704 2.393e+00 — 1.622e-01 — 6.293e-01 — 5.398e-02 — 256 1088 2816 1.224e+00 0.97 8.043e-02 1.01 4.983e-01 0.34 1.337e-02 2.01 1024 4224 11264 6.157e-01 0.99 4.011e-02 1.00 3.494e-01 0.51 3.335e-03 2.00 4096 16640 45056 3.083e-01 1.00 2.004e-02 1.00 1.934e-01 0.85 8.331e-04 2.00 1 64 576 1856 4.951e-01 — 1.829e-02 — 5.117e-01 — 6.955e-03 — 256 2176 7424 1.286e-01 1.94 4.211e-03 2.12 4.186e-01 0.29 7.790e-04 3.16 1024 8448 29696 3.245e-02 1.99 1.026e-03 2.04 1.631e-01 1.36 9.367e-05 3.06 4096 33280 118784 8.131e-03 2.00 2.546e-04 2.01 4.573e-02 1.83 1.159e-05 3.02 2 64 864 3328 5.810e-02 — 1.399e-03 — 4.917e-01 — 7.069e-04 — 256 3264 13312 7.352e-03 2.98 1.481e-04 3.24 2.722e-01 0.85 4.129e-05 4.10 1024 12672 53248 9.223e-04 2.99 1.731e-05 3.10 5.209e-02 2.39 2.533e-06 4.03 4096 49920 212992 1.154e-04 3.00 2.122e-06 3.03 7.240e-03 2.85 1.575e-07 4.01 3 64 1152 5248 5.598e-03 — 9.147e-05 — 4.744e-01 — 6.264e-05 — 256 4352 20992 3.600e-04 3.96 4.127e-06 4.47 1.362e-01 1.80 2.049e-06 4.93 1024 16896 83968 2.272e-05 3.99 2.222e-07 4.21 1.252e-02 3.44 6.492e-08 4.98 4096 66560 335872 1.424e-06 4.00 1.325e-08 4.07 8.610e-04 3.86 2.036e-09 5.00 Mesh D.O.F. $$\|\mathrm{L} - \mathrm{L}^{h} \|_{{\mathcal{T}_{h}}}$$ $$\|\mathbf{u}-\mathbf{u}^{h}\|_{{\mathcal{T}_{h}}}$$ $$\|p - p^{h} \|_{{\mathcal{T}_{h}}}$$ $$\|\mathbf{u}-\mathbf{u}^{*,h}\|_{{\mathcal{T}_{h}}}$$ k Nele Nglobal Nlocal Error Order Error Order Error Order Error Order First test: ν = 1, γ = 1, m = 2. 0 64 288 704 2.393e+00 — 1.622e-01 — 4.133e-01 — 5.398e-02 — 256 1088 2816 1.224e+00 0.97 8.043e-02 1.01 1.300e-01 1.67 1.337e-02 2.01 1024 4224 11264 6.157e-01 0.99 4.011e-02 1.00 4.782e-02 1.44 3.335e-03 2.00 4096 16640 45056 3.083e-01 1.00 2.004e-02 1.00 2.108e-02 1.18 8.331e-04 2.00 1 64 576 1856 4.951e-01 — 1.829e-02 — 1.178e-01 — 6.955e-03 — 256 2176 7424 1.286e-01 1.94 4.211e-03 2.12 1.559e-02 2.92 7.790e-04 3.16 1024 8448 29696 3.245e-02 1.99 1.026e-03 2.04 2.518e-03 2.63 9.367e-05 3.06 4096 33280 118784 8.131e-03 2.00 2.546e-04 2.01 5.171e-04 2.28 1.159e-05 3.02 2 64 864 3328 5.810e-02 — 1.399e-03 — 1.281e-02 — 7.069e-04 — 256 3264 13312 7.352e-03 2.98 1.481e-04 3.24 9.173e-04 3.80 4.129e-05 4.10 1024 12672 53248 9.223e-04 2.99 1.731e-05 3.10 7.743e-05 3.57 2.533e-06 4.03 4096 49920 212992 1.154e-04 3.00 2.122e-06 3.03 8.097e-06 3.26 1.575e-07 4.01 3 64 1152 5248 5.598e-03 — 9.147e-05 — 1.740e-03 — 6.264e-05 — 256 4352 20992 3.600e-04 3.96 4.127e-06 4.47 9.163e-05 4.25 2.049e-06 4.93 1024 16896 83968 2.272e-05 3.99 2.222e-07 4.21 5.203e-06 4.14 6.492e-08 4.98 4096 66560 335872 1.424e-06 4.00 1.325e-08 4.07 3.112e-07 4.06 2.036e-09 5.00 Second test: ν = 1, γ = 1, m = 20. 0 64 288 704 2.393e+00 — 1.622e-01 — 6.293e-01 — 5.398e-02 — 256 1088 2816 1.224e+00 0.97 8.043e-02 1.01 4.983e-01 0.34 1.337e-02 2.01 1024 4224 11264 6.157e-01 0.99 4.011e-02 1.00 3.494e-01 0.51 3.335e-03 2.00 4096 16640 45056 3.083e-01 1.00 2.004e-02 1.00 1.934e-01 0.85 8.331e-04 2.00 1 64 576 1856 4.951e-01 — 1.829e-02 — 5.117e-01 — 6.955e-03 — 256 2176 7424 1.286e-01 1.94 4.211e-03 2.12 4.186e-01 0.29 7.790e-04 3.16 1024 8448 29696 3.245e-02 1.99 1.026e-03 2.04 1.631e-01 1.36 9.367e-05 3.06 4096 33280 118784 8.131e-03 2.00 2.546e-04 2.01 4.573e-02 1.83 1.159e-05 3.02 2 64 864 3328 5.810e-02 — 1.399e-03 — 4.917e-01 — 7.069e-04 — 256 3264 13312 7.352e-03 2.98 1.481e-04 3.24 2.722e-01 0.85 4.129e-05 4.10 1024 12672 53248 9.223e-04 2.99 1.731e-05 3.10 5.209e-02 2.39 2.533e-06 4.03 4096 49920 212992 1.154e-04 3.00 2.122e-06 3.03 7.240e-03 2.85 1.575e-07 4.01 3 64 1152 5248 5.598e-03 — 9.147e-05 — 4.744e-01 — 6.264e-05 — 256 4352 20992 3.600e-04 3.96 4.127e-06 4.47 1.362e-01 1.80 2.049e-06 4.93 1024 16896 83968 2.272e-05 3.99 2.222e-07 4.21 1.252e-02 3.44 6.492e-08 4.98 4096 66560 335872 1.424e-06 4.00 1.325e-08 4.07 8.610e-04 3.86 2.036e-09 5.00 Table 2 History of convergence for the H(div)-conforming HDG method on square meshes Mesh D.O.F. $$\|\mathrm{L} - \mathrm{L}^{h} \|_{{\mathcal{T}_{h}}}$$ $$\|\mathbf{u}-\mathbf{u}^{h}\|_{{\mathcal{T}_{h}}}$$ $$\|p - p^{h} \|_{{\mathcal{T}_{h}}}$$ $$\|\mathbf{u}-\mathbf{u}^{*,h}\|_{{\mathcal{T}_{h}}}$$ k Nele Nglobal Nlocal Error Order Error Order Error Order Error Order First test: ν = 1, γ = 1, m = 2. 0 64 288 704 2.393e+00 — 1.622e-01 — 4.133e-01 — 5.398e-02 — 256 1088 2816 1.224e+00 0.97 8.043e-02 1.01 1.300e-01 1.67 1.337e-02 2.01 1024 4224 11264 6.157e-01 0.99 4.011e-02 1.00 4.782e-02 1.44 3.335e-03 2.00 4096 16640 45056 3.083e-01 1.00 2.004e-02 1.00 2.108e-02 1.18 8.331e-04 2.00 1 64 576 1856 4.951e-01 — 1.829e-02 — 1.178e-01 — 6.955e-03 — 256 2176 7424 1.286e-01 1.94 4.211e-03 2.12 1.559e-02 2.92 7.790e-04 3.16 1024 8448 29696 3.245e-02 1.99 1.026e-03 2.04 2.518e-03 2.63 9.367e-05 3.06 4096 33280 118784 8.131e-03 2.00 2.546e-04 2.01 5.171e-04 2.28 1.159e-05 3.02 2 64 864 3328 5.810e-02 — 1.399e-03 — 1.281e-02 — 7.069e-04 — 256 3264 13312 7.352e-03 2.98 1.481e-04 3.24 9.173e-04 3.80 4.129e-05 4.10 1024 12672 53248 9.223e-04 2.99 1.731e-05 3.10 7.743e-05 3.57 2.533e-06 4.03 4096 49920 212992 1.154e-04 3.00 2.122e-06 3.03 8.097e-06 3.26 1.575e-07 4.01 3 64 1152 5248 5.598e-03 — 9.147e-05 — 1.740e-03 — 6.264e-05 — 256 4352 20992 3.600e-04 3.96 4.127e-06 4.47 9.163e-05 4.25 2.049e-06 4.93 1024 16896 83968 2.272e-05 3.99 2.222e-07 4.21 5.203e-06 4.14 6.492e-08 4.98 4096 66560 335872 1.424e-06 4.00 1.325e-08 4.07 3.112e-07 4.06 2.036e-09 5.00 Second test: ν = 1, γ = 1, m = 20. 0 64 288 704 2.393e+00 — 1.622e-01 — 6.293e-01 — 5.398e-02 — 256 1088 2816 1.224e+00 0.97 8.043e-02 1.01 4.983e-01 0.34 1.337e-02 2.01 1024 4224 11264 6.157e-01 0.99 4.011e-02 1.00 3.494e-01 0.51 3.335e-03 2.00 4096 16640 45056 3.083e-01 1.00 2.004e-02 1.00 1.934e-01 0.85 8.331e-04 2.00 1 64 576 1856 4.951e-01 — 1.829e-02 — 5.117e-01 — 6.955e-03 — 256 2176 7424 1.286e-01 1.94 4.211e-03 2.12 4.186e-01 0.29 7.790e-04 3.16 1024 8448 29696 3.245e-02 1.99 1.026e-03 2.04 1.631e-01 1.36 9.367e-05 3.06 4096 33280 118784 8.131e-03 2.00 2.546e-04 2.01 4.573e-02 1.83 1.159e-05 3.02 2 64 864 3328 5.810e-02 — 1.399e-03 — 4.917e-01 — 7.069e-04 — 256 3264 13312 7.352e-03 2.98 1.481e-04 3.24 2.722e-01 0.85 4.129e-05 4.10 1024 12672 53248 9.223e-04 2.99 1.731e-05 3.10 5.209e-02 2.39 2.533e-06 4.03 4096 49920 212992 1.154e-04 3.00 2.122e-06 3.03 7.240e-03 2.85 1.575e-07 4.01 3 64 1152 5248 5.598e-03 — 9.147e-05 — 4.744e-01 — 6.264e-05 — 256 4352 20992 3.600e-04 3.96 4.127e-06 4.47 1.362e-01 1.80 2.049e-06 4.93 1024 16896 83968 2.272e-05 3.99 2.222e-07 4.21 1.252e-02 3.44 6.492e-08 4.98 4096 66560 335872 1.424e-06 4.00 1.325e-08 4.07 8.610e-04 3.86 2.036e-09 5.00 Mesh D.O.F. $$\|\mathrm{L} - \mathrm{L}^{h} \|_{{\mathcal{T}_{h}}}$$ $$\|\mathbf{u}-\mathbf{u}^{h}\|_{{\mathcal{T}_{h}}}$$ $$\|p - p^{h} \|_{{\mathcal{T}_{h}}}$$ $$\|\mathbf{u}-\mathbf{u}^{*,h}\|_{{\mathcal{T}_{h}}}$$ k Nele Nglobal Nlocal Error Order Error Order Error Order Error Order First test: ν = 1, γ = 1, m = 2. 0 64 288 704 2.393e+00 — 1.622e-01 — 4.133e-01 — 5.398e-02 — 256 1088 2816 1.224e+00 0.97 8.043e-02 1.01 1.300e-01 1.67 1.337e-02 2.01 1024 4224 11264 6.157e-01 0.99 4.011e-02 1.00 4.782e-02 1.44 3.335e-03 2.00 4096 16640 45056 3.083e-01 1.00 2.004e-02 1.00 2.108e-02 1.18 8.331e-04 2.00 1 64 576 1856 4.951e-01 — 1.829e-02 — 1.178e-01 — 6.955e-03 — 256 2176 7424 1.286e-01 1.94 4.211e-03 2.12 1.559e-02 2.92 7.790e-04 3.16 1024 8448 29696 3.245e-02 1.99 1.026e-03 2.04 2.518e-03 2.63 9.367e-05 3.06 4096 33280 118784 8.131e-03 2.00 2.546e-04 2.01 5.171e-04 2.28 1.159e-05 3.02 2 64 864 3328 5.810e-02 — 1.399e-03 — 1.281e-02 — 7.069e-04 — 256 3264 13312 7.352e-03 2.98 1.481e-04 3.24 9.173e-04 3.80 4.129e-05 4.10 1024 12672 53248 9.223e-04 2.99 1.731e-05 3.10 7.743e-05 3.57 2.533e-06 4.03 4096 49920 212992 1.154e-04 3.00 2.122e-06 3.03 8.097e-06 3.26 1.575e-07 4.01 3 64 1152 5248 5.598e-03 — 9.147e-05 — 1.740e-03 — 6.264e-05 — 256 4352 20992 3.600e-04 3.96 4.127e-06 4.47 9.163e-05 4.25 2.049e-06 4.93 1024 16896 83968 2.272e-05 3.99 2.222e-07 4.21 5.203e-06 4.14 6.492e-08 4.98 4096 66560 335872 1.424e-06 4.00 1.325e-08 4.07 3.112e-07 4.06 2.036e-09 5.00 Second test: ν = 1, γ = 1, m = 20. 0 64 288 704 2.393e+00 — 1.622e-01 — 6.293e-01 — 5.398e-02 — 256 1088 2816 1.224e+00 0.97 8.043e-02 1.01 4.983e-01 0.34 1.337e-02 2.01 1024 4224 11264 6.157e-01 0.99 4.011e-02 1.00 3.494e-01 0.51 3.335e-03 2.00 4096 16640 45056 3.083e-01 1.00 2.004e-02 1.00 1.934e-01 0.85 8.331e-04 2.00 1 64 576 1856 4.951e-01 — 1.829e-02 — 5.117e-01 — 6.955e-03 — 256 2176 7424 1.286e-01 1.94 4.211e-03 2.12 4.186e-01 0.29 7.790e-04 3.16 1024 8448 29696 3.245e-02 1.99 1.026e-03 2.04 1.631e-01 1.36 9.367e-05 3.06 4096 33280 118784 8.131e-03 2.00 2.546e-04 2.01 4.573e-02 1.83 1.159e-05 3.02 2 64 864 3328 5.810e-02 — 1.399e-03 — 4.917e-01 — 7.069e-04 — 256 3264 13312 7.352e-03 2.98 1.481e-04 3.24 2.722e-01 0.85 4.129e-05 4.10 1024 12672 53248 9.223e-04 2.99 1.731e-05 3.10 5.209e-02 2.39 2.533e-06 4.03 4096 49920 212992 1.154e-04 3.00 2.122e-06 3.03 7.240e-03 2.85 1.575e-07 4.01 3 64 1152 5248 5.598e-03 — 9.147e-05 — 4.744e-01 — 6.264e-05 — 256 4352 20992 3.600e-04 3.96 4.127e-06 4.47 1.362e-01 1.80 2.049e-06 4.93 1024 16896 83968 2.272e-05 3.99 2.222e-07 4.21 1.252e-02 3.44 6.492e-08 4.98 4096 66560 335872 1.424e-06 4.00 1.325e-08 4.07 8.610e-04 3.86 2.036e-09 5.00 Table 2 Continued Mesh D.O.F. $$\|\mathrm{L} - \mathrm{L}^{h} \|_{{\mathcal{T}_{h}}}$$ $$\|\mathbf{u}-\mathbf{u}^{h}\|_{{\mathcal{T}_{h}}}$$ $$\|p - p^{h} \|_{{\mathcal{T}_{h}}}$$ $$\|\mathbf{u}-\mathbf{u}^{*,h}\|_{{\mathcal{T}_{h}}}$$ k Nele Nglobal Nlocal Error Order Error Order Error Order Error Order Third test: ν = 0.0001, γ = 1, m = 2. 0 64 288 704 2.399e+00 — 1.621e-01 — 1.567e-01 — 5.329e-02 — 256 1088 2816 1.226e+00 0.97 8.039e-02 1.01 7.970e-02 0.98 1.313e-02 2.02 1024 4224 11264 6.160e-01 0.99 4.011e-02 1.00 4.002e-02 0.99 3.268e-03 2.01 4096 16640 45056 3.083e-01 1.00 2.004e-02 1.00 2.003e-02 1.00 8.164e-04 2.00 1 64 576 1856 3.779e-01 — 1.679e-02 — 2.967e-02 — 6.192e-03 — 256 2176 7424 9.967e-02 1.92 4.096e-03 2.04 7.556e-03 1.97 7.509e-04 3.04 1024 8448 29696 2.761e-02 1.85 1.020e-03 2.01 1.898e-03 1.99 9.297e-05 3.01 4096 33280 118784 7.630e-03 1.86 2.544e-04 2.00 4.750e-04 2.00 1.157e-05 3.01 2 64 864 3328 4.844e-02 — 1.223e-03 — 3.755e-03 — 6.990e-04 — 256 3264 13312 6.177e-03 2.97 1.399e-04 3.13 4.773e-04 2.98 4.215e-05 4.05 1024 12672 53248 8.198e-04 2.91 1.708e-05 3.03 5.992e-05 2.99 2.571e-06 4.04 4096 49920 212992 1.099e-04 2.90 2.118e-06 3.01 7.498e-06 3.00 1.584e-07 4.02 3 64 1152 5248 4.973e-03 — 7.545e-05 — 3.567e-04 — 6.160e-05 — 256 4352 20992 3.248e-04 3.94 3.766e-06 4.32 2.264e-05 3.98 2.038e-06 4.92 1024 16896 83968 2.136e-05 3.93 2.173e-07 4.12 1.420e-06 3.99 6.486e-08 4.97 4096 66560 335872 1.390e-06 3.94 1.322e-08 4.04 8.885e-08 4.00 2.035e-09 4.99 Fourth test: ν = 1e−8, γ = 1, m = 2. 0 64 288 704 2.400e+00 — 1.621e-01 — 1.567e-01 — 5.329e-02 — 256 1088 2816 1.226e+00 0.97 8.039e-02 1.01 7.970e-02 0.98 1.313e-02 2.02 1024 4224 11264 6.162e-01 0.99 4.011e-02 1.00 4.002e-02 0.99 3.268e-03 2.01 4096 16640 45056 3.084e-01 1.00 2.004e-02 1.00 2.003e-02 1.00 8.156e-04 2.00 1 64 576 1856 3.717e-01 — 1.678e-02 — 2.967e-02 — 6.166e-03 — 256 2176 7424 9.358e-02 1.99 4.091e-03 2.04 7.556e-03 1.97 7.472e-04 3.04 1024 8448 29696 2.344e-02 2.00 1.018e-03 2.01 1.898e-03 1.99 9.261e-05 3.01 4096 33280 118784 5.863e-03 2.00 2.541e-04 2.00 4.750e-04 2.00 1.155e-05 3.00 2 64 864 3328 4.775e-02 — 1.222e-03 — 3.755e-03 — 6.963e-04 — 256 3264 13312 5.940e-03 3.01 1.396e-04 3.13 4.773e-04 2.98 4.237e-05 4.04 1024 12672 53248 7.406e-04 3.00 1.701e-05 3.04 5.992e-05 2.99 2.629e-06 4.01 4096 49920 212992 9.251e-05 3.00 2.112e-06 3.01 7.498e-06 3.00 1.640e-07 4.00 3 64 1152 5248 4.919e-03 — 7.526e-05 — 3.567e-04 — 6.159e-05 — 256 4352 20992 3.114e-04 3.98 3.737e-06 4.33 2.264e-05 3.98 2.032e-06 4.92 1024 16896 83968 1.955e-05 3.99 2.151e-07 4.12 1.420e-06 3.99 6.451e-08 4.98 4096 66560 335872 1.223e-06 4.00 1.314e-08 4.03 8.885e-08 4.00 2.024e-09 4.99 Mesh D.O.F. $$\|\mathrm{L} - \mathrm{L}^{h} \|_{{\mathcal{T}_{h}}}$$ $$\|\mathbf{u}-\mathbf{u}^{h}\|_{{\mathcal{T}_{h}}}$$ $$\|p - p^{h} \|_{{\mathcal{T}_{h}}}$$ $$\|\mathbf{u}-\mathbf{u}^{*,h}\|_{{\mathcal{T}_{h}}}$$ k Nele Nglobal Nlocal Error Order Error Order Error Order Error Order Third test: ν = 0.0001, γ = 1, m = 2. 0 64 288 704 2.399e+00 — 1.621e-01 — 1.567e-01 — 5.329e-02 — 256 1088 2816 1.226e+00 0.97 8.039e-02 1.01 7.970e-02 0.98 1.313e-02 2.02 1024 4224 11264 6.160e-01 0.99 4.011e-02 1.00 4.002e-02 0.99 3.268e-03 2.01 4096 16640 45056 3.083e-01 1.00 2.004e-02 1.00 2.003e-02 1.00 8.164e-04 2.00 1 64 576 1856 3.779e-01 — 1.679e-02 — 2.967e-02 — 6.192e-03 — 256 2176 7424 9.967e-02 1.92 4.096e-03 2.04 7.556e-03 1.97 7.509e-04 3.04 1024 8448 29696 2.761e-02 1.85 1.020e-03 2.01 1.898e-03 1.99 9.297e-05 3.01 4096 33280 118784 7.630e-03 1.86 2.544e-04 2.00 4.750e-04 2.00 1.157e-05 3.01 2 64 864 3328 4.844e-02 — 1.223e-03 — 3.755e-03 — 6.990e-04 — 256 3264 13312 6.177e-03 2.97 1.399e-04 3.13 4.773e-04 2.98 4.215e-05 4.05 1024 12672 53248 8.198e-04 2.91 1.708e-05 3.03 5.992e-05 2.99 2.571e-06 4.04 4096 49920 212992 1.099e-04 2.90 2.118e-06 3.01 7.498e-06 3.00 1.584e-07 4.02 3 64 1152 5248 4.973e-03 — 7.545e-05 — 3.567e-04 — 6.160e-05 — 256 4352 20992 3.248e-04 3.94 3.766e-06 4.32 2.264e-05 3.98 2.038e-06 4.92 1024 16896 83968 2.136e-05 3.93 2.173e-07 4.12 1.420e-06 3.99 6.486e-08 4.97 4096 66560 335872 1.390e-06 3.94 1.322e-08 4.04 8.885e-08 4.00 2.035e-09 4.99 Fourth test: ν = 1e−8, γ = 1, m = 2. 0 64 288 704 2.400e+00 — 1.621e-01 — 1.567e-01 — 5.329e-02 — 256 1088 2816 1.226e+00 0.97 8.039e-02 1.01 7.970e-02 0.98 1.313e-02 2.02 1024 4224 11264 6.162e-01 0.99 4.011e-02 1.00 4.002e-02 0.99 3.268e-03 2.01 4096 16640 45056 3.084e-01 1.00 2.004e-02 1.00 2.003e-02 1.00 8.156e-04 2.00 1 64 576 1856 3.717e-01 — 1.678e-02 — 2.967e-02 — 6.166e-03 — 256 2176 7424 9.358e-02 1.99 4.091e-03 2.04 7.556e-03 1.97 7.472e-04 3.04 1024 8448 29696 2.344e-02 2.00 1.018e-03 2.01 1.898e-03 1.99 9.261e-05 3.01 4096 33280 118784 5.863e-03 2.00 2.541e-04 2.00 4.750e-04 2.00 1.155e-05 3.00 2 64 864 3328 4.775e-02 — 1.222e-03 — 3.755e-03 — 6.963e-04 — 256 3264 13312 5.940e-03 3.01 1.396e-04 3.13 4.773e-04 2.98 4.237e-05 4.04 1024 12672 53248 7.406e-04 3.00 1.701e-05 3.04 5.992e-05 2.99 2.629e-06 4.01 4096 49920 212992 9.251e-05 3.00 2.112e-06 3.01 7.498e-06 3.00 1.640e-07 4.00 3 64 1152 5248 4.919e-03 — 7.526e-05 — 3.567e-04 — 6.159e-05 — 256 4352 20992 3.114e-04 3.98 3.737e-06 4.33 2.264e-05 3.98 2.032e-06 4.92 1024 16896 83968 1.955e-05 3.99 2.151e-07 4.12 1.420e-06 3.99 6.451e-08 4.98 4096 66560 335872 1.223e-06 4.00 1.314e-08 4.03 8.885e-08 4.00 2.024e-09 4.99 Table 2 Continued Mesh D.O.F. $$\|\mathrm{L} - \mathrm{L}^{h} \|_{{\mathcal{T}_{h}}}$$ $$\|\mathbf{u}-\mathbf{u}^{h}\|_{{\mathcal{T}_{h}}}$$ $$\|p - p^{h} \|_{{\mathcal{T}_{h}}}$$ $$\|\mathbf{u}-\mathbf{u}^{*,h}\|_{{\mathcal{T}_{h}}}$$ k Nele Nglobal Nlocal Error Order Error Order Error Order Error Order Third test: ν = 0.0001, γ = 1, m = 2. 0 64 288 704 2.399e+00 — 1.621e-01 — 1.567e-01 — 5.329e-02 — 256 1088 2816 1.226e+00 0.97 8.039e-02 1.01 7.970e-02 0.98 1.313e-02 2.02 1024 4224 11264 6.160e-01 0.99 4.011e-02 1.00 4.002e-02 0.99 3.268e-03 2.01 4096 16640 45056 3.083e-01 1.00 2.004e-02 1.00 2.003e-02 1.00 8.164e-04 2.00 1 64 576 1856 3.779e-01 — 1.679e-02 — 2.967e-02 — 6.192e-03 — 256 2176 7424 9.967e-02 1.92 4.096e-03 2.04 7.556e-03 1.97 7.509e-04 3.04 1024 8448 29696 2.761e-02 1.85 1.020e-03 2.01 1.898e-03 1.99 9.297e-05 3.01 4096 33280 118784 7.630e-03 1.86 2.544e-04 2.00 4.750e-04 2.00 1.157e-05 3.01 2 64 864 3328 4.844e-02 — 1.223e-03 — 3.755e-03 — 6.990e-04 — 256 3264 13312 6.177e-03 2.97 1.399e-04 3.13 4.773e-04 2.98 4.215e-05 4.05 1024 12672 53248 8.198e-04 2.91 1.708e-05 3.03 5.992e-05 2.99 2.571e-06 4.04 4096 49920 212992 1.099e-04 2.90 2.118e-06 3.01 7.498e-06 3.00 1.584e-07 4.02 3 64 1152 5248 4.973e-03 — 7.545e-05 — 3.567e-04 — 6.160e-05 — 256 4352 20992 3.248e-04 3.94 3.766e-06 4.32 2.264e-05 3.98 2.038e-06 4.92 1024 16896 83968 2.136e-05 3.93 2.173e-07 4.12 1.420e-06 3.99 6.486e-08 4.97 4096 66560 335872 1.390e-06 3.94 1.322e-08 4.04 8.885e-08 4.00 2.035e-09 4.99 Fourth test: ν = 1e−8, γ = 1, m = 2. 0 64 288 704 2.400e+00 — 1.621e-01 — 1.567e-01 — 5.329e-02 — 256 1088 2816 1.226e+00 0.97 8.039e-02 1.01 7.970e-02 0.98 1.313e-02 2.02 1024 4224 11264 6.162e-01 0.99 4.011e-02 1.00 4.002e-02 0.99 3.268e-03 2.01 4096 16640 45056 3.084e-01 1.00 2.004e-02 1.00 2.003e-02 1.00 8.156e-04 2.00 1 64 576 1856 3.717e-01 — 1.678e-02 — 2.967e-02 — 6.166e-03 — 256 2176 7424 9.358e-02 1.99 4.091e-03 2.04 7.556e-03 1.97 7.472e-04 3.04 1024 8448 29696 2.344e-02 2.00 1.018e-03 2.01 1.898e-03 1.99 9.261e-05 3.01 4096 33280 118784 5.863e-03 2.00 2.541e-04 2.00 4.750e-04 2.00 1.155e-05 3.00 2 64 864 3328 4.775e-02 — 1.222e-03 — 3.755e-03 — 6.963e-04 — 256 3264 13312 5.940e-03 3.01 1.396e-04 3.13 4.773e-04 2.98 4.237e-05 4.04 1024 12672 53248 7.406e-04 3.00 1.701e-05 3.04 5.992e-05 2.99 2.629e-06 4.01 4096 49920 212992 9.251e-05 3.00 2.112e-06 3.01 7.498e-06 3.00 1.640e-07 4.00 3 64 1152 5248 4.919e-03 — 7.526e-05 — 3.567e-04 — 6.159e-05 — 256 4352 20992 3.114e-04 3.98 3.737e-06 4.33 2.264e-05 3.98 2.032e-06 4.92 1024 16896 83968 1.955e-05 3.99 2.151e-07 4.12 1.420e-06 3.99 6.451e-08 4.98 4096 66560 335872 1.223e-06 4.00 1.314e-08 4.03 8.885e-08 4.00 2.024e-09 4.99 Mesh D.O.F. $$\|\mathrm{L} - \mathrm{L}^{h} \|_{{\mathcal{T}_{h}}}$$ $$\|\mathbf{u}-\mathbf{u}^{h}\|_{{\mathcal{T}_{h}}}$$ $$\|p - p^{h} \|_{{\mathcal{T}_{h}}}$$ $$\|\mathbf{u}-\mathbf{u}^{*,h}\|_{{\mathcal{T}_{h}}}$$ k Nele Nglobal Nlocal Error Order Error Order Error Order Error Order Third test: ν = 0.0001, γ = 1, m = 2. 0 64 288 704 2.399e+00 — 1.621e-01 — 1.567e-01 — 5.329e-02 — 256 1088 2816 1.226e+00 0.97 8.039e-02 1.01 7.970e-02 0.98 1.313e-02 2.02 1024 4224 11264 6.160e-01 0.99 4.011e-02 1.00 4.002e-02 0.99 3.268e-03 2.01 4096 16640 45056 3.083e-01 1.00 2.004e-02 1.00 2.003e-02 1.00 8.164e-04 2.00 1 64 576 1856 3.779e-01 — 1.679e-02 — 2.967e-02 — 6.192e-03 — 256 2176 7424 9.967e-02 1.92 4.096e-03 2.04 7.556e-03 1.97 7.509e-04 3.04 1024 8448 29696 2.761e-02 1.85 1.020e-03 2.01 1.898e-03 1.99 9.297e-05 3.01 4096 33280 118784 7.630e-03 1.86 2.544e-04 2.00 4.750e-04 2.00 1.157e-05 3.01 2 64 864 3328 4.844e-02 — 1.223e-03 — 3.755e-03 — 6.990e-04 — 256 3264 13312 6.177e-03 2.97 1.399e-04 3.13 4.773e-04 2.98 4.215e-05 4.05 1024 12672 53248 8.198e-04 2.91 1.708e-05 3.03 5.992e-05 2.99 2.571e-06 4.04 4096 49920 212992 1.099e-04 2.90 2.118e-06 3.01 7.498e-06 3.00 1.584e-07 4.02 3 64 1152 5248 4.973e-03 — 7.545e-05 — 3.567e-04 — 6.160e-05 — 256 4352 20992 3.248e-04 3.94 3.766e-06 4.32 2.264e-05 3.98 2.038e-06 4.92 1024 16896 83968 2.136e-05 3.93 2.173e-07 4.12 1.420e-06 3.99 6.486e-08 4.97 4096 66560 335872 1.390e-06 3.94 1.322e-08 4.04 8.885e-08 4.00 2.035e-09 4.99 Fourth test: ν = 1e−8, γ = 1, m = 2. 0 64 288 704 2.400e+00 — 1.621e-01 — 1.567e-01 — 5.329e-02 — 256 1088 2816 1.226e+00 0.97 8.039e-02 1.01 7.970e-02 0.98 1.313e-02 2.02 1024 4224 11264 6.162e-01 0.99 4.011e-02 1.00 4.002e-02 0.99 3.268e-03 2.01 4096 16640 45056 3.084e-01 1.00 2.004e-02 1.00 2.003e-02 1.00 8.156e-04 2.00 1 64 576 1856 3.717e-01 — 1.678e-02 — 2.967e-02 — 6.166e-03 — 256 2176 7424 9.358e-02 1.99 4.091e-03 2.04 7.556e-03 1.97 7.472e-04 3.04 1024 8448 29696 2.344e-02 2.00 1.018e-03 2.01 1.898e-03 1.99 9.261e-05 3.01 4096 33280 118784 5.863e-03 2.00 2.541e-04 2.00 4.750e-04 2.00 1.155e-05 3.00 2 64 864 3328 4.775e-02 — 1.222e-03 — 3.755e-03 — 6.963e-04 — 256 3264 13312 5.940e-03 3.01 1.396e-04 3.13 4.773e-04 2.98 4.237e-05 4.04 1024 12672 53248 7.406e-04 3.00 1.701e-05 3.04 5.992e-05 2.99 2.629e-06 4.01 4096 49920 212992 9.251e-05 3.00 2.112e-06 3.01 7.498e-06 3.00 1.640e-07 4.00 3 64 1152 5248 4.919e-03 — 7.526e-05 — 3.567e-04 — 6.159e-05 — 256 4352 20992 3.114e-04 3.98 3.737e-06 4.33 2.264e-05 3.98 2.032e-06 4.92 1024 16896 83968 1.955e-05 3.99 2.151e-07 4.12 1.420e-06 3.99 6.451e-08 4.98 4096 66560 335872 1.223e-06 4.00 1.314e-08 4.03 8.885e-08 4.00 2.024e-09 4.99 In Table 2, we present the L2-convergence rates for Lh, uh, ph and u*, h for the HDG method (2.5) with polynomial degree varying from k = 0 to k = 3 on rectangular meshes. The first-level mesh consists of 8 × 8 congruent squares, and the consequent meshes are obtained by uniform refinements. Here, the local postprocessing u*, h ∈ Pk+1(K) is defined elementwise by the following set of equations: \begin{align*} ({\boldsymbol{\nabla}} \mathbf{u}^{*,h},{\boldsymbol{\nabla}} \mathbf{v})_{K} = &\; (\mathrm{L}^{h}, {\boldsymbol{\nabla}} \mathbf{v})_{K}\quad \forall \, \mathbf{v}\in\boldsymbol{\mathcal{P}}_{k+1}(K),\\ (\mathbf{u}^{*,h},\mathbf{w})_{K} = &\; (\mathbf{u}^{h}, \mathbf{w})_{K}\quad \forall \, \mathbf{w}\in\boldsymbol{\mathcal{P}}_{0}(K). \end{align*} It is quite easy to show (cf. Stenberg, 1991; Cockburn et al., 2010) that u*, h convergence with an order of k + 2 − δ0, k, where $$\delta _{0,k}=\begin{cases} 1 & if\, k=0\\ 0 & if\, k>0\\ \end{cases} $$ is the Kronecker delta, if u has enough regularity. Table 3 History of convergence for the H(div)-conforming HDG method on triangular meshes Mesh D.O.F. $$\|\mathrm{L} - \mathrm{L}^{h} \|_{{\mathcal{T}_{h}}}$$ $$\|\mathbf{u}-\mathbf{u}^{h}\|_{{\mathcal{T}_{h}}}$$ $$\|p - p^{h} \|_{{\mathcal{T}_{h}}}$$ $$\|\mathbf{u}-\mathbf{u}^{*,h}\|_{{\mathcal{T}_{h}}}$$ k Nele Nglobal Nlocal Error Order Error Order Error Order Error Order First test: ν = 1, γ = 1, m = 2. 1 32 256 555 1.567e+00 — 8.253e-02 — 5.144e-01 — 5.985e-02 — 128 960 2203 3.378e-01 2.21 3.220e-02 1.36 1.158e-01 2.15 6.449e-03 3.21 512 3712 8763 8.757e-02 1.95 8.073e-03 2.00 2.712e-02 2.09 8.455e-04 2.93 2048 14592 34939 2.213e-02 1.98 2.018e-03 2.00 6.559e-03 2.05 1.073e-04 2.98 8192 57856 139515 5.550e-03 2.00 5.045e-04 2.00 1.615e-03 2.02 1.348e-05 2.99 2 32 368 1163 9.679e-02 — 3.553e-02 — 4.949e-02 — 2.407e-03 — 128 1376 4635 3.471e-02 1.48 3.432e-03 3.37 1.183e-02 2.07 4.712e-04 2.35 512 5312 18491 4.381e-03 2.99 4.359e-04 2.98 1.488e-03 2.99 2.964e-05 3.99 2048 20864 73851 5.488e-04 3.00 5.472e-05 2.99 1.862e-04 3.00 1.854e-06 4.00 8192 82688 295163 6.864e-05 3.00 6.847e-06 3.00 2.325e-05 3.00 1.159e-07 4.00 3 32 480 1995 3.551e-02 — 1.557e-03 — 2.159e-02 — 1.760e-03 — 128 1792 7963 1.815e-03 4.29 .245e-04 2.79 9.946e-04 4.44 4.237e-05 5.38 512 6912 31803 1.172e-04 3.95 1.418e-05 3.99 6.099e-05 4.03 1.356e-06 4.97 2048 27136 127099 7.398e-06 3.99 8.883e-07 4.00 3.774e-06 4.01 4.266e-08 4.99 8192 107520 508155 4.638e-07 4.00 5.555e-08 4.00 2.348e-07 4.01 1.336e-09 5.00 Second test: ν = 1, γ = 1, m = 20. 1 32 256 555 1.582e+00 — 8.376e-02 — 1.022e+00 — 6.085e-02 — 128 960 2203 3.652e-01 2.12 3.256e-02 1.36 6.395e-01 0.68 7.492e-03 3.02 512 3712 8763 8.758e-02 2.06 8.073e-03 2.01 3.010e-01 1.09 8.457e-04 3.15 2048 14592 34939 2.213e-02 1.98 2.018e-03 2.00 1.091e-01 1.46 1.073e-04 2.98 8192 57856 139515 5.550e-03 2.00 5.045e-04 2.00 3.012e-02 1.86 1.348e-05 2.99 2 32 368 1163 1.813e-01 — 3.599e-02 — 6.015e-01 — 5.208e-03 — 128 1376 4635 3.471e-02 2.39 3.432e-03 3.39 4.004e-01 0.59 4.715e-04 3.47 512 5312 18491 4.381e-03 2.99 4.359e-04 2.98 1.741e-01 1.20 2.964e-05 3.99 2048 20864 73851 5.488e-04 3.00 5.472e-05 2.99 3.076e-02 2.50 1.854e-06 4.00 8192 82688 295163 6.864e-05 3.00 6.847e-06 3.00 4.192e-03 2.88 1.159e-07 4.00 3 32 480 1995 4.193e-02 — 1.793e-03 — 5.592e-01 — 1.907e-03 — 128 1792 7963 1.815e-03 4.53 2.245e-04 3.00 3.068e-01 0.87 4.237e-05 5.49 512 6912 31803 1.172e-04 3.95 1.418e-05 3.99 3.201e-02 3.26 1.356e-06 4.97 2048 27136 127099 7.398e-06 3.99 8.883e-07 4.00 1.507e-03 4.41 4.266e-08 4.99 8192 107520 508155 4.638e-07 4.00 5.555e-08 4.00 6.589e-05 4.52 1.336e-09 5.00 Mesh D.O.F. $$\|\mathrm{L} - \mathrm{L}^{h} \|_{{\mathcal{T}_{h}}}$$ $$\|\mathbf{u}-\mathbf{u}^{h}\|_{{\mathcal{T}_{h}}}$$ $$\|p - p^{h} \|_{{\mathcal{T}_{h}}}$$ $$\|\mathbf{u}-\mathbf{u}^{*,h}\|_{{\mathcal{T}_{h}}}$$ k Nele Nglobal Nlocal Error Order Error Order Error Order Error Order First test: ν = 1, γ = 1, m = 2. 1 32 256 555 1.567e+00 — 8.253e-02 — 5.144e-01 — 5.985e-02 — 128 960 2203 3.378e-01 2.21 3.220e-02 1.36 1.158e-01 2.15 6.449e-03 3.21 512 3712 8763 8.757e-02 1.95 8.073e-03 2.00 2.712e-02 2.09 8.455e-04 2.93 2048 14592 34939 2.213e-02 1.98 2.018e-03 2.00 6.559e-03 2.05 1.073e-04 2.98 8192 57856 139515 5.550e-03 2.00 5.045e-04 2.00 1.615e-03 2.02 1.348e-05 2.99 2 32 368 1163 9.679e-02 — 3.553e-02 — 4.949e-02 — 2.407e-03 — 128 1376 4635 3.471e-02 1.48 3.432e-03 3.37 1.183e-02 2.07 4.712e-04 2.35 512 5312 18491 4.381e-03 2.99 4.359e-04 2.98 1.488e-03 2.99 2.964e-05 3.99 2048 20864 73851 5.488e-04 3.00 5.472e-05 2.99 1.862e-04 3.00 1.854e-06 4.00 8192 82688 295163 6.864e-05 3.00 6.847e-06 3.00 2.325e-05 3.00 1.159e-07 4.00 3 32 480 1995 3.551e-02 — 1.557e-03 — 2.159e-02 — 1.760e-03 — 128 1792 7963 1.815e-03 4.29 .245e-04 2.79 9.946e-04 4.44 4.237e-05 5.38 512 6912 31803 1.172e-04 3.95 1.418e-05 3.99 6.099e-05 4.03 1.356e-06 4.97 2048 27136 127099 7.398e-06 3.99 8.883e-07 4.00 3.774e-06 4.01 4.266e-08 4.99 8192 107520 508155 4.638e-07 4.00 5.555e-08 4.00 2.348e-07 4.01 1.336e-09 5.00 Second test: ν = 1, γ = 1, m = 20. 1 32 256 555 1.582e+00 — 8.376e-02 — 1.022e+00 — 6.085e-02 — 128 960 2203 3.652e-01 2.12 3.256e-02 1.36 6.395e-01 0.68 7.492e-03 3.02 512 3712 8763 8.758e-02 2.06 8.073e-03 2.01 3.010e-01 1.09 8.457e-04 3.15 2048 14592 34939 2.213e-02 1.98 2.018e-03 2.00 1.091e-01 1.46 1.073e-04 2.98 8192 57856 139515 5.550e-03 2.00 5.045e-04 2.00 3.012e-02 1.86 1.348e-05 2.99 2 32 368 1163 1.813e-01 — 3.599e-02 — 6.015e-01 — 5.208e-03 — 128 1376 4635 3.471e-02 2.39 3.432e-03 3.39 4.004e-01 0.59 4.715e-04 3.47 512 5312 18491 4.381e-03 2.99 4.359e-04 2.98 1.741e-01 1.20 2.964e-05 3.99 2048 20864 73851 5.488e-04 3.00 5.472e-05 2.99 3.076e-02 2.50 1.854e-06 4.00 8192 82688 295163 6.864e-05 3.00 6.847e-06 3.00 4.192e-03 2.88 1.159e-07 4.00 3 32 480 1995 4.193e-02 — 1.793e-03 — 5.592e-01 — 1.907e-03 — 128 1792 7963 1.815e-03 4.53 2.245e-04 3.00 3.068e-01 0.87 4.237e-05 5.49 512 6912 31803 1.172e-04 3.95 1.418e-05 3.99 3.201e-02 3.26 1.356e-06 4.97 2048 27136 127099 7.398e-06 3.99 8.883e-07 4.00 1.507e-03 4.41 4.266e-08 4.99 8192 107520 508155 4.638e-07 4.00 5.555e-08 4.00 6.589e-05 4.52 1.336e-09 5.00 Table 3 History of convergence for the H(div)-conforming HDG method on triangular meshes Mesh D.O.F. $$\|\mathrm{L} - \mathrm{L}^{h} \|_{{\mathcal{T}_{h}}}$$ $$\|\mathbf{u}-\mathbf{u}^{h}\|_{{\mathcal{T}_{h}}}$$ $$\|p - p^{h} \|_{{\mathcal{T}_{h}}}$$ $$\|\mathbf{u}-\mathbf{u}^{*,h}\|_{{\mathcal{T}_{h}}}$$ k Nele Nglobal Nlocal Error Order Error Order Error Order Error Order First test: ν = 1, γ = 1, m = 2. 1 32 256 555 1.567e+00 — 8.253e-02 — 5.144e-01 — 5.985e-02 — 128 960 2203 3.378e-01 2.21 3.220e-02 1.36 1.158e-01 2.15 6.449e-03 3.21 512 3712 8763 8.757e-02 1.95 8.073e-03 2.00 2.712e-02 2.09 8.455e-04 2.93 2048 14592 34939 2.213e-02 1.98 2.018e-03 2.00 6.559e-03 2.05 1.073e-04 2.98 8192 57856 139515 5.550e-03 2.00 5.045e-04 2.00 1.615e-03 2.02 1.348e-05 2.99 2 32 368 1163 9.679e-02 — 3.553e-02 — 4.949e-02 — 2.407e-03 — 128 1376 4635 3.471e-02 1.48 3.432e-03 3.37 1.183e-02 2.07 4.712e-04 2.35 512 5312 18491 4.381e-03 2.99 4.359e-04 2.98 1.488e-03 2.99 2.964e-05 3.99 2048 20864 73851 5.488e-04 3.00 5.472e-05 2.99 1.862e-04 3.00 1.854e-06 4.00 8192 82688 295163 6.864e-05 3.00 6.847e-06 3.00 2.325e-05 3.00 1.159e-07 4.00 3 32 480 1995 3.551e-02 — 1.557e-03 — 2.159e-02 — 1.760e-03 — 128 1792 7963 1.815e-03 4.29 .245e-04 2.79 9.946e-04 4.44 4.237e-05 5.38 512 6912 31803 1.172e-04 3.95 1.418e-05 3.99 6.099e-05 4.03 1.356e-06 4.97 2048 27136 127099 7.398e-06 3.99 8.883e-07 4.00 3.774e-06 4.01 4.266e-08 4.99 8192 107520 508155 4.638e-07 4.00 5.555e-08 4.00 2.348e-07 4.01 1.336e-09 5.00 Second test: ν = 1, γ = 1, m = 20. 1 32 256 555 1.582e+00 — 8.376e-02 — 1.022e+00 — 6.085e-02 — 128 960 2203 3.652e-01 2.12 3.256e-02 1.36 6.395e-01 0.68 7.492e-03 3.02 512 3712 8763 8.758e-02 2.06 8.073e-03 2.01 3.010e-01 1.09 8.457e-04 3.15 2048 14592 34939 2.213e-02 1.98 2.018e-03 2.00 1.091e-01 1.46 1.073e-04 2.98 8192 57856 139515 5.550e-03 2.00 5.045e-04 2.00 3.012e-02 1.86 1.348e-05 2.99 2 32 368 1163 1.813e-01 — 3.599e-02 — 6.015e-01 — 5.208e-03 — 128 1376 4635 3.471e-02 2.39 3.432e-03 3.39 4.004e-01 0.59 4.715e-04 3.47 512 5312 18491 4.381e-03 2.99 4.359e-04 2.98 1.741e-01 1.20 2.964e-05 3.99 2048 20864 73851 5.488e-04 3.00 5.472e-05 2.99 3.076e-02 2.50 1.854e-06 4.00 8192 82688 295163 6.864e-05 3.00 6.847e-06 3.00 4.192e-03 2.88 1.159e-07 4.00 3 32 480 1995 4.193e-02 — 1.793e-03 — 5.592e-01 — 1.907e-03 — 128 1792 7963 1.815e-03 4.53 2.245e-04 3.00 3.068e-01 0.87 4.237e-05 5.49 512 6912 31803 1.172e-04 3.95 1.418e-05 3.99 3.201e-02 3.26 1.356e-06 4.97 2048 27136 127099 7.398e-06 3.99 8.883e-07 4.00 1.507e-03 4.41 4.266e-08 4.99 8192 107520 508155 4.638e-07 4.00 5.555e-08 4.00 6.589e-05 4.52 1.336e-09 5.00 Mesh D.O.F. $$\|\mathrm{L} - \mathrm{L}^{h} \|_{{\mathcal{T}_{h}}}$$ $$\|\mathbf{u}-\mathbf{u}^{h}\|_{{\mathcal{T}_{h}}}$$ $$\|p - p^{h} \|_{{\mathcal{T}_{h}}}$$ $$\|\mathbf{u}-\mathbf{u}^{*,h}\|_{{\mathcal{T}_{h}}}$$ k Nele Nglobal Nlocal Error Order Error Order Error Order Error Order First test: ν = 1, γ = 1, m = 2. 1 32 256 555 1.567e+00 — 8.253e-02 — 5.144e-01 — 5.985e-02 — 128 960 2203 3.378e-01 2.21 3.220e-02 1.36 1.158e-01 2.15 6.449e-03 3.21 512 3712 8763 8.757e-02 1.95 8.073e-03 2.00 2.712e-02 2.09 8.455e-04 2.93 2048 14592 34939 2.213e-02 1.98 2.018e-03 2.00 6.559e-03 2.05 1.073e-04 2.98 8192 57856 139515 5.550e-03 2.00 5.045e-04 2.00 1.615e-03 2.02 1.348e-05 2.99 2 32 368 1163 9.679e-02 — 3.553e-02 — 4.949e-02 — 2.407e-03 — 128 1376 4635 3.471e-02 1.48 3.432e-03 3.37 1.183e-02 2.07 4.712e-04 2.35 512 5312 18491 4.381e-03 2.99 4.359e-04 2.98 1.488e-03 2.99 2.964e-05 3.99 2048 20864 73851 5.488e-04 3.00 5.472e-05 2.99 1.862e-04 3.00 1.854e-06 4.00 8192 82688 295163 6.864e-05 3.00 6.847e-06 3.00 2.325e-05 3.00 1.159e-07 4.00 3 32 480 1995 3.551e-02 — 1.557e-03 — 2.159e-02 — 1.760e-03 — 128 1792 7963 1.815e-03 4.29 .245e-04 2.79 9.946e-04 4.44 4.237e-05 5.38 512 6912 31803 1.172e-04 3.95 1.418e-05 3.99 6.099e-05 4.03 1.356e-06 4.97 2048 27136 127099 7.398e-06 3.99 8.883e-07 4.00 3.774e-06 4.01 4.266e-08 4.99 8192 107520 508155 4.638e-07 4.00 5.555e-08 4.00 2.348e-07 4.01 1.336e-09 5.00 Second test: ν = 1, γ = 1, m = 20. 1 32 256 555 1.582e+00 — 8.376e-02 — 1.022e+00 — 6.085e-02 — 128 960 2203 3.652e-01 2.12 3.256e-02 1.36 6.395e-01 0.68 7.492e-03 3.02 512 3712 8763 8.758e-02 2.06 8.073e-03 2.01 3.010e-01 1.09 8.457e-04 3.15 2048 14592 34939 2.213e-02 1.98 2.018e-03 2.00 1.091e-01 1.46 1.073e-04 2.98 8192 57856 139515 5.550e-03 2.00 5.045e-04 2.00 3.012e-02 1.86 1.348e-05 2.99 2 32 368 1163 1.813e-01 — 3.599e-02 — 6.015e-01 — 5.208e-03 — 128 1376 4635 3.471e-02 2.39 3.432e-03 3.39 4.004e-01 0.59 4.715e-04 3.47 512 5312 18491 4.381e-03 2.99 4.359e-04 2.98 1.741e-01 1.20 2.964e-05 3.99 2048 20864 73851 5.488e-04 3.00 5.472e-05 2.99 3.076e-02 2.50 1.854e-06 4.00 8192 82688 295163 6.864e-05 3.00 6.847e-06 3.00 4.192e-03 2.88 1.159e-07 4.00 3 32 480 1995 4.193e-02 — 1.793e-03 — 5.592e-01 — 1.907e-03 — 128 1792 7963 1.815e-03 4.53 2.245e-04 3.00 3.068e-01 0.87 4.237e-05 5.49 512 6912 31803 1.172e-04 3.95 1.418e-05 3.99 3.201e-02 3.26 1.356e-06 4.97 2048 27136 127099 7.398e-06 3.99 8.883e-07 4.00 1.507e-03 4.41 4.266e-08 4.99 8192 107520 508155 4.638e-07 4.00 5.555e-08 4.00 6.589e-05 4.52 1.336e-09 5.00 Table 3 Continued Mesh D.O.F. $$\|\mathrm{L} - \mathrm{L}^{h} \|_{{\mathcal{T}_{h}}}$$ $$\|\mathbf{u}-\mathbf{u}^{h}\|_{{\mathcal{T}_{h}}}$$ $$\|p - p^{h} \|_{{\mathcal{T}_{h}}}$$ $$\|\mathbf{u}-\mathbf{u}^{*,h}\|_{{\mathcal{T}_{h}}}$$ k Nele Nglobal Nlocal Error Order Error Order Error Order Error Order Third test: ν = 0.0001, γ = 1, m = 2. 1 32 256 555 1.436e+00 — 7.825e-02 — 3.891e-02 — 5.242e-02 — 128 960 2203 3.932e-01 1.87 3.013e-02 1.38 1.949e-02 1.00 8.254e-03 2.67 512 3712 8763 1.241e-01 1.66 7.719e-03 1.96 4.951e-03 1.98 1.497e-03 2.46 2048 14592 34939 3.414e-02 1.86 1.962e-03 1.98 1.243e-03 1.99 2.153e-04 2.80 8192 57856 139515 7.387e-03 2.21 4.987e-04 1.98 3.110e-04 2.00 2.215e-05 3.28 2 32 368 1163 3.442e-01 — 3.288e-02 — 2.266e-02 — 1.065e-02 — 128 1376 4635 6.978e-02 2.30 3.132e-03 3.39 2.169e-03 3.39 9.514e-04 3.48 512 5312 18491 1.042e-02 2.74 4.049e-04 2.95 2.748e-04 2.98 7.304e-05 3.70 2048 20864 73851 1.085e-03 3.26 5.285e-05 2.94 3.447e-05 3.00 4.209e-06 4.12 8192 82688 295163 9.842e-05 3.46 6.770e-06 2.96 4.313e-06 3.00 1.938e-07 4.44 3 32 480 1995 3.231e-02 — 1.545e-03 — 6.370e-04 — 1.717e-03 — 128 1792 7963 6.311e-03 2.36 1.928e-04 3.00 2.490e-05 4.68 6.757e-05 4.67 512 6912 31803 3.905e-04 4.01 1.295e-05 3.90 1.348e-06 4.21 2.318e-06 4.87 2048 27136 127099 1.768e-05 4.46 8.583e-07 3.91 8.055e-08 4.07 6.261e-08 5.21 8192 107520 508155 7.284e-07 4.60 5.500e-08 3.96 4.974e-09 4.02 2.111e-09 4.89 Fourth test: ν = 1e −8, γ = 1, m = 2. 1 32 256 555 1.434e+00 — 7.825e-02 — 3.891e-02 — 5.233e-02 — 128 960 2203 4.167e-01 1.78 3.011e-02 1.38 1.949e-02 1.00 8.738e-03 2.58 512 3712 8763 1.581e-01 1.40 7.687e-03 1.97 4.951e-03 1.98 1.919e-03 2.19 2048 14592 34939 7.048e-02 1.17 1.932e-03 1.99 1.243e-03 1.99 4.633e-04 2.05 8192 57856 139515 3.411e-02 1.05 4.836e-04 2.00 3.110e-04 2.00 1.151e-04 2.01 2 32 368 1163 3.778e-01 — 3.286e-02 — 2.266e-02 — 1.139e-02 — 128 1376 4635 9.041e-02 2.06 3.110e-03 3.40 2.169e-03 3.39 1.172e-03 3.28 512 5312 18491 2.349e-02 1.94 3.895e-04 3.00 2.748e-04 2.98 1.440e-04 3.02 2048 20864 73851 5.980e-03 1.97 4.859e-05 3.00 3.447e-05 3.00 1.805e-05 3.00 8192 82688 295163 1.504e-03 1.99 6.065e-06 3.00 4.313e-06 3.00 2.261e-06 3.00 3 32 480 1995 3.228e-02 — 1.551e-03 — 6.370e-04 — 1.711e-03 — 128 1792 7963 1.094e-02 1.56 1.882e-04 3.04 2.489e-05 4.68 9.261e-05 4.21 512 6912 31803 1.314e-03 3.06 1.191e-05 3.98 1.348e-06 4.21 5.506e-06 4.07 2048 27136 127099 1.612e-04 3.03 7.466e-07 4.00 8.055e-08 4.06 3.403e-07 4.02 8192 107520 508155 1.995e-05 3.01 4.669e-08 4.00 4.974e-09 4.02 2.122e-08 4.00 Mesh D.O.F. $$\|\mathrm{L} - \mathrm{L}^{h} \|_{{\mathcal{T}_{h}}}$$ $$\|\mathbf{u}-\mathbf{u}^{h}\|_{{\mathcal{T}_{h}}}$$ $$\|p - p^{h} \|_{{\mathcal{T}_{h}}}$$ $$\|\mathbf{u}-\mathbf{u}^{*,h}\|_{{\mathcal{T}_{h}}}$$ k Nele Nglobal Nlocal Error Order Error Order Error Order Error Order Third test: ν = 0.0001, γ = 1, m = 2. 1 32 256 555 1.436e+00 — 7.825e-02 — 3.891e-02 — 5.242e-02 — 128 960 2203 3.932e-01 1.87 3.013e-02 1.38 1.949e-02 1.00 8.254e-03 2.67 512 3712 8763 1.241e-01 1.66 7.719e-03 1.96 4.951e-03 1.98 1.497e-03 2.46 2048 14592 34939 3.414e-02 1.86 1.962e-03 1.98 1.243e-03 1.99 2.153e-04 2.80 8192 57856 139515 7.387e-03 2.21 4.987e-04 1.98 3.110e-04 2.00 2.215e-05 3.28 2 32 368 1163 3.442e-01 — 3.288e-02 — 2.266e-02 — 1.065e-02 — 128 1376 4635 6.978e-02 2.30 3.132e-03 3.39 2.169e-03 3.39 9.514e-04 3.48 512 5312 18491 1.042e-02 2.74 4.049e-04 2.95 2.748e-04 2.98 7.304e-05 3.70 2048 20864 73851 1.085e-03 3.26 5.285e-05 2.94 3.447e-05 3.00 4.209e-06 4.12 8192 82688 295163 9.842e-05 3.46 6.770e-06 2.96 4.313e-06 3.00 1.938e-07 4.44 3 32 480 1995 3.231e-02 — 1.545e-03 — 6.370e-04 — 1.717e-03 — 128 1792 7963 6.311e-03 2.36 1.928e-04 3.00 2.490e-05 4.68 6.757e-05 4.67 512 6912 31803 3.905e-04 4.01 1.295e-05 3.90 1.348e-06 4.21 2.318e-06 4.87 2048 27136 127099 1.768e-05 4.46 8.583e-07 3.91 8.055e-08 4.07 6.261e-08 5.21 8192 107520 508155 7.284e-07 4.60 5.500e-08 3.96 4.974e-09 4.02 2.111e-09 4.89 Fourth test: ν = 1e −8, γ = 1, m = 2. 1 32 256 555 1.434e+00 — 7.825e-02 — 3.891e-02 — 5.233e-02 — 128 960 2203 4.167e-01 1.78 3.011e-02 1.38 1.949e-02 1.00 8.738e-03 2.58 512 3712 8763 1.581e-01 1.40 7.687e-03 1.97 4.951e-03 1.98 1.919e-03 2.19 2048 14592 34939 7.048e-02 1.17 1.932e-03 1.99 1.243e-03 1.99 4.633e-04 2.05 8192 57856 139515 3.411e-02 1.05 4.836e-04 2.00 3.110e-04 2.00 1.151e-04 2.01 2 32 368 1163 3.778e-01 — 3.286e-02 — 2.266e-02 — 1.139e-02 — 128 1376 4635 9.041e-02 2.06 3.110e-03 3.40 2.169e-03 3.39 1.172e-03 3.28 512 5312 18491 2.349e-02 1.94 3.895e-04 3.00 2.748e-04 2.98 1.440e-04 3.02 2048 20864 73851 5.980e-03 1.97 4.859e-05 3.00 3.447e-05 3.00 1.805e-05 3.00 8192 82688 295163 1.504e-03 1.99 6.065e-06 3.00 4.313e-06 3.00 2.261e-06 3.00 3 32 480 1995 3.228e-02 — 1.551e-03 — 6.370e-04 — 1.711e-03 — 128 1792 7963 1.094e-02 1.56 1.882e-04 3.04 2.489e-05 4.68 9.261e-05 4.21 512 6912 31803 1.314e-03 3.06 1.191e-05 3.98 1.348e-06 4.21 5.506e-06 4.07 2048 27136 127099 1.612e-04 3.03 7.466e-07 4.00 8.055e-08 4.06 3.403e-07 4.02 8192 107520 508155 1.995e-05 3.01 4.669e-08 4.00 4.974e-09 4.02 2.122e-08 4.00 Table 3 Continued Mesh D.O.F. $$\|\mathrm{L} - \mathrm{L}^{h} \|_{{\mathcal{T}_{h}}}$$ $$\|\mathbf{u}-\mathbf{u}^{h}\|_{{\mathcal{T}_{h}}}$$ $$\|p - p^{h} \|_{{\mathcal{T}_{h}}}$$ $$\|\mathbf{u}-\mathbf{u}^{*,h}\|_{{\mathcal{T}_{h}}}$$ k Nele Nglobal Nlocal Error Order Error Order Error Order Error Order Third test: ν = 0.0001, γ = 1, m = 2. 1 32 256 555 1.436e+00 — 7.825e-02 — 3.891e-02 — 5.242e-02 — 128 960 2203 3.932e-01 1.87 3.013e-02 1.38 1.949e-02 1.00 8.254e-03 2.67 512 3712 8763 1.241e-01 1.66 7.719e-03 1.96 4.951e-03 1.98 1.497e-03 2.46 2048 14592 34939 3.414e-02 1.86 1.962e-03 1.98 1.243e-03 1.99 2.153e-04 2.80 8192 57856 139515 7.387e-03 2.21 4.987e-04 1.98 3.110e-04 2.00 2.215e-05 3.28 2 32 368 1163 3.442e-01 — 3.288e-02 — 2.266e-02 — 1.065e-02 — 128 1376 4635 6.978e-02 2.30 3.132e-03 3.39 2.169e-03 3.39 9.514e-04 3.48 512 5312 18491 1.042e-02 2.74 4.049e-04 2.95 2.748e-04 2.98 7.304e-05 3.70 2048 20864 73851 1.085e-03 3.26 5.285e-05 2.94 3.447e-05 3.00 4.209e-06 4.12 8192 82688 295163 9.842e-05 3.46 6.770e-06 2.96 4.313e-06 3.00 1.938e-07 4.44 3 32 480 1995 3.231e-02 — 1.545e-03 — 6.370e-04 — 1.717e-03 — 128 1792 7963 6.311e-03 2.36 1.928e-04 3.00 2.490e-05 4.68 6.757e-05 4.67 512 6912 31803 3.905e-04 4.01 1.295e-05 3.90 1.348e-06 4.21 2.318e-06 4.87 2048 27136 127099 1.768e-05 4.46 8.583e-07 3.91 8.055e-08 4.07 6.261e-08 5.21 8192 107520 508155 7.284e-07 4.60 5.500e-08 3.96 4.974e-09 4.02 2.111e-09 4.89 Fourth test: ν = 1e −8, γ = 1, m = 2. 1 32 256 555 1.434e+00 — 7.825e-02 — 3.891e-02 — 5.233e-02 — 128 960 2203 4.167e-01 1.78 3.011e-02 1.38 1.949e-02 1.00 8.738e-03 2.58 512 3712 8763 1.581e-01 1.40 7.687e-03 1.97 4.951e-03 1.98 1.919e-03 2.19 2048 14592 34939 7.048e-02 1.17 1.932e-03 1.99 1.243e-03 1.99 4.633e-04 2.05 8192 57856 139515 3.411e-02 1.05 4.836e-04 2.00 3.110e-04 2.00 1.151e-04 2.01 2 32 368 1163 3.778e-01 — 3.286e-02 — 2.266e-02 — 1.139e-02 — 128 1376 4635 9.041e-02 2.06 3.110e-03 3.40 2.169e-03 3.39 1.172e-03 3.28 512 5312 18491 2.349e-02 1.94 3.895e-04 3.00 2.748e-04 2.98 1.440e-04 3.02 2048 20864 73851 5.980e-03 1.97 4.859e-05 3.00 3.447e-05 3.00 1.805e-05 3.00 8192 82688 295163 1.504e-03 1.99 6.065e-06 3.00 4.313e-06 3.00 2.261e-06 3.00 3 32 480 1995 3.228e-02 — 1.551e-03 — 6.370e-04 — 1.711e-03 — 128 1792 7963 1.094e-02 1.56 1.882e-04 3.04 2.489e-05 4.68 9.261e-05 4.21 512 6912 31803 1.314e-03 3.06 1.191e-05 3.98 1.348e-06 4.21 5.506e-06 4.07 2048 27136 127099 1.612e-04 3.03 7.466e-07 4.00 8.055e-08 4.06 3.403e-07 4.02 8192 107520 508155 1.995e-05 3.01 4.669e-08 4.00 4.974e-09 4.02 2.122e-08 4.00 Mesh D.O.F. $$\|\mathrm{L} - \mathrm{L}^{h} \|_{{\mathcal{T}_{h}}}$$ $$\|\mathbf{u}-\mathbf{u}^{h}\|_{{\mathcal{T}_{h}}}$$ $$\|p - p^{h} \|_{{\mathcal{T}_{h}}}$$ $$\|\mathbf{u}-\mathbf{u}^{*,h}\|_{{\mathcal{T}_{h}}}$$ k Nele Nglobal Nlocal Error Order Error Order Error Order Error Order Third test: ν = 0.0001, γ = 1, m = 2. 1 32 256 555 1.436e+00 — 7.825e-02 — 3.891e-02 — 5.242e-02 — 128 960 2203 3.932e-01 1.87 3.013e-02 1.38 1.949e-02 1.00 8.254e-03 2.67 512 3712 8763 1.241e-01 1.66 7.719e-03 1.96 4.951e-03 1.98 1.497e-03 2.46 2048 14592 34939 3.414e-02 1.86 1.962e-03 1.98 1.243e-03 1.99 2.153e-04 2.80 8192 57856 139515 7.387e-03 2.21 4.987e-04 1.98 3.110e-04 2.00 2.215e-05 3.28 2 32 368 1163 3.442e-01 — 3.288e-02 — 2.266e-02 — 1.065e-02 — 128 1376 4635 6.978e-02 2.30 3.132e-03 3.39 2.169e-03 3.39 9.514e-04 3.48 512 5312 18491 1.042e-02 2.74 4.049e-04 2.95 2.748e-04 2.98 7.304e-05 3.70 2048 20864 73851 1.085e-03 3.26 5.285e-05 2.94 3.447e-05 3.00 4.209e-06 4.12 8192 82688 295163 9.842e-05 3.46 6.770e-06 2.96 4.313e-06 3.00 1.938e-07 4.44 3 32 480 1995 3.231e-02 — 1.545e-03 — 6.370e-04 — 1.717e-03 — 128 1792 7963 6.311e-03 2.36 1.928e-04 3.00 2.490e-05 4.68 6.757e-05 4.67 512 6912 31803 3.905e-04 4.01 1.295e-05 3.90 1.348e-06 4.21 2.318e-06 4.87 2048 27136 127099 1.768e-05 4.46 8.583e-07 3.91 8.055e-08 4.07 6.261e-08 5.21 8192 107520 508155 7.284e-07 4.60 5.500e-08 3.96 4.974e-09 4.02 2.111e-09 4.89 Fourth test: ν = 1e −8, γ = 1, m = 2. 1 32 256 555 1.434e+00 — 7.825e-02 — 3.891e-02 — 5.233e-02 — 128 960 2203 4.167e-01 1.78 3.011e-02 1.38 1.949e-02 1.00 8.738e-03 2.58 512 3712 8763 1.581e-01 1.40 7.687e-03 1.97 4.951e-03 1.98 1.919e-03 2.19 2048 14592 34939 7.048e-02 1.17 1.932e-03 1.99 1.243e-03 1.99 4.633e-04 2.05 8192 57856 139515 3.411e-02 1.05 4.836e-04 2.00 3.110e-04 2.00 1.151e-04 2.01 2 32 368 1163 3.778e-01 — 3.286e-02 — 2.266e-02 — 1.139e-02 — 128 1376 4635 9.041e-02 2.06 3.110e-03 3.40 2.169e-03 3.39 1.172e-03 3.28 512 5312 18491 2.349e-02 1.94 3.895e-04 3.00 2.748e-04 2.98 1.440e-04 3.02 2048 20864 73851 5.980e-03 1.97 4.859e-05 3.00 3.447e-05 3.00 1.805e-05 3.00 8192 82688 295163 1.504e-03 1.99 6.065e-06 3.00 4.313e-06 3.00 2.261e-06 3.00 3 32 480 1995 3.228e-02 — 1.551e-03 — 6.370e-04 — 1.711e-03 — 128 1792 7963 1.094e-02 1.56 1.882e-04 3.04 2.489e-05 4.68 9.261e-05 4.21 512 6912 31803 1.314e-03 3.06 1.191e-05 3.98 1.348e-06 4.21 5.506e-06 4.07 2048 27136 127099 1.612e-04 3.03 7.466e-07 4.00 8.055e-08 4.06 3.403e-07 4.02 8192 107520 508155 1.995e-05 3.01 4.669e-08 4.00 4.974e-09 4.02 2.122e-08 4.00 In Table 3, we present the same convergence study with polynomial degree varying from k = 1 to k = 3 on triangular meshes. The first level mesh consists of 2 × 4 × 4 congruent triangles, and the consequent meshes are obtained by uniform refinements. In both tables, Nele denotes the number of elements, Nglobal denotes the number of globally coupled degrees of freedom and Nlocal denotes the number of local (static-condensed) degrees of freedom. From the results for the first test in Tables 2 and 3, we observe an optimal convergence order of k + 1 for all the three variables Lh, uh and ph, and a superconvergence order of k + 2 for the postprocessing u*, h. The convergence results for Lh, uh and ph are in full agreement with the theoretical predictions in Corollary 2.5 and Theorem 2.6. The superconvergence for u*, h is in agreement with the theoretical predictions in Theorem 2.6 for k ≥ 1, while the superconvergence of u*, h for k = 0 on rectangular meshes is not covered by our analysis in Theorem 2.6. From the results for the second test in Tables 2 and 3, we observe the same L2-errors in Lh, uh and u*, h as the corresponding ones in the first test. This indicates that velocity error is independent of the pressure, in full agreement with the estimates in Corollary 2.5. We also observe that the L2-error for ph is significantly larger than that for the first test. It is clear that, in this test, convergence for pressure is not in the asymptotic regime yet. From the results for the Darcy-dominated regimes in the third and fourth tests in Tables 2 and 3, we observe a similar L2-error in the velocity. This indicates the uniform stability of the proposed HDG method since the L2-error of the velocity does not depend on the ratio γ/ν, which is in full agreement with the velocity estimate in Corollary 2.5. We also observe a suboptimal convergence of one order less for the velocity gradient Lh, and the loss of superconvergence of u*, h for the fourth test on triangular meshes in Table 2. This is expected since in the limiting case when $$\nu \rightarrow 0$$, the regularity constant Cr for the dual problem (2.12) will blow up, and the Brinkman equations will collapse to the Darcy equations, in which the control of velocity gradient in Corollary 2.5 and that of the velocity error in Theorem 2.6 will be lost. However, such losses of convergence do not appear in Table 2 on rectangular meshes, which are better than our analysis in Section 3 indicates. 6. Conclusion We present and analyse a class of parameter-free superconvergent H(div)-conforming HDG methods on both simplicial and rectangular meshes for the Brinkman equations. Numerical results in two dimensions are presented to validate the theoretical findings. Funding Research Grants Council of the Hong Kong Special Administrative Region, China (Project No. City U 11304017) to W.Q. Acknowledgements G. Fu would like to thank Matthias Maier from the University of Minnesota for providing the general framework of the HDG code in deal.II and for many helpful discussions on numerical computations with deal.II. He would also like to thank Christoph Lehrenfeld from the University of Göttingen for many helpful discussions and hands-on tutorials on numerical computation using NGSolve’s Python interface. References Anaya , V. , Gatica , G. N. , Mora , D. & Ruiz-Baier , R. ( 2015 ) An augmented velocity-vorticity-pressure formulation for the Brinkman equations . Internat. J. Numer. Methods Fluids , 79 , 109 -- 137 . Google Scholar CrossRef Search ADS Arnold , D. N. & Awanou , G. ( 2014 ) Finite element differential forms on cubical meshes . Math. Comp. , 83 , 1551 -- 1570 . Google Scholar CrossRef Search ADS Badia , S. & Codina , R. ( 2009 ) Unified stabilized finite element formulations for the Stokes and the Darcy problems . SIAM J. Numer. Anal. , 47 , 1971 -- 2000 . 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Anal. , 52 , 258 -- 281 . Google Scholar CrossRef Search ADS Xie, X. , Xu , J. & Xue , G. ( 2008 ) Uniformly-stable finite element methods for Darcy-Stokes-Brinkman models . J. Comput. Math. , 26 , 437 -- 455 . © 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 http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png IMA Journal of Numerical Analysis Oxford University Press

Parameter-free superconvergent H(div)-conforming HDG methods for the Brinkman equations

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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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Abstract

Abstract In this paper, we present new parameter-free superconvergent H(div)-conforming hybridizable discontinuous Galerkin (HDG) methods for the Brinkman equations on both simplicial and rectangular meshes. The methods are based on a velocity gradient–velocity–pressure formulation, which can be considered a natural extension of the H(div)-conforming HDG method (defined on simplicial meshes) for the Stokes flow (Cockburn, B. & Sayas, F.-J. (2014) Divergence-conforming HDG methods for Stokes flow. Math. Comp., 83, 1571–1598). We obtain an optimal L2-error estimate for the velocity in both the Stokes-dominated regime (high viscosity/permeability ratio) and Darcy-dominated regime (low viscosity/permeability ratio). We also obtain a superconvergent L2-estimate of one order higher for a suitable projection of the velocity error in the Stokes-dominated regime. Moreover, thanks to H(div)-conformity of the velocity, our velocity error estimates are independent of the pressure regularity. Furthermore, we provide a discrete H1-stability result for the velocity field, which is essential in the error analysis of the natural generalization of these new HDG methods to the incompressible Navier–Stokes equations. Preliminary numerical results on both triangular and rectangular meshes in two dimensions confirm our theoretical predictions. 1. Introduction In this paper, we devise a superconvergent H(div)-conforming hybridizable discontinuous Galerkin (HDG) method for the following Brinkman equations in a velocity gradient–velocity–pressure formulation: \begin{align} \mathrm{L}=\nabla\mathbf{u}\;\quad\, \textrm{in}\,\,\,\varOmega, \end{align} (1.1a) \begin{align} -\nu\nabla\cdot \mathrm{L}+\gamma\mathbf{u}+\nabla p=\mathbf{f}\;\qquad\,\textrm{in}\,\,\,\varOmega, \end{align} (1.1b) \begin{align} \nabla\cdot\mathbf{u}=g\;\qquad \textrm{in}\,\,\, \varOmega, \end{align} (1.1c) \begin{align} \mathbf{u}\cdot\mathbf{n}=0\;\qquad \textrm{on}\,\,\,\, \partial\varOmega, \end{align} (1.1d) \begin{align} \ \nu(I_{d}-\mathbf{n}\otimes\mathbf{n})\mathbf{u}=0\;\qquad \textrm{on}\quad \partial\varOmega, \end{align} (1.1e) \begin{align} \int_{\varOmega} p=0, \end{align} (1.1f) where L is the velocity gradient, u is the velocity, p is the pressure, ν is the effective viscosity constant, $$\gamma \in L^{\infty }(\varOmega )^{d\times d}$$ is the inverse of the permeability tensor, f ∈ L2$$(\varOmega)$$d is the external body force, $$g\in{L_{0}^{2}}(\varOmega ) := \{ q\in L^{2}(\varOmega ): (q, 1)_{\varOmega } = 0 \}$$ and n is the unit outward normal vector along ∂$$\varOmega$$. The domain $$\varOmega \subset \mathbb{R}^{d}$$ is a polygon (d = 2) or polyhedron (d = 3). Here (1.1e) indicates that we impose a homogeneous tangential trace of $$\boldsymbol{u} $$ on ∂$$\varOmega$$. We notice that when ν = 0, (1.1e) vanishes such that equations (1.1) become the Darcy equations. One challenging aspect of numerical discretization of the Brinkman equations is the construction of stable finite element methods in both Stokes-dominated and Darcy-dominated regimes. We refer to such methods as uniformly stable methods. Uniformly stable methods for the Brinkman equations have been extensively studied for the classical velocity–pressure formulation, including nonconforming methods with an H(div)-conforming velocity field (Mardal et al., 2002; Tai & Winther, 2006; Xie et al., 2008Guzmán & Neilan, 2012), conforming methods ( Xie et al., 2008Juntunen & Stenberg, 2010), stabilized methods ( Xie et al., 2008; Badia & Codina, 2009; Juntunen & Stenberg, 2010), the H(div)-conforming discontinuous Galerkin method (Könnö & Stenberg, 2011) and the hybridized H(div)-conforming discontinuous Galerkin method (Könnö & Stenberg, 2012), and other alternative formulations, including the vorticity–velocity–pressure formulation (Vassilevski & Villa, 2014,Anaya et al., 2015), the pseudostress-based formulation (Gatica et al., 2015) and a dual-mixed formulation (Howell & Neilan, 2016). In this paper, we propose and study a class of high-order, parameter-free, H(div)-conforming HDG methods for the Brinkman equations (1.1) on both simplicial and rectangular meshes. This is the first HDG method for the Brinkman equations based on a velocity gradient–velocity–pressure formulation. Our method can be considered a natural, stable extension to the Brinkman equations of the high-order, parameter-free, H(div)-conforming HDG method for the Stokes problem on simplicial meshes (Cockburn & Sayas, 2014). Three distinctive properties of the method make it attractive. Firstly, our method provides an optimal error estimate in L2-norms for the velocity that is robust with respect to viscosity/permeability ratio ν/γ (Theorem 2.4, Corollary 2.5), and a superconvergent error estimate in the L2-norm of one order higher for a suitable projection of the velocity error (under a regularity assumption on the dual problem). To the best of our knowledge, this is the first superconvergent velocity estimate for the Brinkman equations. Secondly, thanks to H(div)-conformity of the velocity, our velocity error estimates are independent of the pressure regularity (see Corollary 2.5 and Theorem 2.6). Such a pressure-robustness property is highly appreciated for incompressible flow problems (Linke, 2014; Linke & Merdon, 2016). Finally, our error analysis, which is quite different from and more straightforward than that in the study by Cockburn & Sayas (2014) for the Stokes flow, is based on a so-called discrete H1-stability result (see Theorem 2.1), which is the essential ingredient in the analysis of the velocity gradient–velocity–pressure HDG formulation of the incompressible Navier–Stokes equations. We specifically remark that no stabilization parameter enters our method, which has to be compared with the hybridized H(div)-conforming discontinuous Galerkin method (Könnö & Stenberg, 2012) in the classical velocity–pressure formulation, where Nitsche’s penalty method is used to impose tangential continuity of the velocity field and the stabilization parameter needs to be ‘sufficiently large’. The organization of the paper is as follows. In Section 2, we introduce the parameter-free H(div)-conforming HDG method and give the main results on a priori error estimates. In Section 3, we prove our main results in Section 2. In Section 4, we discuss the hybridization of the H(div)-conforming HDG method. In Section 5, we provide preliminary two-dimensional numerical experiments on triangular and rectangular meshes to validate our theoretical results. We end in Section 6 with some concluding remarks. 2. Main results: superconvergent H(div)-conforming HDG In this section, we first introduce the notation that will be used throughout the paper, and then present the finite element spaces that define the H(div)-conforming HDG methods. We conclude with an a priori error estimate along with a key inequality that we call discrete H1-stability. 2.1. Meshes and trace operators We denote by $${\mathcal{T}_{h}}:=\{K\}$$ (the mesh) a shape-regular conforming triangulation of the domain $$\varOmega \subset \mathbb{R}^{d}$$ into affine-mapped simplices (triangles if d = 2, tetrahedra if d = 3) or hypercubes (squares if d = 2, cubes if d = 3), and by $$\mathcal{E}_{h}$$ (the mesh skeleton) the set of facets F (edges if d = 2, faces if d = 3) of the elements $$K \in{\mathcal{T}_{h}}$$. Let $$\mathcal{F}(K)$$ denote the set of facets F of the element $$\textit{K} $$. We set hF := diam(F), hK := diam(K) and $$h := \max _{K\in{\mathcal{T}_{h}}}h_{K}$$. Let $$\underline{\textsf{K}}$$ be the reference element (d-dimensional simplex or hypercube), and $$\underline{\textsf{F}}$$ be the reference facet (d−1-dimensional simplex or hypercube). We denote by $$\varPhi _{K}: \underline{\textsf{K}}\rightarrow K$$ and $$\varPhi _{F}: \underline{\textsf{F}}\rightarrow F$$ the associated affine mappings. For a d-dimensional vector-valued function v on an element $$K\subset \mathbb{R}^{d}$$ with sufficient regularity, we denote by \begin{align} \textrm{tr}_{t}^{F}(\boldsymbol\upsilon):= \left.\left(\boldsymbol\upsilon-(\boldsymbol\upsilon\cdot\boldsymbol{n}_{F})\,\boldsymbol{n}_{F})\right)\right|{}_{F}\quad \textrm{and}\quad \textrm{tr}_{n}^{F}(\boldsymbol\upsilon):= \left.\left(\boldsymbol\upsilon\cdot\boldsymbol{n}_{F}\right)\boldsymbol{n}_{F} \right|{}_{F} \end{align} (2.1) the tangential and normal traces of v on the facet $$F\in \mathcal{F}(K)$$, where $$\boldsymbol{n} $$F is the unit normal vector to F. Note that the above trace operators are independent of the direction of the normal nF. Whenever there is no confusion, we suppress the superscript and denote by trt(v) and trn(v) the related tangential and normal traces, respectively. With an abuse of notation, we also denote $$ \textrm{tr}_{t}(\widehat{\mathbf{v}}):= \left.\left(\widehat{\mathbf{v}}-(\widehat{\mathbf{v}}\cdot\mathbf{n}_{F})\,\mathbf{n}_{F})\right)\right|{}_{F}\quad \textrm{and}\quad \textrm{tr}_{n}(\widehat{\mathbf{v}}):= \left.\left(\widehat{\mathbf{v}}\cdot\mathbf{n}_{F}\right)\mathbf{n}_{F} \right|{}_{F} $$ for a d-dimensional vector-valued function v on a facet $$F\subset \mathbb{R}^{d-1}$$ with sufficient regularity. 2.2. The finite element spaces Now, we define the finite element spaces associated with the mesh $${\mathcal{T}_{h}}$$ and mesh skeleton $$\mathcal{E}_{h}$$ via appropriate mappings (cf. Brenner & Scott, 2008) from (polynomial) spaces on the reference elements. We use the following mapped finite element spaces on the mapped element K and facet F: \begin{align} {\mathcal{G}}^{\mathrm{r}\mathrm{o}\mathrm{w}}(K):=\;\left\{\mathbf{v}\in{L}^{2}(K)^{d}:\;\;\mathbf{v}= \frac{1}{\mathrm{d}\mathrm{e}\mathrm{t}\,\varPhi_{K}^{\prime}}\varPhi_{K}^{\prime}\,\underline{\boldsymbol{\textsf{ v}}}\circ\varPhi_{K}^{-1},\;\;\underline{\boldsymbol{\textsf{ v}}}\in{\mathcal{G}}^{\mathrm{r}\mathrm{o}\mathrm{w}}(\underline{\textsf{K}})\right\}\!, \end{align} (2.2a) \begin{align} {\boldsymbol V}(K):=\;\left\{\mathbf{v}\in{L}^{2}(K)^{d}:\; \mathbf{v}= \frac{1}{\mathrm{d}\mathrm{e}\mathrm{t}\,\varPhi_{K}^{\prime}}\varPhi_{K}^{\prime}\,\underline{\boldsymbol{\textsf{ v}}}\circ\varPhi_{K}^{-1},\;\;\underline{\boldsymbol{\mathbf{ v}}}\in{\mathbf V}(\underline{\textsf{K}})\right\}\!, \end{align} (2.2b) \begin{align} Q(K):=\;\left\{q\in{L}^{2}(K):\;q=\underline{{\textsf{ q}}}\circ\varPhi_{K}^{-1},\;\;\underline{{\textsf{ q}}}\in{Q}(\underline{\textsf{K}})\right\}\!, \end{align} (2.2c) \begin{align} {\boldsymbol{M}}(F):=\;\left\{\widehat{\mathbf{v}}\in{L}^{2}(F)^{d}:\;\widehat{\mathbf{v}}=\underline{\widehat{\boldsymbol{\mathbf{v}}}}\circ\varPhi_{F}^{-1},\;\;\underline{\widehat{\boldsymbol{\mathbf{v}}}}\in{\boldsymbol{M}}(\underline{\textsf{F}})\right\}\!. \end{align} (2.2d) Here $$\varPhi$$K and $$\varPhi$$F are the affine mappings introduced above, and $$\varPhi _{K}^{\prime }$$ is the Jacobian matrix of the mapping $$\varPhi$$K. Note that the vector spaces in (2.2a) and (2.2b) are obtained from the well-known Piola transformation which preserves normal continuity (cf. Durán, 2008). The polynomial spaces on the reference elements are given in Table 1. Table 1 The reference finite element spaces Element $${\mathcal{G}}^{\mathrm{r}\mathrm{o}\mathrm{w}}(\underline{\textsf{K}})$$ $${\boldsymbol V}(\underline{\textsf{K}})$$ $${Q}(\underline{\textsf{K}})$$ $${\boldsymbol{M}}(\underline{\textsf{F}})$$ Simplex $$\mathcal{P}_{k}(\underline{\textsf{K}})^{d}$$ $$\boldsymbol{\mathrm{R}\mathrm{T}}_{k}(\underline{\textsf{K}})$$ $$\mathcal{P}_{k}(\underline{\textsf{K}})$$ $$\mathcal{P}_{k}(\underline{\textsf{F}})^{d}$$ Hypercube $$\boldsymbol{\mathrm{B}\mathrm{D}\mathrm{M}}_{k}(\underline{\textsf{K}})$$ $$\boldsymbol{\mathrm{B}\mathrm{D}\mathrm{F}\mathrm{M}}_{k}(\underline{\textsf{K}})$$ $$\mathcal{P}_{k}(\underline{\textsf{K}})$$ $$\mathcal{P}_{k}(\underline{\textsf{F}})^{d}$$ Element $${\mathcal{G}}^{\mathrm{r}\mathrm{o}\mathrm{w}}(\underline{\textsf{K}})$$ $${\boldsymbol V}(\underline{\textsf{K}})$$ $${Q}(\underline{\textsf{K}})$$ $${\boldsymbol{M}}(\underline{\textsf{F}})$$ Simplex $$\mathcal{P}_{k}(\underline{\textsf{K}})^{d}$$ $$\boldsymbol{\mathrm{R}\mathrm{T}}_{k}(\underline{\textsf{K}})$$ $$\mathcal{P}_{k}(\underline{\textsf{K}})$$ $$\mathcal{P}_{k}(\underline{\textsf{F}})^{d}$$ Hypercube $$\boldsymbol{\mathrm{B}\mathrm{D}\mathrm{M}}_{k}(\underline{\textsf{K}})$$ $$\boldsymbol{\mathrm{B}\mathrm{D}\mathrm{F}\mathrm{M}}_{k}(\underline{\textsf{K}})$$ $$\mathcal{P}_{k}(\underline{\textsf{K}})$$ $$\mathcal{P}_{k}(\underline{\textsf{F}})^{d}$$ Table 1 The reference finite element spaces Element $${\mathcal{G}}^{\mathrm{r}\mathrm{o}\mathrm{w}}(\underline{\textsf{K}})$$ $${\boldsymbol V}(\underline{\textsf{K}})$$ $${Q}(\underline{\textsf{K}})$$ $${\boldsymbol{M}}(\underline{\textsf{F}})$$ Simplex $$\mathcal{P}_{k}(\underline{\textsf{K}})^{d}$$ $$\boldsymbol{\mathrm{R}\mathrm{T}}_{k}(\underline{\textsf{K}})$$ $$\mathcal{P}_{k}(\underline{\textsf{K}})$$ $$\mathcal{P}_{k}(\underline{\textsf{F}})^{d}$$ Hypercube $$\boldsymbol{\mathrm{B}\mathrm{D}\mathrm{M}}_{k}(\underline{\textsf{K}})$$ $$\boldsymbol{\mathrm{B}\mathrm{D}\mathrm{F}\mathrm{M}}_{k}(\underline{\textsf{K}})$$ $$\mathcal{P}_{k}(\underline{\textsf{K}})$$ $$\mathcal{P}_{k}(\underline{\textsf{F}})^{d}$$ Element $${\mathcal{G}}^{\mathrm{r}\mathrm{o}\mathrm{w}}(\underline{\textsf{K}})$$ $${\boldsymbol V}(\underline{\textsf{K}})$$ $${Q}(\underline{\textsf{K}})$$ $${\boldsymbol{M}}(\underline{\textsf{F}})$$ Simplex $$\mathcal{P}_{k}(\underline{\textsf{K}})^{d}$$ $$\boldsymbol{\mathrm{R}\mathrm{T}}_{k}(\underline{\textsf{K}})$$ $$\mathcal{P}_{k}(\underline{\textsf{K}})$$ $$\mathcal{P}_{k}(\underline{\textsf{F}})^{d}$$ Hypercube $$\boldsymbol{\mathrm{B}\mathrm{D}\mathrm{M}}_{k}(\underline{\textsf{K}})$$ $$\boldsymbol{\mathrm{B}\mathrm{D}\mathrm{F}\mathrm{M}}_{k}(\underline{\textsf{K}})$$ $$\mathcal{P}_{k}(\underline{\textsf{K}})$$ $$\mathcal{P}_{k}(\underline{\textsf{F}})^{d}$$ Here we denote by $$\mathcal{P}_{k}(D)$$ and $$\widetilde{\mathcal{P}}_{k}(D)$$ the polynomials of degree no greater than k, and homogeneous polynomials of degree k, respectively, on the domain D. The vector space $$\boldsymbol{\mathrm{R}\mathrm{T}}_{k}(\underline{\textsf{K}})$$ on the reference simplex is the Raviart–Thomas–Nedéléc space (see Raviart & Thomas 1977; Nédélec 1980) $$ \boldsymbol{\mathrm{R}\mathrm{T}}_{k}(\underline{\textsf{K}}):=\mathcal{P}_{k}(\underline{\textsf{K}})^{d}\oplus \mathbf{x}\,\widetilde{\mathcal{P}}_{k}(\underline{\textsf{K}}), $$ the vector space $$\boldsymbol{\mathrm{B}\mathrm{D}\mathrm{M}}_{k}(\underline{\textsf{K}})$$ on the reference hypercube is the Brezzi–Douglas–Marini space, (see Brezzi et al., 1985; Brezzi et al., 1987a;Arnold & Awanou 2014) $$ \boldsymbol{\mathrm{B}\mathrm{D}\mathrm{M}}_{k}(\underline{\textsf{K}}):=\begin{cases} \mathcal{P}_{k}(\underline{\textsf{K}})^{d}\oplus{{\nabla\times}}\{x\,y^{k+1},y\,x^{k+1}\} & \textrm{if}\ d=2, \\ \mathcal{P}_{k}(\underline{\textsf{K}})^{d}\oplus{{\nabla\times}}\left\{ \begin{array}{c} x\,\widetilde{\mathcal{P}}_{k}(y,z)(y{\nabla} z-z{\nabla} y),\\ y\,\widetilde{\mathcal{P}}_{k}(z,x)(z{\nabla} x-x{\nabla} z),\\ z\,\widetilde{\mathcal{P}}_{k}(x,y)(x{\nabla} y-y{\nabla} x)\\ \end{array} \right\}& \textrm{if}\ d=3, \end{cases} $$ and the vector space $$\boldsymbol{\mathrm{B}\mathrm{D}\mathrm{F}\mathrm{M}}_{k}(\underline{\textsf{K}})$$ on the reference hypercube is the Brezzi–Douglas–Fortin–Marini space, (see Brezzi et al., 1987b) $$ \boldsymbol{\mathrm{B}\mathrm{D}\mathrm{F}\mathrm{M}}_{k}(\underline{\textsf{K}}):=\begin{cases} \mathcal{P}_{k}(\underline{\textsf{K}})^{d}\oplus \left[ \begin{array}{c} x\,\widetilde{\mathcal{P}}_{k}(\underline{\textsf{K}}) \\ y\,\widetilde{\mathcal{P}}_{k}(\underline{\textsf{K}}) \end{array} \right] & \textrm{if}\ d=2, \\ \mathcal{P}_{k}(\underline{\textsf{K}})^{d}\oplus \left[ \begin{array}{c} x\,\widetilde{\mathcal{P}}_{k}(\underline{\textsf{K}}) \\ y\,\widetilde{\mathcal{P}}_{k}(\underline{\textsf{K}}) \\ z\,\widetilde{\mathcal{P}}_{k}(\underline{\textsf{K}}) \end{array} \right] & \textrm{if}\ d=3. \end{cases} $$ Next, for the vector-valued finite element space $${\mathcal{G}}^{\mathrm{r}\mathrm{o}\mathrm{w}}(K)$$ given in (2.2a), we denote by \begin{align} {\mathcal{G}}(K):= \left[{\mathcal{G}}^{\mathrm{r}\mathrm{o}\mathrm{w}}(K)\right]^{d} \end{align} (2.3) the tensor-valued space such that each row is the space $${\mathcal{G}}^{\mathrm{r}\mathrm{o}\mathrm{w}}(K)$$. We use the following finite element spaces on the mesh $${\mathcal{T}_{h}}$$ and mesh skeleton $$\mathcal{E}_{h}$$ to define the H(div)-conforming HDG method in the next section: \begin{align} {\mathcal{G}}_{h}:=\ \left\{\mathrm{g}\in L^{2}({\mathcal{T}_{h}})^{d\times d}:\quad\mathrm{g}|_{K}\in{\mathcal{G}}(K),\;\; K\in{\mathcal{T}_{h}}\right\}\!, \end{align} (2.4a) \begin{align}{\boldsymbol{V}}_{h}:=\, \left\{\mathbf{v}\in L^{2}({\mathcal{T}_{h}})^{d}:\quad\quad\mathbf{v}|_{K} \in{\textbf V}(K),\;\; K\in{\mathcal{T}_{h}}\right\}\!,\end{align} (2.4b) \begin{align} {\textbf V}_{h}^{\textrm{div}}:=\; \left\{\mathbf{v}\in{\textbf V}_{h}:\,\,\quad\quad\quad\quad\mathbf{v}\in H(\textrm{div};\varOmega)\right\}\!, \end{align} (2.4c) \begin{align}{\boldsymbol V}_{h}^{\textrm{div}}(0):=\; \left\{\mathbf{v}\in{\textbf V}_{h}^{\textrm{div}}:\,\,\, \quad\quad\quad\textrm{tr}_{n}(\mathbf{v})|_{\partial \varOmega}=0 \right\}\!, \end{align} (2.4d) \begin{align} Q_{h}:=\;\left\{q\in L^{2}({\mathcal{T}_{h}}): q|_{K}\in{Q}(K),\;\; K\in{\mathcal{T}_{h}}\right\}\!, \end{align} (2.4e) \begin{align} \mathring{Q_{h}}:=\;\left\{q\in Q_{h}:\qquad\qquad ( q\,,\,1 )_{{\mathcal{T}_{h}}} = 0 \right\}\!,\end{align} (2.4f) \begin{align}{\boldsymbol{M}}_{h}:=\ \left\{\widehat{\mathbf{v}}\in L^{2}(\mathcal{E}_{h})^{d}:\qquad\;\widehat{\mathbf{v}}|_{F}\in{\boldsymbol{M}}(F),\quad F\in\mathcal{E}_{h}\right\}\!, \end{align} (2.4g) \begin{align} \boldsymbol{M}_{h}(0):=\;\left\{\widehat{\mathbf{v}}\in{\boldsymbol{M}}_{h}:\qquad\qquad\,\widehat{\mathbf{v}}|_{\partial\varOmega} = \mathbf{0} \right\}\!, \end{align} (2.4h) \begin{align} {\boldsymbol{M}}^{t}_{h}:=\;\left\{\widehat{\mathbf{v}}\in \boldsymbol{M}_{h}:\textrm{tr}_{n}(\widehat{\mathbf{v}})|_{F}=0,\;\; F\in\mathcal{E}_{h}\right\}\!, \end{align} (2.4i) \begin{align} {\boldsymbol{M}}^{t}_{h}(0):=\, \left\{\widehat{\mathbf{v}}\in{\boldsymbol{M}}^{t}_{h}:\qquad\;\;\;\;\;\;\;\textrm{tr}_{t}(\widehat{\mathbf{v}})|_{\partial \varOmega}=\mathbf{0}\right\}\!.\end{align} (2.4j) 2.3. The H(div)-conforming HDG method Now, we are ready to present the H(div)-conforming HDG method for the Brinkman equations (1.1). It is defined as the unique element $$(\mathrm{L}^{h},\mathbf \;{u}^{h},\;p^{h},\;{\widehat{\mathbf{u}}_{t}}^{h})\in{\mathcal{G}}_{h}\times{\boldsymbol V}_{h}^{\textrm{div}}(0)\times \mathring{{Q}_{h}}\times{\boldsymbol{M}}^{t}_{h}(0)$$ such that the following weak formulation holds: \begin{align} ( \mathrm{L}^{h},\,\nu\,\mathrm{g}^{h} )_{{\mathcal{T}_{h}}}-( {\boldsymbol{\nabla}} \mathbf{u}^{h},\,\nu\,\mathrm{g}^{h} )_{{\mathcal{T}_{h}}} + \left\langle \textrm{tr}_{t}(\mathbf{u}^{h})-\widehat{\mathbf{u}}_{t}^{h},\, \textrm{tr}_{t}(\nu\,\mathrm{g}^{h}\, \mathbf{n}) \right\rangle_{\partial{{\mathcal{T}_{h}}}} = 0, \end{align} (2.5a) \begin{align} ( \nu\,\mathrm{L}^{h},\,{\boldsymbol{\nabla}} \mathbf{v}^{h} )_{{\mathcal{T}_{h}}} - \left\langle \textrm{tr}_{t}(\nu\,\mathrm{L}^{h}\,\mathbf{n}),\,\textrm{tr}_{t}(\mathbf{v}^{h})-\widehat{\mathbf{v}}_{t}^{h} \right\rangle_{\partial{{\mathcal{T}_{h}}}} \end{align} (2.5b) \begin{align} - ( p^{h},\,{\nabla\cdot} \mathbf{v}^{h} )_{{\mathcal{T}_{h}}}+( \gamma\, \mathbf{u}^{h},\,\mathbf{v}^{h} )_{{\mathcal{T}_{h}}}&=(\mathbf{f},\mathbf{v}^{h})_{\mathcal{T}_{h}},\nonumber \\ ( {\nabla\cdot}\mathbf{u}^{h},\,q^{h} )_{{\mathcal{T}_{h}}} & = (g, q^{h})_{\mathcal{T}_{h}}, \end{align} (2.5c) for all $$(\mathrm{g}^{h},\mathbf{v}^{h},q^{h},{\widehat{\mathbf{v}}_{t}}^{h})\in{\mathcal{G}}_{h}\times{\boldsymbol V}_{h}^{\textrm{div}}(0)\times \mathring{{Q}_{h}}\times{\boldsymbol{M}}^{t}_{h}(0)$$. Here we write $$(\eta, \, \zeta )_{{\mathcal{T}_{h}}} := \sum _{K \in{\mathcal{T}_{h}}} (\eta , \zeta )_{K},$$ where (η, ζ)K denotes the integral of ηζ over the domain $$K \subset \mathbb{R}^{n}$$. We also write $$\langle \eta,\,\zeta \rangle _{\partial{{\mathcal{T}_{h}}}}:= \sum _{K \in{\mathcal{T}_{h}}}\langle \eta,\,\zeta \rangle _{{\partial K}}$$, where $$\langle \eta,\,\zeta \rangle _{{\partial K}}:=\sum _{F \in \mathcal{F}(K)} \langle \eta,\,\zeta \rangle _{F},$$ and ⟨η, ζ⟩F denotes the integral of ηζ over the facet $$F \subset \mathbb{R}^{n-1}$$ and where $$\partial{\mathcal{T}_{h}} := \{ \partial K: K \subset{\mathcal{T}_{h}} \}$$. When vector-valued or tensor-valued functions are involved, we use similar notation. We specifically remark that, when γ = 0 and g = 0 in $$\varOmega$$, our method on simplicial meshes is identical to the one for the Stokes equations introduced in the study by Cockburn & Sayas (2014). As mentioned in the introduction, we postpone to Section 4 discussing the efficient implementation of the above method via hybridization. Here we focus on the presentation of its (superconvergent) a priori error estimates. 2.3.1. Discrete H1-stability We first obtain a key result, which will be used to prove the error estimates presented in the Section 2.3.3, on the control of a discrete H1-norm of the pair $$(\mathbf{u}^{h}, \widehat{\mathbf{u}}_{t}^{h})\in{\boldsymbol V}_{h}^{\textrm{div}}\times{\boldsymbol{M}}^{t}_{h}$$ by the L2-norm of a tensor field. For a pair $$(\mathbf{v}^{h}, \widehat{\mathbf{v}}_{t}^{h})\in{\boldsymbol V}_{h}^{\textrm{div}}\times{\boldsymbol{M}}^{t}_{h}$$, we denote its discrete H1-norm as: \begin{align} {\left\vert\kern-0.25ex\left\vert\kern-0.25ex\left\vert (\mathbf{u}^{h}, \widehat{\mathbf{u}}_{t}^{h}) \right\vert\kern-0.25ex\right\vert\kern-0.25ex\right\vert}_{1,{\mathcal{T}_{h}}} :=\left( \sum_{K\in{\mathcal{T}_{h}}} \|{\boldsymbol{\nabla}} \mathbf{u}^{h}\|_{K}^{2} + \sum_{F\in\mathcal{E}_{h}}h_{F}^{-1}\| \textrm{tr}_{t}(\mathbf{u}^{h})-\widehat{\mathbf{u}}_{t}^{h}\|_{F}^{2} \right)^{1/2} .\end{align} (2.6) Theorem 2.1 (Discrete H1-stability). Let $$(\mathrm{r}, \mathbf{z}^{h},\; \widehat{\mathbf{z}}_{t}^{h})\in L^{2}({\mathcal{T}_{h}})^{d\times d}\times{\boldsymbol V}_{h}^{\textrm{div}}\times{\boldsymbol{M}}^{t}_{h}\ $$ satisfy the following equation \begin{align} ( \mathrm{r},\,\mathrm{g}^{h} )_{{\mathcal{T}_{h}}}-( {\boldsymbol{\nabla}} \mathbf{z}^{h},\,\mathrm{g}^{h} )_{{\mathcal{T}_{h}}} + \left\langle \textrm{tr}_{t}(\mathbf{z}^{h})-\widehat{\mathbf{z}}_{t}^{h},\, \textrm{tr}_{t}(\mathrm{g}^{h}\, \mathbf{n}) \right\rangle_{\partial{{\mathcal{T}_{h}}}} = 0 \end{align} (2.7) for all $$\mathrm{g}^{h}\in{\mathcal{G}}_{h}$$; then we have \begin{align} {\left\vert\kern-0.25ex\left\vert\kern-0.25ex\left\vert \left(\mathbf{z}^{h},\; \widehat{\mathbf{z}}_{t}^{h}\right) \right\vert\kern-0.25ex\right\vert\kern-0.25ex\right\vert}_{1,{\mathcal{T}_{h}}}\le C\,\|\mathrm{r}\|_{\mathcal{T}_{h}}, \end{align} (2.8) with a constant C depending only on the polynomial degree k and the shape regularity of the elements $$K\in{\mathcal{T}_{h}}$$. Here $$\Vert \cdot \Vert _{{\mathcal{T}_{h}}}$$ is the standard L2-norm on $$\varOmega$$. 2.3.2. Well-posedness of the HDG method Theorem 2.2 shows the well-posedness of the HDG method (2.5), which is a direct consequence of Theorem 2.4. Theorem 2.2 For any $$(\mathbf{f}, g) \in L^{2}(\varOmega )^{d} \times{L_{0}^{2}}(\varOmega )$$, the HDG method (2.5) has a unique solution $$(\mathrm{L}^{h},\mathbf{u}^{h},p^{h},{\widehat{\mathbf{u}}_{t}}^{h})\in{\mathcal{G}}_{h}\times{\boldsymbol V}_{h}^{\textrm{div}}(0)\times \mathring{{Q}_{h}}\times{\boldsymbol{M}}^{t}_{h}(0)$$. 2.3.3. A priori error estimates We are now ready to present the a priori error estimates for the method (2.5). We compare the numerical solution against suitably chosen projections. The projections In the following, we denote by $$P_{\mathcal{G}}$$, PV, PQ and $$P_{\boldsymbol{M}^{t}}$$ the L2-projections onto $${\mathcal{G}}_{h}$$, Vh, $$\mathring{{Q}_{h}} $$ and $${\boldsymbol{M}}^{t}_{h}$$, respectively. Moreover, we set \begin{align*} \mathrm{e}_{L} &= P_{\mathcal{G}} \mathrm{L} - \mathrm{L}^{h},\;\; \mathbf{e}_{u} = \Pi_{{\boldsymbol V}} \mathbf{u} - \mathbf{u}^{h}, \;\; {e}_{p} = P_{Q} p - p^{h}, \;\;\mathbf{e}_{\widehat u_{t}} = P_{\boldsymbol{M}^{t}} \mathbf{u} - \widehat{\mathbf{u}}_{t}^{h},\\{\delta}_{L} &= \mathrm{L} -P_{\mathcal{G}} \mathrm{L}, \;\; \boldsymbol{\delta}_{u} \;= \mathbf{u} - \Pi_{{\boldsymbol V}} \mathbf{u}, \;\; {\delta}_{p} = p -P_{Q} p, \;\;\boldsymbol{\delta}_{\widehat u_{t}} \;=\textrm{tr}_{t}(\mathbf{u}) - P_{\boldsymbol{M}^{t}} \mathbf{u}. \end{align*} Here the projection ΠVu ∈ Vh whose restriction to an element K is the unique function in V(K) such that \begin{align} (\Pi_{{\boldsymbol V}} \mathbf{u}, \mathbf{v})_{K} = \; (\mathbf{u},\mathbf{v})_{K} \quad \forall\,\mathbf{v}\in{\boldsymbol{\nabla\cdot}} {\mathcal{G}}(K), \end{align} (2.9a) \begin{align} \left\langle \textrm{tr}_{n}(\Pi_{{\boldsymbol V}} \mathbf{u}),\,\textrm{tr}_{n}(\widehat{\mathbf{v}}) \right\rangle_{F} = \; \left\langle \textrm{tr}_{n}(\mathbf{u}),\,\textrm{tr}_{n}(\widehat{\mathbf{v}}) \right\rangle_{F} \quad \forall\, \widehat{\mathbf{v}}\in{\boldsymbol{M}}(F),\quad \forall\, F\in \mathcal{F}(K). \end{align} (2.9b) Recall that the spaces V(K), M(F) and $${\mathcal{G}}(K)$$ are defined in (2.2) and (2.3), respectively. When K is a simplex, the above projection is nothing but the Raviart–Thomas projection (see Raviart & Thomas, 1977: Nédélec, 1980); when K is a hypercube, the above projection is nothing but the Brezzi–Douglas–Fortin–Marini projection (see Brezzi et al., 1987b). The following approximation property of the above projection is well known; see Boffi et al. (2013, Chapter 2). Lemma 2.3 There exists a unique function $$\Pi _{{\boldsymbol V}} \mathbf{u}\in{\boldsymbol V}_{h}^{\textrm{div}}$$ defined elementwise by equations (2.9). Moreover, there exists a constant C depending only on the polynomial degree and shape regularity of the elements $$K\in{\mathcal{T}_{h}}$$ such that \begin{align} \|\Pi_{{\boldsymbol V}} \mathbf{u} - \mathbf{u}\|_{\mathcal{T}_{h}}\le \;C\, \left(\|P_{{\boldsymbol V}} \mathbf{u} - \mathbf{u}\|_{\mathcal{T}_{h}} +\sum_{K\in{\mathcal{T}_{h}}}h_{K}^{1/2}\|P_{{\boldsymbol V}} \mathbf{u} -\mathbf{u}\|_{\partial K} \right). \end{align} (2.10) The projection errors Now, we state our main results on the superconvergent error estimates. Theorem 2.4 Let $$(\mathrm{L}^{h},\mathbf{u}^{h},p^{h}, \widehat{\mathbf{u}}_{t}^{h})\in{\mathcal{G}}_{h}\times{\boldsymbol V}_{h}^{\textrm{div}}(0)\times \mathring{{Q}_{h}}\times{\boldsymbol{M}}^{t}_{h}(0)$$ be the numerical solution of (2.5); then there exists a constant C, depending only on the polynomial degree k, the shape regularity of the mesh $${\mathcal{T}_{h}}$$ and the domain Ω, such that \begin{align}{2} \|\mathbf{e}_{u}\|_{{\mathcal{T}_{h}}} \le \;C\, {\left\vert\kern-0.25ex\left\vert\kern-0.25ex\left\vert (\mathbf{e}_{u},\mathbf{e}_{\widehat u_{t}}) \right\vert\kern-0.25ex\right\vert\kern-0.25ex\right\vert}_{1,{\mathcal{T}_{h}}}, \end{align} (2.11a) \begin{align} {\left\vert\kern-0.25ex\left\vert\kern-0.25ex\left\vert (\mathbf{e}_{u},\mathbf{e}_{\widehat u_{t}}) \right\vert\kern-0.25ex\right\vert\kern-0.25ex\right\vert}_{1,{\mathcal{T}_{h}}} \le \; C\,\|\mathrm{e}_{L}\|_{{\mathcal{T}_{h}}}, \end{align} (2.11b) \begin{align} \nu\|\mathrm{e}_{L}\|_{{\mathcal{T}_{h}}}^{2} +\|\gamma^{1/2}\,\mathbf{e}_{u}\|_{\mathcal{T}_{h}}^{2} \le \;C\,\left(\sum_{F\in\mathcal{E}_{h}}\nu\,h_{F}\|{\delta}_{L}\,\mathbf{n}\|_{F}^{2}+\|\gamma^{1/2}\,\boldsymbol{\delta}_{u}\|_{\mathcal{T}_{h}}^{2} \right)\!\!. \end{align} (2.11c) Combining this result with Lemma 2.3, we immediately obtain optimal convergence of the L2-error for Lh and uh, and superconvergent discrete H1-error for the pair $$(\mathbf{u}^{h},\widehat{\mathbf{u}}_{t}^{h})$$ comparing with the projection $$(\Pi _{{\boldsymbol V}} \mathbf{u}, P_{\mathbf{M}^{t}} \mathbf{u})$$; see the following corollary. We omit the proof due to its simplicity. We specifically remark that the errors below are independent of the regularity of the pressure. Corollary 2.5 Let $$(\mathrm{L}^{h},\mathbf{u}^{h},p^{h}, \widehat{\mathbf{u}}_{t}^{h})\in{\mathcal{G}}_{h}\times{\boldsymbol V}_{h}^{\textrm{div}}(0)\times \mathring{{Q}_{h}}\times{\boldsymbol{M}}^{t}_{h}(0)$$ be the numerical solution of (2.5); then there exists a constant C, depending only on the polynomial degree k, the shape regularity of the mesh $${\mathcal{T}_{h}}$$ and the domain $$\varOmega$$, such that $$ \nu^{1/2}\left(\|\mathrm{e}_{L}\|_{{\mathcal{T}_{h}}} +{\left\vert\kern-0.25ex\left\vert\kern-0.25ex\left\vert (\mathbf{e}_{u},\mathbf{e}_{\widehat u_{t}}) \right\vert\kern-0.25ex\right\vert\kern-0.25ex\right\vert}_{1,{\mathcal{T}_{h}}}\right) + \max\{ \nu^{1/2}\|\mathbf{e}_{u}\|_{\mathcal{T}_{h}}, \|\gamma^{1/2}\,\mathbf{e}_{u}\|_{\mathcal{T}_{h}}\} \le \;C\,\varTheta\, h^{k+1}, $$ where $$ \varTheta:= \nu^{1/2}\,\|\mathrm{L}\|_{k+1,\varOmega}+ \gamma_{\max}^{1/2}\,\|\mathbf{u}\|_{k+1,\varOmega}, $$ and $$\gamma _{\max }$$ is the maximum eigenvalue of the inverse permeability tensor γ, and ∥⋅∥m, denotes the Hm-norm on $$\varOmega$$. Next we obtain optimal L2-estimates for pressure for k ≥ 0 and superconvergent L2-estimates for the projection error eu for k ≥ 1 (with an H2-regularity assumption for the dual problem). We assume that the regularity estimate \begin{align} \|\varPhi\|_{1,\varOmega}+\|\phi\|_{2,\varOmega}+\|\varphi\|_{1,\varOmega}\le C_{r}\|\boldsymbol{\theta}\|_{\varOmega} \end{align} (2.12) holds for the dual problem \begin{align} \varPhi-{\boldsymbol{\nabla}}\phi=0\;\qquad \textrm{in}\;\varOmega, \end{align} (2.13a) \begin{align} -\nu{\boldsymbol{\nabla\cdot}} \varPhi+\gamma\phi-\nabla \varphi=\boldsymbol{\theta}\;\qquad\textrm{in}\;\varOmega, \end{align} (2.13b) \begin{align} {\boldsymbol{\nabla\cdot}}\phi=0\;\qquad \textrm{in}\;\varOmega, \end{align} (2.13c) \begin{align} \phi=0\;\qquad \textrm{on}\; \partial\varOmega. \end{align} (2.13d) We notice that it is easy to see that the dual problem (2.13) is well posed. Obviously, ($$\varPhi$$, ϕ, φ) is the solution of the Stokes problem with the source term θ − γϕ. So, the regularity estimate (2.12) comes from that of the Stokes problem (see Girault & Raviart, 1986). Theorem 2.6 Let $$(\mathrm{L}^{h},\mathbf{u}^{h},p^{h}, \widehat{\mathbf{u}}_{t}^{h})\in{\mathcal{G}}_{h}\times{\boldsymbol V}_{h}^{\textrm{div}}(0)\times{\mathring{Q}_{h}}\times{\boldsymbol{M}}^{t}_{h}(0)$$ be the numerical solution of (2.5); then there exists a constant C, depending only on the polynomial degree k, the shape regularity of the mesh $${\mathcal{T}_{h}}$$ and the domain $$\varOmega$$, such that \begin{align} \|{e}_{p}\|_{{\mathcal{T}_{h}}}\le C\left(\nu^{1/2}+\gamma_{\max}^{1/2}\right)\,\varTheta\,h^{k+1}. \end{align} (2.14) Here $$\gamma _{\max }$$ and $$\varTheta $$ are defined in Corollary 2.5. In addition, if k ≥ 1, the regularity assumption (2.12) holds and $$\gamma \in W^{1,\infty }(\varOmega )^{d\times d}$$, then we have \begin{align} \|\mathbf{e}_{u}\|_{{\mathcal{T}_{h}}}\le C\,C_{r}\,\left(\left(\nu^{1/2}+\gamma_{\max}^{1/2}\right)\,\varTheta+ \|\gamma\|_{1,\infty}\|\mathbf{u}\|_{k+1}\right)h^{k+2}. \end{align} (2.15) 3. Proofs of Theorems 2.1, 2.4 and 2.6 In this section, we prove the main results in Section 2, namely, Theorems 2.1, 2.4 and 2.6. The following results, Lemmas 3.1 and 3.2, are key ingredients to prove Theorem 2.1. The proof of Lemma 3.1 comes directly from Lemma 2.3 and the usual scaling argument. Lemma 3.1 Given $$(\mathrm{r}^{h},\; \widehat{\mathbf{z}}^{h})\in{\mathcal{G}}(K)\times{\boldsymbol{M}}({\partial K})$$ where $$ {\boldsymbol{M}}({\partial K}):=\left\{\widehat{\mathbf{v}}\in L^{2}({\partial K})^{d}:\;\widehat{\mathbf{v}}|_{F}\in{\boldsymbol{M}}(F)\quad \forall \, F\in\mathcal{F}(K)\right\}\!, $$ there exists a unique function wh ∈ V(K) such that \begin{align*} ({\mathbf{w}^{h}},{\mathbf{v}^{h}})_{K} = &\; ({{\boldsymbol{\nabla\cdot}} \mathrm{r}^{h}},{\mathbf{v}^{h}})_{K}\quad \forall \, \mathbf{v}^{h}\in{\boldsymbol{\nabla\cdot}}{\mathcal{G}}(K),\\ \left\langle \textrm{tr}_{n}(\mathbf{w}^{h}),\,\mathrm{t}\mathrm{r}_{n}(\widehat{\mathbf{v}}) \right\rangle_{\partial{K}} = &\; \left\langle \textrm{tr}_{n}(\widehat{\mathbf{z}}^{h}),\,\textrm{tr}_{n}(\widehat{\mathbf{v}}) \right\rangle_{\partial{K}}\quad \forall \, \widehat{\mathbf{v}}^{h}\in{\boldsymbol{M}}({\partial K}). \end{align*} Moreover, there exists a constant C depending only on the shape regularity of the element K such that \begin{align} \|\mathbf{w}^{h}\|_{K}\le C\left(\|{\boldsymbol{\nabla\cdot}} \mathrm{r}^{h}\|_{K}^{2} + \sum_{F\in\mathcal{F}(K)}h_{F}\|\textrm{tr}_{n}(\,\widehat{\mathbf{z}}^{h})\|_{F}^{2}\right)^{1/2} .\end{align} (3.1) Lemma 3.2 Given $$(\mathbf{z}^{h},\; \widehat{\mathbf{z}}^{h})\in{\boldsymbol V}(K)\times{\boldsymbol{M}}({\partial K})$$ where $$ {\boldsymbol{M}}({\partial K}):=\left\{\widehat{\mathbf{v}}\in L^{2}({\partial K})^{d}:\;\widehat{\mathbf{v}}|_{F}\in{\boldsymbol{M}}(F)\quad \forall \, F\in\mathcal{F}(K)\right\}\!, $$ there exists a unique function $$\mathrm{r}^{h}\in{\mathcal{G}}(K)$$ such that \begin{align} ({\mathrm{r}^{h}},{\mathrm{g}^{h}})_{K} = \; ({{\boldsymbol{\nabla}} \mathbf{z}^{h}},{\mathrm{g}^{h}})_{K}\,\quad \forall \, \mathrm{g}^{h}\in{\boldsymbol{\nabla}}{\boldsymbol V}(K)\oplus{\mathcal{G}}_{\mathrm{s}\mathrm{b}\mathrm{b}}(K), \end{align} (3.2a) \begin{align} \left\langle \textrm{tr}_{t}(\mathrm{r}^{h}\,\mathbf{n}),\,\textrm{tr}_{t}(\widehat{\mathbf{v}}) \right\rangle_{\partial{K}} = \; \left\langle \textrm{tr}_{t}(\widehat{\mathbf{z}}^{h}),\,\textrm{tr}_{t}(\widehat{\mathbf{v}}) \right\rangle_{\partial{K}}\quad \forall \, \widehat{\mathbf{v}}^{h}\in{\textbf{M}}({\partial K}), \end{align} (3.2b) where $$ {\mathcal{G}}_{\mathrm{s}\mathrm{b}\mathrm{b}}(K):=\left\{\mathrm{g}\in{\mathcal{G}}(K):\;\;{\boldsymbol{\nabla\cdot}} \mathrm{g} = 0,\;\;\textrm{tr}_{n}^{F}(\mathrm{g}\,\mathbf{n})=0 \quad \forall \, F\in\mathcal{F}(K)\right\}\!. $$ Moreover, there exists a constant C depending only on the shape regularity of the element K such that \begin{align} \|\mathrm{r}^{h}\|_{K}\le C\left(\|{\boldsymbol{\nabla}} \mathbf{z}^{h}\|_{K}^{2} + \sum_{F\in\mathcal{F}(K)}h_{F}\|\textrm{tr}_{t}(\,\widehat{\mathbf{z}}^{h})\|_{F}^{2}\right)^{1/2}. \end{align} (3.3) Proof. We prove only the existence and uniqueness of the function $$\mathrm{r}^{h}\in{\mathcal{G}}(K)$$ satisfying equations (3.2) on the reference element $$K=\underline{\textsf{K}}$$; the result on an affine-mapped element K can be easily obtained from that on the reference element (cf. Boffi et al., 2013, Chapter 2), and the estimate (3.3) is a direct consequence of the usual scaling argument and equivalence of norms on finite-dimensional spaces. We first show that (3.2) defines a square system. We use the concept of an M-decomposition (Cockburn & Fu, 2017a,b; Cockburn et al., 2017) to prove it. By the choice of $${\mathcal{G}}^{\mathrm{r}\mathrm{o}\mathrm{w}}(K)$$ in Table 1, we obtain that the pair $$ {\mathcal{G}}^{\mathrm{r}\mathrm{o}\mathrm{w}}(K)\times \mathcal{P}_{k}(K)$$ admits an M-decomposition with the trace space $$ M({\partial K}):=\left\{\widehat{w}\in L^{2}({\partial K}):\;\;\widehat{w}|_{F}\in\mathcal{P}_{k}(F)\quad \forall \, F\in\mathcal{F}(K)\right\}\!. $$ Hence \begin{align*} \dim{\mathcal{G}}^{\mathrm{r}\mathrm{o}\mathrm{w}}(K)+ \dim\mathcal{P}_{k}(K) = &\; \dim{\mathcal{G}}^{\mathrm{r}\mathrm{o}\mathrm{w}}_{\mathrm{s}\mathrm{b}\mathrm{b}}(K) + \dim{\nabla\cdot}{\mathcal{G}}^{\mathrm{r}\mathrm{o}\mathrm{w}}(K)\\ & +\dim{\nabla} \mathcal{P}_{k}(K) +\dim M({\partial K}). \end{align*} Here $$ {\mathcal{G}}^{\mathrm{r}\mathrm{o}\mathrm{w}}_{\mathrm{s}\mathrm{b}\mathrm{b}}(K):= \left \{ \mathbf{v}\in{\mathcal{G}}^{\mathrm{r}\mathrm{o}\mathrm{w}}(K):\;{\nabla \cdot } \mathbf{v}=0,\; \textrm{tr}_{n}(\mathbf{v})=0\ \textrm{on}\ {\partial K} \right \}. $$ This immediately implies that \begin{align} \dim{\mathcal{G}}(K)+ \dim\mathcal{P}_{k}(K)^{d} = & \dim{\mathcal{G}}_{\mathrm{s}\mathrm{b}\mathrm{b}}(K) + \dim{\boldsymbol{\nabla\cdot}}{\mathcal{G}}(K)\\ & +\dim{\boldsymbol{\nabla}} \mathcal{P}_{k}(K)^{d} +\dim{\boldsymbol{M}}({\partial K}).\nonumber \end{align} (3.4) By Lemma 2.3, we have $$ \dim{\boldsymbol V}(K) = \dim{\boldsymbol{\nabla\cdot}}{\mathcal{G}}(K)+\dim \textrm{tr}_{n}({\boldsymbol{M}}({\partial K})). $$ Combining the above equality with (3.4) and reordering the terms, we get \begin{align} \dim{\mathcal{G}}(K) = & \dim{\mathcal{G}}_{\mathrm{s}\mathrm{b}\mathrm{b}}(K) +\dim \textrm{tr}_{t}({\boldsymbol{M}}({\partial K})) \\ & +\dim{\boldsymbol V}(K)- \dim\mathcal{P}_{k}(K)^{d} +\dim{\boldsymbol{\nabla}} \mathcal{P}_{k}(K)^{d}.\nonumber \end{align} (3.5) Since it is trivial to prove that $$ \dim{\boldsymbol V}(K)- \dim\mathcal{P}_{k}(K)^{d} +\dim{\boldsymbol{\nabla}} \mathcal{P}_{k}(K)^{d} = \dim{\boldsymbol{\nabla}} {\boldsymbol V}(K) $$ for the vector space V(K) in Table 1, we conclude that equation (3.2) is indeed a square system. Hence, we are left to prove the uniqueness. To this end, we take $$\mathbf{z}^{h}=0, \widehat{\mathbf{z}}^{h}=0$$ in (3.2). By (3.2b), we have \begin{align} \textrm{tr}_{t}(\mathrm{r}^{h}\mathbf{n})=0. \end{align} (3.6) By (3.2a), we have, for all v ∈ V(K), \begin{align*} 0= (\mathrm{r}^{h}, {\boldsymbol{\nabla}} \mathbf{v})_{K} &= -({\boldsymbol{\nabla\cdot}}\mathrm{r}^{h}, \mathbf{v})_{K} +\left\langle \textrm{tr}_{n}(\mathrm{r}^{h}\mathbf{n}),\,\textrm{tr}_{n}(\mathbf{v}) \right\rangle_{\partial{K}} +\left\langle \textrm{tr}_{t}(\mathrm{r}^{h}\mathbf{n}),\,\textrm{tr}_{t}(\mathbf{v}) \right\rangle_{\partial{K}}\\ &= -({\boldsymbol{\nabla\cdot}}\mathrm{r}^{h}, \mathbf{v})_{K} +\left\langle \textrm{tr}_{n}(\mathrm{r}^{h}\mathbf{n}),\,\textrm{tr}_{n}(\mathbf{v}) \right\rangle_{\partial{K}}. \end{align*} Then, by Lemma 3.1, there exists a function v ∈ V(K) such that $$ -({\boldsymbol{\nabla\cdot}}\mathrm{r}^{h}, \mathbf{v})_{K} +\left\langle \textrm{tr}_{n}(\mathrm{r}^{h}\mathbf{n}),\,\textrm{tr}_{n}(\mathbf{v}) \right\rangle_{\partial{K}} =({\boldsymbol{\nabla\cdot}}\mathrm{r}^{h}, {\boldsymbol{\nabla\cdot}}\mathrm{r}^{h})_{K} +\left\langle \textrm{tr}_{n}(\mathrm{r}^{h}\mathbf{n}),\,\textrm{tr}_{n}(\mathrm{r}^{h}\mathbf{n}) \right\rangle_{\partial{K}}.$$ Hence ∇⋅rh = 0 and trn(rhn) = 0. This implies that $$\mathrm{r}^{h}\in{\mathcal{G}}_{\mathrm{s}\mathrm{b}\mathrm{b}}(K)$$. Then taking $$\mathrm{g}^{h} :=\mathrm{r}^{h}\in{\mathcal{G}}_{\mathrm{s}\mathrm{b}\mathrm{b}}(K)$$ in (3.2a), we conclude that rh = 0. This concludes the proof of Lemma 3.2. Now, we are ready to prove Theorem 2.1. 3.1. Proof of Theorem 2.1 Proof. By Lemma 3.2, for any zh ∈ V(K) and $$\widehat{\mathbf{z}}_{t}^{h}\in \{\widehat{\mathbf{v}}\in{\boldsymbol{M}}({\partial K}):\; \textrm{tr}_{n}(\widehat{\mathbf{v}})=0\}$$, there exists $$\mathrm{g}^{h}\in{\mathcal{G}}(K)$$ such that \begin{align*} ({\boldsymbol{\nabla}} \mathbf{z}^{h}, \mathrm{g}^{h})_{K} - \left\langle \textrm{tr}_{t}(\mathbf{z}^{h})-\widehat{\mathbf{z}}_{t}^{h},\,\textrm{tr}_{t}(\mathrm{g}^{h}\,\mathbf{n}) \right\rangle_{\partial{K}} &=\|{\boldsymbol{\nabla}} \mathbf{z}^{h}\|_{K}^{2}\\ &\quad+\sum_{F\in\mathcal{F}(K)}h_{F}^{-1}\|P_{\boldsymbol{M}^{t}}\left(\textrm{tr}_{t}(\mathbf{z}^{h})\right)-\widehat{\mathbf{z}}_{t}^{h}\|_{F}^{2} \end{align*} and $$\|\mathrm{g}^{h}\|_{K}\le C\, \left (\|{\boldsymbol{\nabla }} \mathbf{z}^{h}\|_{K}^{2}+\sum _{F\in \mathcal{F}(K)}h_{F}^{-1}\|P_{\boldsymbol{M}^{t}}(\textrm{tr}_{t}(\mathbf{z}^{h}))-\widehat{\mathbf{z}}_{t}^{h}\|_{F}^{2}\right )^{1/2}$$. Taking such gh in (2.7), we get \begin{align*} \|{\boldsymbol{\nabla}} \mathbf{z}^{h}\|_{K}^{2}+\sum_{F\in\mathcal{F}(K)}h_{F}^{-1}\|P_{\boldsymbol{M}^{t}}(\textrm{tr}_{t}(\mathbf{z}^{h}))-\widehat{\mathbf{z}}_{t}^{h}\|_{F}^{2} &=(\mathrm{r}, \mathrm{g}^{h})_{K}\\ &\le C\, \left(\|{\boldsymbol{\nabla}} \mathbf{z}^{h}\|_{K}^{2}+\sum_{F\in\mathcal{F}(K)}h_{F}^{-1}\|P_{\boldsymbol{M}^{t}}(\textrm{tr}_{t}(\mathbf{z}^{h}))-\widehat{\mathbf{z}}_{t}^{h}\|_{F}^{2}\right)^{1/2}\,\|\mathrm{r}\|_{K}. \end{align*} Hence \begin{align} \left(\|{\boldsymbol{\nabla}} \mathbf{z}^{h}\|_{K}^{2}+\sum_{F\in\mathcal{F}(K)}h_{F}^{-1}\|P_{\boldsymbol{M}^{t}}(\textrm{tr}_{t}(\mathbf{z}^{h}))-\widehat{\mathbf{z}}_{t}^{h}\|_{F}^{2}\right)^{1/2}\le C\,\|\mathrm{r}\|_{K}. \end{align} (3.7) Moreover, on each facet $$F\in \mathcal{F}(K)$$, we have $$ \| \textrm{tr}_{t}(\mathbf{z}^{h}) -P_{\boldsymbol{M}^{t}}(\textrm{tr}_{t}(\mathbf{z}^{h}))\|_{F} = \| \mathbf{z}^{h} -P_{\boldsymbol{M}}(\mathbf{z}^{h})\|_{F} \le \,\| \mathbf{z}^{h} -\overline{\mathbf{z}^{h}}\,\|_{F} \le C\,h_{K}^{1/2}\| {\boldsymbol{\nabla}} \mathbf{z}^{h}\,\|_{K}, $$ where $$\overline{\mathbf{z}^{h}}$$ is the average of zh in the element K and the last inequality is the inverse inequality. Combining the above result with (3.7), we obtain $$ {\left\vert\kern-0.25ex\left\vert\kern-0.25ex\left\vert \left(\mathbf{z}^{h}, \widehat{\mathbf{z}}_{t}^{h}\right) \right\vert\kern-0.25ex\right\vert\kern-0.25ex\right\vert}_{1,K}\le C\,\|\mathrm{r}\|_{K}. $$ The proof of Theorem 2.1 is completed by summing the above estimate over all the elements $$K\in{\mathcal{T}_{h}}$$. We use the following error equation to prove Theorem 2.4. To simplify notation, we denote \begin{align} B_{h}\left(\mathrm{L}, \mathbf{u}, \;p,\;\widehat{\mathbf{u}}_{t};\; \mathrm{g}, \mathbf{v},\; q,\;\widehat{\mathbf{v}}_{t}\right)&:= ( \mathrm{L}\,,\,\nu\,\mathrm{g} )_{{\mathcal{T}_{h}}}-( {\boldsymbol{\nabla}} \mathbf{u}\,,\,\nu\,\mathrm{g} )_{{\mathcal{T}_{h}}} \\ &\quad + \left\langle \textrm{tr}_{t}(\mathbf{u})-\widehat{\mathbf{u}}_{t},\, \textrm{tr}_{t}(\nu\,\mathrm{g}\, \mathbf{n}) \right\rangle_{\partial{{\mathcal{T}_{h}}}} \nonumber\\ &\quad + ( \nu\,\mathrm{L}\,,\,{\boldsymbol{\nabla}} \mathbf{v} )_{{\mathcal{T}_{h}}} - \left\langle \textrm{tr}_{t}(\nu\,\mathrm{L}\,\mathbf{n}),\,\textrm{tr}_{t}(\mathbf{v})-\widehat{\mathbf{v}}_{t} \right\rangle_{\partial{{\mathcal{T}_{h}}}}\nonumber\\ &\quad- ( p,\,{\nabla\cdot} \mathbf{v} )_{{\mathcal{T}_{h}}}+( \gamma\, \mathbf{u},\,\mathbf{v} )_{{\mathcal{T}_{h}}}\nonumber\\ &\quad + ( {\nabla\cdot}\mathbf{u},\,q )_{{\mathcal{T}_{h}}}.\nonumber \end{align} (3.8) Lemma 3.3 Let (L, u, p) be the solution to (1.1), and $$(\mathrm{L}^{h},\; \mathbf{u}^{h}, \;p^{h},\;\widehat{\mathbf{u}}_{t}^{h})$$ be the numerical solution to (2.5). Then we have \begin{align} B_{h}\left(\mathrm{e}_{L},\; \mathbf{e}_{u},\; {e}_{p},\;\mathbf{e}_{\widehat u_{t}}; \mathrm{g}^{h},\; \mathbf{v}^{h}, q^{h},\;\widehat{\mathbf{v}}_{t}^{h}\right) &= \left\langle \textrm{tr}_{t}(\nu\,{\delta}_{L}\,\mathbf{n}),\,\textrm{tr}_{t}(\mathbf{v}^{h})-\widehat{\mathbf{v}}_{t}^{h} \right\rangle_{\partial{{\mathcal{T}_{h}}}}\\[-5pt] & \quad-( \gamma\,\boldsymbol{\delta}_{u},\,\mathbf{v}^{h} )_{{\mathcal{T}_{h}}}\nonumber \end{align} (3.9) for all $$(\mathrm{g}^{h},\mathbf{v}^{h},q^{h},{\widehat{\mathbf{v}}_{t}}^{h})\in{\mathcal{G}}_{h}\times{\boldsymbol V}_{h}^{\textrm{div}}(0)\times \mathring{{Q}_{h}}\times{\boldsymbol{M}}^{t}_{h}(0)$$. Proof. By (1.1), (2.5) and (3.8), we have \begin{align*} B_{h}\left(\mathrm{L}^{h},\; \mathbf{u}^{h},\; p^{h},\widehat{\mathbf{u}}_{t}^{h};\; \mathrm{g}^{h}, \mathbf{v}^{h},\; q^{h},\;\widehat{\mathbf{v}}_{t}^{h}\right) &=(\mathbf{f},\mathbf{v}^{h})_{\mathcal{T}_{h}} + (g,q^{h})_{\mathcal{T}_{h}},\\ B_{h}\left(\mathrm{L},\; \mathbf{u},\; p,\;\textrm{tr}_{t}(\mathbf{u});\; \mathrm{g}^{h}, \mathbf{v}^{h},\; q^{h},\;\widehat{\mathbf{v}}_{t}^{h}\right) &=(\mathbf{f},\mathbf{v}^{h})_{\mathcal{T}_{h}} + (g,q^{h})_{\mathcal{T}_{h}}, \end{align*} for all $$(\mathrm{g}^{h},\;\mathbf{v}^{h},\;q^{h},\;{\widehat{\mathbf{v}}_{t}}^{h})\in{\mathcal{G}}_{h}\times{\boldsymbol V}_{h}^{\textrm{div}}(0)\times \mathring{{Q}_{h}}\times{\boldsymbol{M}}^{t}_{h}(0)$$. Hence $$ B_{h}\left(\mathrm{e}_{L},\; \mathbf{e}_{u},\; {e}_{p},\;\mathbf{e}_{\widehat u_{t}};\; \mathrm{g}^{h}, \mathbf{v}^{h},\; q^{h},\;\widehat{\mathbf{v}}_{t}^{h}\right) = -{B}_{h}\left({\delta}_{L},\; \boldsymbol{\delta}_{u},\; {\delta}_{p},\;\boldsymbol{\delta}_{\widehat u_{t}};\; \mathrm{g}^{h},\; \mathbf{v}^{h},\; q^{h},\;\widehat{\mathbf{v}}_{t}^{h}\right). $$ Using orthogonality properties of the projections, we easily obtain $$ B_{h}\left({\delta}_{L},\; \boldsymbol{\delta}_{u},\; {\delta}_{p},\;\boldsymbol{\delta}_{\widehat u_{t}};\; \mathrm{g}^{h},\; \mathbf{v}^{h},\; q^{h},\;\widehat{\mathbf{v}}_{t}^{h}\right) = -\left\langle \textrm{tr}_{t}(\nu\,{\delta}_{L}\,\mathbf{n}),\,\textrm{tr}_{t}(\mathbf{v}^{h})-\widehat{\mathbf{v}}_{t}^{h} \right\rangle_{\partial{{\mathcal{T}_{h}}}}+( \gamma\,\boldsymbol{\delta}_{u},\,\mathbf{v}^{h} )_{{\mathcal{T}_{h}}}. $$ This completes the proof. Now we are ready to prove Theorem 2.4. 3.2. Proof of Theorem 2.4 Proof. By Di Pietro & Ern (2010, Theorem 2.1), we have $$ \|\mathbf{e}_{u}\|_{\mathcal{T}_{h}} \le C\, \left( \|{\boldsymbol{\nabla}} \mathbf{e}_{u}\|_{\mathcal{T}_{h}} + \sum_{F\in\mathcal{F}(K)}h_{F}^{-1} \left\|\,[\![\mathbf{e}_{u}]\!]\right\|_{F}^{2} \right)^{1/2}. $$ Here $$\,[\![\mathbf{e}_{u}]\!]:=\mathbf{e}_{u}^{+}-\mathbf{e}_{u}^{-}$$ denotes the jump of $$\mathbf{e}_{u}\in{\boldsymbol V}_{h}^{\textrm{div}}(0)$$ on an interior facet F := K+ ∩ K−, and $$[\![\mathbf{e}_u]\!] :=\mathbf{e}_u $$ on a boundary facet F ⊂ ∂$$\varOmega $$, where $$\mathbf{e}_{u}^{\pm } = \mathbf{e}_{u}|_{K^{\pm }}$$. Since eu is H(div)-conforming and has vanishing normal trace on the boundary, we have trn$$([\![\mathbf{e}]\!]) = 0 $$ for all facets $$F\in \mathcal{E}_{h}$$. Hence $$ \,[\![\mathbf{e}_{u}]\!]=\textrm{tr}_{t}(\,[\![\mathbf{e}_{u}]\!]).$$ By the triangle inequality, we have $$ \|\mathrm{t}\mathrm{r}_{t}(\,[\![\mathbf{e}_{u}]\!])\|_{F}\le \|\mathrm{t}\mathrm{r}_{t}(\mathbf{e}_{u}^{+})-\mathbf{e}_{\widehat u_{t}}\|_{F} +\|\mathrm{t}\mathrm{r}_{t}(\mathbf{e}_{u}^{-})-\mathbf{e}_{\widehat u_{t}}\|_{F}. $$ Combining the above estimates, we finish the proof of the first error estimate (2.11a). The second error estimate (2.11b) comes directly from Theorem 2.1. Now, let us prove the last error estimate (2.11c). Taking $$(\mathrm{g}^{h}, \mathbf{v}^{h}, q^{h},\widehat{\mathbf{v}}_{t}^{h}):=(\mathrm{e}_{L}, \mathbf{e}_{u}, {e}_{p},\mathbf{e}_{\widehat u_{t}})$$, we obtain \begin{align*} \nu\|\mathrm{e}_{L}\|_{\mathcal{T}_{h}}^{2}+ \|\gamma^{1/2}\mathbf{e}_{u}\|_{\mathcal{T}_{h}}^{2} &= -\left\langle \mathrm{t}\mathrm{r}_{t}(\nu\,{\delta}_{L}\,\mathbf{n}),\,\mathrm{t}\mathrm{r}_{t}(\mathbf{e}_{u})-\mathbf{e}_{\widehat u_{t}} \right\rangle_{\partial{{\mathcal{T}_{h}}}}+( \gamma\,\boldsymbol{\delta}_{u},\,\mathbf{e}_{u} )_{{\mathcal{T}_{h}}}\\ &\le \sum_{F\in\mathcal{E}_{h}}\left(h_{F}^{1/2}\|\mathrm{t}\mathrm{r}_{t}(\nu\,{\delta}_{L}\,\mathbf{n})\|_{F}\,h_{F}^{-1/2}\|{\mathrm{t}\mathrm{r}_{t}(\mathbf{e}_{u})-\mathbf{e}_{\widehat u_{t}}}\|_{F} \right) \\ &\quad +\|\gamma^{1/2}\boldsymbol{\delta}_{u}\|_{\mathcal{T}_{h}}\|\gamma^{1/2}\mathbf{e}_{u}\|_{\mathcal{T}_{h}}\\ &\le \;C\, \left(\sum_{F\in\mathcal{E}_{h}}\nu\,h_{F}\|{\delta}_{L}\,\mathbf{n}\|_{F}^{2}+\|\gamma^{1/2}\,\boldsymbol{\delta}_{u}\|_{\mathcal{T}_{h}}^{2} \right)^{{1/2}} \,\left( \nu\|\mathrm{e}_{L}\|_{\mathcal{T}_{h}}^{2}+ \|\gamma^{1/2}\mathbf{e}_{u}\|_{\mathcal{T}_{h}}^{2} \right)^{1/2} \end{align*} by the Cauchy–Schwarz inequality. This completes the proof of Theorem 2.4. The following result is used to prove the velocity estimate in Theorem 2.6. Lemma 3.4 Let ($$\varPhi$$, ϕ, φ) be the solution to the dual problem (2.13) for $$\boldsymbol{\theta }\in L^{2}({\mathcal{T}_{h}})^{d}$$. We have \begin{align} ( \mathbf{e}_{u},\,\boldsymbol{\theta} )_{{\mathcal{T}_{h}}}&=\left\langle \nu\,\mathrm{e}_{L}\,\mathbf{n},\,\boldsymbol{\delta}_{\phi} \right\rangle_{\partial{{\mathcal{T}_{h}}}}+\left\langle \mathrm{t}\mathrm{r}_{t}(\nu\,{\delta}_{L}\,\mathbf{n})+\mathrm{t}\mathrm{r}_{t}(\nu\,\mathrm{e}_{L}\,\mathbf{n}),\,\Pi_{{\boldsymbol V}}\phi-P_{M}\phi \right\rangle_{\partial{{\mathcal{T}_{h}}}}\nonumber\\ &\quad+\left\langle \mathrm{t}\mathrm{r}_{t}(\mathbf{e}_{u})-\mathbf{e}_{\widehat u_{t}},\,\nu\,\delta_{\varPhi}\mathbf{n} \right\rangle_{\partial{{\mathcal{T}_{h}}}}+( \gamma\,\mathbf{e}_{u},\,\boldsymbol{\delta}_{\phi} )_{{\mathcal{T}_{h}}}-( \gamma\,\boldsymbol{\delta}_{u},\,\Pi_{{\boldsymbol V}}\phi )_{{\mathcal{T}_{h}}}\nonumber\\ &=:T_{1}+T_{2}+T_{3}+T_{4}+T_{5}, \end{align} (3.10) where $$\delta _{\varPhi }=\varPhi -P_{\mathcal{G}}\varPhi ,\boldsymbol{\delta }_{\phi }=\phi -\Pi _{{\boldsymbol V}}\phi ,\delta _{\varphi }=\varphi -P_{Q}\varphi $$. Proof. By (2.13a)–(2.13c), we have \begin{align*} ( \mathbf{e}_{u},\,\boldsymbol{\theta} )_{{\mathcal{T}_{h}}}=&-( \mathbf{e}_{u},\,\nu\,{\boldsymbol{\nabla\cdot}}\varPhi )_{{\mathcal{T}_{h}}}+( \mathbf{e}_{u},\,\nu\phi )_{{\mathcal{T}_{h}}}-( \mathbf{e}_{u},\,{\nabla}\varphi )_{{\mathcal{T}_{h}}}\\ &-( \nu\mathrm{e}_{L},\,\varPhi )_{{\mathcal{T}_{h}}}+( \nu\,\mathrm{e}_{L},\,{\boldsymbol{\nabla}}\phi )_{{\mathcal{T}_{h}}}-( {e}_{p},\,{\boldsymbol{\nabla\cdot}}\phi )_{{\mathcal{T}_{h}}}\\ =&-( \mathbf{e}_{u},\,\nu\,{\boldsymbol{\nabla\cdot}}P_{\mathcal{G}}\varPhi )_{{\mathcal{T}_{h}}}-( \mathbf{e}_{u},\,\nu\,{\boldsymbol{\nabla\cdot}}\delta_{\varPhi} )_{{\mathcal{T}_{h}}}-( \mathbf{e}_{u},\,{\nabla}P_{Q}\varphi )_{{\mathcal{T}_{h}}}-( \mathbf{e}_{u},\,{\nabla}\delta_{\varphi} )_{{\mathcal{T}_{h}}}\\ &\quad+( \mathbf{e}_{u},\,\gamma\phi )_{{\mathcal{T}_{h}}}-( \nu\,\mathrm{e}_{L},\,P_{\mathcal{G}}\varPhi )_{{\mathcal{T}_{h}}}+( \nu\,\mathrm{e}_{L},\,{\boldsymbol{\nabla}}\phi )_{{\mathcal{T}_{h}}}-( {e}_{p},\,{\boldsymbol{\nabla\cdot}}\phi )_{{\mathcal{T}_{h}}}. \end{align*} Taking $$(\mathrm{g}^{h}, \mathbf{v}^{h}, q^{h},\widehat{\mathbf{v}}_{t}^{h}):=(P_{\mathcal{G}}\varPhi, \boldsymbol{0}, -P_{Q}\varphi ,0)$$ in the error equation (3.9), putting the resulting identity into the above expression and simplifying, we have \begin{align*} ( \mathbf{e}_{u},\,\boldsymbol{\theta} )_{{\mathcal{T}_{h}}}=&-\left\langle \mathbf{e}_{u},\,\nu\,P_{\mathcal{G}}\varPhi\mathbf{n} \right\rangle_{\partial{{\mathcal{T}_{h}}}}-\left\langle \mathbf{e}_{u},\,P_{Q}\varphi\mathbf{n} \right\rangle_{\partial{{\mathcal{T}_{h}}}}\\ &+\left\langle \mathrm{t}\mathrm{r}_{t}(\mathbf{e}_{u})-\mathbf{e}_{\widehat u_{t}},\,\mathrm{t}\mathrm{r}_{t}(\nu\,P_{\mathcal{G}}\varPhi\mathbf{n}) \right\rangle_{\partial{{\mathcal{T}_{h}}}}-( \mathbf{e}_{u},\,\nu\,{\boldsymbol{\nabla\cdot}}\delta_{\varPhi} )_{{\mathcal{T}_{h}}}-( \mathbf{e}_{u},\,{\nabla}\delta_{\varphi} )_{{\mathcal{T}_{h}}}\\ &+( \mathbf{e}_{u},\,\gamma\phi )_{{\mathcal{T}_{h}}}+( \nu\,\mathrm{e}_{L},\,{\boldsymbol{\nabla}}\phi )_{{\mathcal{T}_{h}}}-( {e}_{p},\,{\boldsymbol{\nabla\cdot}}\phi )_{{\mathcal{T}_{h}}}\\ =&-\langle \mathbf{e}_{u},\,\nu\,P_{\mathcal{G}}\varPhi\mathbf{n} \rangle_{\partial{{\mathcal{T}_{h}}}}-\langle \mathbf{e}_{u},\,P_{Q}\varphi\mathbf{n} \rangle_{\partial{{\mathcal{T}_{h}}}}\\ &+\left\langle \mathrm{t}\mathrm{r}_{t}(\mathbf{e}_{u})-\mathbf{e}_{\widehat u_{t}},\,\mathrm{t}\mathrm{r}_{t}(\nu\,P_{\mathcal{G}}\varPhi\mathbf{n}) \right\rangle_{\partial{{\mathcal{T}_{h}}}}-\langle \mathbf{e}_{u},\,\nu\,\delta_{\varPhi}\mathbf{n} \rangle_{\partial{{\mathcal{T}_{h}}}}-\langle \mathbf{e}_{u},\,\delta_{\varphi}\mathbf{n} \rangle_{\partial{{\mathcal{T}_{h}}}}\\ &+( \mathbf{e}_{u},\,\gamma\phi )_{{\mathcal{T}_{h}}}+( \nu\,\mathrm{e}_{L},\,{\boldsymbol{\nabla}}\phi )_{{\mathcal{T}_{h}}}-( {e}_{p},\,{\boldsymbol{\nabla\cdot}}\phi )_{{\mathcal{T}_{h}}}\\ =&-\langle \mathbf{e}_{u},\,\nu\,\varPhi\mathbf{n} \rangle_{\partial{{\mathcal{T}_{h}}}}+\langle \mathrm{t}\mathrm{r}_{t}(\mathbf{e}_{u})-\mathbf{e}_{\widehat u_{t}},\,\nu\,P_{\mathcal{G}}\varPhi\mathbf{n} \rangle_{\partial{{\mathcal{T}_{h}}}}\\ &+( \mathbf{e}_{u},\,\gamma\phi )_{{\mathcal{T}_{h}}}+( \nu\,\mathrm{e}_{L},\,{\boldsymbol{\nabla}}\phi )_{{\mathcal{T}_{h}}}-( {e}_{p},\,{\boldsymbol{\nabla\cdot}}\phi )_{{\mathcal{T}_{h}}}\\ =&-\left\langle \mathrm{t}\mathrm{r}_{t}(\mathbf{e}_{u})-\mathbf{e}_{\widehat u_{t}},\,\nu\,\delta_{\varPhi}\mathbf{n} \right\rangle_{\partial{{\mathcal{T}_{h}}}}\\ &+( \mathbf{e}_{u},\,\gamma\phi )_{{\mathcal{T}_{h}}}+( \nu\,\mathrm{e}_{L},\,{\boldsymbol{\nabla}}\phi )_{{\mathcal{T}_{h}}}-( {e}_{p},\,{\boldsymbol{\nabla\cdot}}\phi )_{{\mathcal{T}_{h}}}, \end{align*} by inserting the zero term $$\langle \mathbf{e}_{\widehat u_{t}},\,\nu \,\varPhi \mathbf{n} \rangle _{\partial{{\mathcal{T}_{h}}}}$$ and using the fact that $$\langle \mathbf{e}_{u},\,\nu \,\varPhi \mathbf{n} \rangle _{\partial{{\mathcal{T}_{h}}}}=\langle \mathrm{t}\mathrm{r}_{t}(\mathbf{e}_{u}),\,\nu \,\varPhi \mathbf{n} \rangle _{\partial{{\mathcal{T}_{h}}}}$$ and $$\langle \mathbf{e}_{u},\,\varphi \mathbf{n} \rangle _{\partial{{\mathcal{T}_{h}}}}=0$$. Take $$(\mathrm{g}^{h}, \mathbf{v}^{h}, q^{h},\widehat{\mathbf{v}}_{t}^{h}):=(0, \Pi _{{\boldsymbol V}}\phi , 0, P_{\textbf{M}^{t}}\phi )$$ in the error equation (3.9). Denoting by $$I:=( \mathbf{e}_{u},\,\gamma \phi )_{{\mathcal{T}_{h}}}+( \nu \,\mathrm{e}_{L},\,{\boldsymbol{\nabla }}\phi )_{{\mathcal{T}_{h}}}-( {e}_{p},\,{\boldsymbol{\nabla \cdot }}\phi )_{{\mathcal{T}_{h}}}$$, we obtain \begin{align*} I&=( \mathbf{e}_{u},\,\gamma\boldsymbol{\delta}_{\phi} )_{{\mathcal{T}_{h}}}+( \nu\,\mathrm{e}_{L},\,{\boldsymbol{\nabla}}\boldsymbol{\delta}_{\phi} )_{{\mathcal{T}_{h}}}-( {e}_{p},\,{\boldsymbol{\nabla\cdot}}\boldsymbol{\delta}_{\phi} )_{{\mathcal{T}_{h}}}\\ &\quad+( \mathbf{e}_{u},\,\gamma\Pi_{{\boldsymbol V}}\phi )_{{\mathcal{T}_{h}}}+( \nu\,\mathrm{e}_{L},\,{\boldsymbol{\nabla}}\Pi_{{\boldsymbol V}}\phi )_{{\mathcal{T}_{h}}}-( {e}_{p},\,{\boldsymbol{\nabla\cdot}}\Pi_{{\boldsymbol V}}\phi )_{{\mathcal{T}_{h}}}\\ &=( \mathbf{e}_{u},\,\gamma\boldsymbol{\delta}_{\phi} )_{{\mathcal{T}_{h}}}+( \nu\,\mathrm{e}_{L},\,{\boldsymbol{\nabla}}\boldsymbol{\delta}_{\phi} )_{{\mathcal{T}_{h}}}-( {e}_{p},\,{\boldsymbol{\nabla\cdot}}\boldsymbol{\delta}_{\phi} )_{{\mathcal{T}_{h}}}\\ &\quad\left\langle \mathrm{t}\mathrm{r}_{t}(\nu\,{\delta}_{L}\mathbf{n})+\mathrm{t}\mathrm{r}_{t}(\nu\,\mathrm{e}_{L}\mathbf{n}),\,\mathrm{t}\mathrm{r}_{t}(\Pi_{{\boldsymbol V}}\phi)-P_{\textbf{M}^{t}}\phi \right\rangle_{\partial{{\mathcal{T}_{h}}}}-( \gamma\boldsymbol{\delta}_{u},\,\Pi_{{\boldsymbol V}}\phi )_{{\mathcal{T}_{h}}}\\ &=( \mathbf{e}_{u},\,\gamma\boldsymbol{\delta}_{\phi} )_{{\mathcal{T}_{h}}}+\langle \nu\,\mathrm{e}_{L}\mathbf{n},\,\boldsymbol{\delta}_{\phi} \rangle_{\partial{{\mathcal{T}_{h}}}}\\ &\quad\left\langle \mathrm{t}\mathrm{r}_{t}(\nu\,{\delta}_{L}\mathbf{n})+\mathrm{t}\mathrm{r}_{t}(\nu\,\mathrm{e}_{L}\mathbf{n}),\,\Pi_{{\boldsymbol V}}\phi-P_{M}\phi \right\rangle_{\partial{{\mathcal{T}_{h}}}}-( \gamma\boldsymbol{\delta}_{u},\,\Pi_{{\boldsymbol V}}\phi )_{{\mathcal{T}_{h}}}. \end{align*} This completes the proof of Lemma 3.4. Now we are ready to prove Theorem 2.6. 3.3. Proof of Theorem 2.6 Proof. We first present the optimal error estimate for ep by applying an inf–sup argument. It is well known that the following inf–sup condition holds for a positive constant κ (cf. Girault & Raviart, 1986, Chapter 1, Corollary 2.4): \begin{align} \sup_{\boldsymbol{\omega}\in{H^{1}_{0}}(\varOmega)^{d}\backslash\{0\}}\frac{({\boldsymbol{\nabla\cdot}}\boldsymbol{\omega},q)_{\varOmega}}{\|\boldsymbol{\omega}\|_{1,\varOmega}}\ge\kappa\|q\|_{\varOmega}. \end{align} (3.11) Here ∥⋅∥$$1{,\varOmega}$$ is the standard H1-norm on $$\varOmega$$. Since $${e}_{p}\in{L^{2}_{0}}(\varOmega )$$, we have by (3.11), \begin{align} \|e_{p}\|_{\varOmega}\le\frac{1}{\kappa}\sup_{\boldsymbol{\omega}\in{H^{1}_{0}}(\varOmega)^{d}\backslash\{0\}}\frac{({\boldsymbol{\nabla\cdot}}\boldsymbol{\omega},e_{p})_{\varOmega}}{\|\boldsymbol{\omega}\|_{1,\varOmega}}. \end{align} (3.12) Taking $$(\mathrm{g}^{h},\; \mathbf{v}^{h},\; q^{h},\;\widehat{\mathbf{v}}_{t}^{h}):=(0,\; \Pi _{{\boldsymbol V}}\boldsymbol{\omega },\; 0,\; P_{\boldsymbol{M}^{t}}\boldsymbol{\omega })$$ in the error equation (3.9) and applying integration by parts, we can rewrite the numerator as: \begin{align*} ( {\boldsymbol{\nabla\cdot}} \boldsymbol{\omega},\,{e}_{p} )_{{\mathcal{T}_{h}}}&=( {\boldsymbol{\nabla\cdot}}\Pi_{{\boldsymbol V}}\boldsymbol{\omega},\,{e}_{p} )_{{\mathcal{T}_{h}}}+( {\boldsymbol{\nabla\cdot}}(\boldsymbol{\omega}-\Pi_{{\boldsymbol V}}\boldsymbol{\omega}),\,{e}_{p} )_{{\mathcal{T}_{h}}}=( {\boldsymbol{\nabla\cdot}}\Pi_{{\boldsymbol V}}\boldsymbol{\omega},\,{e}_{p} )_{{\mathcal{T}_{h}}}\\ &=( \nu\,\mathrm{e}_{L},\,{\boldsymbol{\nabla}}\Pi_{{\boldsymbol V}}\boldsymbol{\omega} )_{{\mathcal{T}_{h}}}-\langle \mathrm{t}\mathrm{r}_{t}(\nu\,\mathrm{e}_{L}\mathbf{n})+\mathrm{t}\mathrm{r}_{t}(\nu\,{\delta}_{L}\mathbf{n}),\,\mathrm{t}\mathrm{r}_{t}(\Pi_{{\boldsymbol V}}\boldsymbol{\omega})-P_{\boldsymbol{M}^{t}}\boldsymbol{\omega} \rangle_{\partial{{\mathcal{T}_{h}}}}\\ &\quad+( \gamma\mathbf{e}_{u},\,\Pi_{{\boldsymbol V}}\boldsymbol{\omega} )_{{\mathcal{T}_{h}}}+( \gamma\boldsymbol{\delta}_{u},\,\Pi_{{\boldsymbol V}}\boldsymbol{\omega} )_{{\mathcal{T}_{h}}}\\ &=( \nu\,\mathrm{e}_{L},\,{\boldsymbol{\nabla}}\Pi_{{\boldsymbol V}}\boldsymbol{\omega} )_{{\mathcal{T}_{h}}}-\langle \mathrm{t}\mathrm{r}_{t}(\nu\,\mathrm{e}_{L}\mathbf{n})+\mathrm{t}\mathrm{r}_{t}(\nu\,{\delta}_{L}\mathbf{n}),\,\Pi_{{\boldsymbol V}}\boldsymbol{\omega}-P_{M}\boldsymbol{\omega} \rangle_{\partial{{\mathcal{T}_{h}}}}\\ &\quad+( \gamma\mathbf{e}_{u},\,\Pi_{{\boldsymbol V}}\boldsymbol{\omega} )_{{\mathcal{T}_{h}}}+( \gamma\boldsymbol{\delta}_{u},\,\Pi_{{\boldsymbol V}}\boldsymbol{\omega} )_{{\mathcal{T}_{h}}}\\ &=:\,I_{1}+I_{2}+I_{3}+I_{4}. \end{align*} Then we will bound I1 to I4 by Corollary 2.5 as: \begin{align*} I_{1}\le&\;\nu\|\mathrm{e}_{L}\|_{{\mathcal{T}_{h}}}\|{\boldsymbol{\nabla}}\Pi_{{\boldsymbol V}}\boldsymbol{\omega}\|_{{\mathcal{T}_{h}}}\le C\nu^{1/2}\varTheta h^{k+1}\|\boldsymbol{\omega}\|_{1,\varOmega},\\ I_{2}\le&\;\nu\left(\|\mathrm{e}_{L}\mathbf{n}\|_{\partial{\mathcal{T}_{h}}}+\|{\delta}_{L}\mathbf{n}\|_{\partial{\mathcal{T}_{h}}}\right)\|\Pi_{{\boldsymbol V}}\boldsymbol{\omega}-P_{M}\boldsymbol{\omega}\|_{\partial{\mathcal{T}_{h}}}\\ \le&\;C(\nu^{1/2}\varTheta h^{k+1/2}+\nu\|\mathrm{L}\|_{k+1}h^{k+1/2})h^{1/2}\|\boldsymbol{\omega}\|_{1,\varOmega}\le C\nu^{1/2}\varTheta h^{k+1}\|\boldsymbol{\omega}\|_{1,\varOmega},\\ I_{3}\le&\;C\gamma^{1/2}_{\max}\|\gamma^{1/2}\mathbf{e}_{u}\|_{{\mathcal{T}_{h}}}\|\Pi_{{\boldsymbol V}}\boldsymbol{\omega}\|_{{\mathcal{T}_{h}}}\le C\gamma^{1/2}_{\max}\varTheta h^{k+1}\|\boldsymbol{\omega}\|_{1,\varOmega},\\ I_{4}\le&\;C\gamma_{\max}\|\boldsymbol{\delta}_{u}\|_{{\mathcal{T}_{h}}}\|\Pi_{{\boldsymbol V}}\boldsymbol{\omega}\|_{{\mathcal{T}_{h}}}\le C\gamma_{\max}\|\mathbf{u}\|_{k+1} h^{k+1}\|\boldsymbol{\omega}\|_{1,\varOmega}\\ \le&\;C\gamma^{1/2}_{\max}\varTheta h^{k+1}\|\boldsymbol{\omega}\|_{1,\varOmega}. \end{align*} Then we have $$ ( {\boldsymbol{\nabla\cdot}} \boldsymbol{\omega},\,{e}_{p} )_{{\mathcal{T}_{h}}}\le C\left(\nu^{1/2}+\gamma^{1/2}_{\max}\right)\varTheta h^{k+1}\|\boldsymbol{\omega}\|_{1,\varOmega}.$$ By (3.12), we obtain the estimate for ep. Now we give a superconvergent estimate for eu. By (3.10), it suffices to estimate the terms T1 to T5. We apply Corollary 2.5, the regularity assumption (2.12) and the inverse inequality to bound these terms: \begin{align*} T_{1}\le&\;\nu\|\mathrm{e}_{L}\mathbf{n}\|_{\partial{\mathcal{T}_{h}}}\|\boldsymbol{\delta}_{\phi}\|_{\partial{\mathcal{T}_{h}}}\le C\nu h^{-1/2}\|\mathrm{e}_{L}\|_{{\mathcal{T}_{h}}}h^{3/2}\|\phi\|_{2}\\ \le&\;C\nu^{1/2}\varTheta h^{k+2}\|\boldsymbol{\theta}\|_{{\mathcal{T}_{h}}},\\ T_{2}\le&\;\nu\left(\|{\delta}_{L}\mathbf{n}\|_{\partial{\mathcal{T}_{h}}}+\|\mathrm{e}_{L}\mathbf{n}\|_{\partial{\mathcal{T}_{h}}}\right)\|\Pi_{{\boldsymbol V}}\phi-P_{M}\phi\|_{\partial{\mathcal{T}_{h}}}\\ \le&\;C\left(\nu\|\mathrm{L}\|_{k+1}h^{k+1/2}+\nu^{1/2}\varTheta h^{k+1/2}\right)h^{3/2}\|\phi\|_{2}\le C\nu^{1/2}\varTheta h^{k+2}\|\boldsymbol{\theta}\|_{{\mathcal{T}_{h}}},\\ T_{3}\le&\;\nu h^{-1/2}\|\mathrm{t}\mathrm{r}_{t}(\mathbf{e}_{u})-\mathbf{e}_{\widehat u_{t}}\|_{\partial{\mathcal{T}_{h}}}h^{1/2}\|\delta_{\varPhi}\mathbf{n}\|_{\partial{\mathcal{T}_{h}}}\\ \le&\;C\nu^{1/2}{\left\vert\kern-0.25ex\left\vert\kern-0.25ex\left\vert (\mathbf{e}_{u},\mathbf{e}_{\widehat u_{t}}) \right\vert\kern-0.25ex\right\vert\kern-0.25ex\right\vert}_{1,{\mathcal{T}_{h}}}h\|\varPhi\|_{1,\varOmega}\le C\nu^{1/2}\varTheta h^{k+2}\|\boldsymbol{\theta}\|_{{\mathcal{T}_{h}}},\\ T_{4}\le&\;\gamma^{1/2}_{\max}\|\gamma^{1/2}\mathbf{e}_{u}\|_{{\mathcal{T}_{h}}}\|\boldsymbol{\delta}_{\phi}\|_{{\mathcal{T}_{h}}}\le C\gamma^{1/2}_{\max}\varTheta h^{k+2}\|\boldsymbol{\theta}\|_{{\mathcal{T}_{h}}},\\ T_{5}=&\;\left( (\gamma-P_{0,h}\gamma)\boldsymbol{\delta}_{u},\,\Pi_{{\boldsymbol V}}\phi \right)_{{\mathcal{T}_{h}}}+( P_{0,h}\gamma\boldsymbol{\delta}_{u},\,\Pi_{{\boldsymbol V}}\phi- \bar{\phi} )_{{\mathcal{T}_{h}}}\\ \le&\;\|\gamma-P_{0,h}\gamma\|_{\infty}\|\boldsymbol{\delta}_{u}\|_{{\mathcal{T}_{h}}}\|\Pi_{{\boldsymbol V}}\phi\|_{{\mathcal{T}_{h}}}+|P_{0,h}\gamma|\|\boldsymbol{\delta}_{u}\|_{{\mathcal{T}_{h}}}\|\Pi_{{\boldsymbol V}}(\phi-\bar{\phi})\|_{{\mathcal{T}_{h}}}\\ \le&\;Ch\|\gamma\|_{1,\infty}h^{k+1}\|\mathbf{u}\|_{k+1}\|\phi\|_{2}+C\|\gamma\|_{0,\infty}\,h^{k+1}\|\mathbf{u}\|_{k+1}h\|{\boldsymbol{\nabla}}\phi\|_{{\mathcal{T}_{h}}}\\ \le&\;C \|\gamma\|_{1,\infty}\|\mathbf{u}\|_{k+1}h^{k+2}\|\boldsymbol{\theta}\|_{{\mathcal{T}_{h}}}, \end{align*} where P0, h is the L2- orthogonal projection onto $$\mathcal{P}_{0}({\mathcal{T}_{h}})^{d\times d}$$ and $$\bar{\phi }$$ is defined as $$ \bar{\phi}=\frac{1}{|K|}( \phi,\,1 )_{K}\quad \forall \, K\in{\mathcal{T}_{h}}.$$ Combining all the above estimates, we have $$ \|\mathbf{e}_{u}\|_{{\mathcal{T}_{h}}}\le C\left(\nu^{1/2}\varTheta+\gamma_{\max}^{1/2}\varTheta+\|\gamma\|_{1,\infty}\|\mathbf{u}\|_{k+1}\right)h^{k+2}.$$ This completes the proof of Theorem 2.6. 4. Hybridization In this section, we hybridize the H(div)-conforming HDG method (2.5) by relaxing the H(div)-conformity of the velocity field via Lagrange multipliers; similar treatment was used in the study by Cockburn & Sayas (2014). The resulting global linear system is a saddle point system for $$(\widehat{\mathbf{u}}_{t}^{h},\; \widehat{\mathbf{u}}_{n}^{h},\; \bar{p}^{h}) \in{\boldsymbol{M}}^{t}_{h}(0)\times{\boldsymbol{M}}^{n}_{h}(0) \times \overline{Q}_{h}$$, where \begin{align} {\boldsymbol{M}}^{n}_{h}(0):= \left\{ \widehat{\mathbf{v}} \in \boldsymbol{M}_{h}(0) :\;\; \mathrm{t}\mathrm{r}_{t} (\widehat{\mathbf{v}})|_{F} = \boldsymbol{0}\quad \forall \, F\in\mathcal{E}_{h}\right\}\!, \end{align} (4.1a) \begin{align} \overline{Q}_{h} := \left\{ q\in L^{2}({\mathcal{T}_{h}}):\; q|_{K} \in \mathcal{P}_{0}(K)\quad \forall \, K\in{\mathcal{T}_{h}}\right\}\!. \end{align} (4.1b) We show that $$\widehat{\mathbf{u}}_{t}^{h}$$ here is the same as that in (2.5), $$\widehat{\mathbf{u}}_{n}^{h} = \mathrm{t}\mathrm{r}_{n} (\mathbf{u}^{h})$$ on $$\mathcal{E}_{h}$$, $$\bar{p}^{h}$$ is equal to the average of ph on each element of $${\mathcal{T}_{h}}$$. Here we first relax H(div)-conformity of the velocity field in (2.5) to obtain the following result. Theorem 4.1 There exists a unique element $$(\mathrm{L}^{h},\,\mathbf{u}^{h},\,p_{\perp }^{h},\, \bar{p}^{h},\, {\widehat{\mathbf{u}}_{t}}^{h},\,{\widehat{\mathbf{u}}_{n}}^{h},\, \lambda ^{h})\in{\mathcal{G}}_{h}\times \textbf{V}_{h}\times Q^{\perp }_{h} \times \overline{Q}_{h} \times{\boldsymbol{M}}^{t}_{h}(0)\times{\boldsymbol{M}}^{n}_{h}(0) \times M_{h}^{\partial }$$ such that the following weak formulation holds: \begin{align} ( \mathrm{L}^{h},\,\nu\,\mathrm{g}^{h} )_{{\mathcal{T}_{h}}}+\left( \mathbf{u}^{h},\,{\boldsymbol{\nabla}} \cdot (\nu\,\mathrm{g}^{h}) \right)_{{\mathcal{T}_{h}}} - \left\langle \widehat{\mathbf{u}}_{t}^{h} + \widehat{\mathbf{u}}_{n}^{h},\, \nu\,\mathrm{g}^{h}\, \mathbf{n} \right\rangle_{\partial{{\mathcal{T}_{h}}}} = 0, \end{align} (4.2a) \begin{align} \left( \nu\,\mathrm{L}^{h} - (p_{\perp}^{h}+\bar{p}^{h})I_{d},\,{\boldsymbol{\nabla}} \mathbf{v}^{h} \right)_{{\mathcal{T}_{h}}} +( \gamma\, \mathbf{u}^{h},\,\mathbf{v}^{h} )_{{\mathcal{T}_{h}}} \end{align} (4.2b) \begin{align} - \left\langle \nu\,\mathrm{L}^{h} \mathbf{n} - (p_{\perp}^{h}+\bar{p}^{h}) \mathbf{n} +\lambda^{h}\mathbf{n},\,\mathbf{v}^{h} \right\rangle_{\partial{{\mathcal{T}_{h}}}} &=(\mathbf{f},\mathbf{v}^{h} )_{\mathcal{T}_{h}},\nonumber \\ \left( {\nabla\cdot}\mathbf{u}^{h},\,q_{\perp}^{h} + \bar{q}^{h} \right)_{{\mathcal{T}_{h}}} & = \left(g, q_{\perp}^{h} + \bar{q}^{h}\right)_{\mathcal{T}_{h}}, \end{align} (4.2c) \begin{align} \left\langle \nu\,\mathrm{L}^{h} \mathbf{n} - (p_{\perp}^{h}+\bar{p}^{h}) \mathbf{n} +\lambda^{h}\mathbf{n},\,\widehat{\mathbf{v}}_{t}^{h}+\widehat{\mathbf{v}}_{n}^{h} \right\rangle_{\partial{{\mathcal{T}_{h}}}} = 0, \end{align} (4.2d) \begin{align} \left\langle (\mathbf{u}^{h} - \widehat{\mathbf{u}}_{n}^{h})\cdot \mathbf{n},\,\mu^{h} \right\rangle_{\partial{{\mathcal{T}_{h}}}}=0, \end{align} (4.2e) \begin{equation} (\bar{p}^{h}, 1)_{\mathcal{T}_{h}} =0, \end{equation} (4.2f) for all $$(\mathrm{g}^{h},\mathbf{v}^{h},\, q_{\perp }^{h},\,\bar{q}^{h},\,{\widehat{\mathbf{v}}_{t}}^{h},\,\widehat{\mathbf{v}}_{n}^{h},\,\mu ^{h}) \in{\mathcal{G}}_{h}\times \boldsymbol{V}_{h}\times Q^{\perp }_{h} \times \overline{Q}_{h} \times{\boldsymbol{M}}^{t}_{h}(0)\times{\boldsymbol{M}}^{n}_{h}(0) \times M_{h}^{\partial }$$, where \begin{align*} Q_{h}^{\perp} := & \left\{ q\in L^{2}({\mathcal{T}_{h}}):\; (q,1)_{K}=0\quad \forall \, K\in{\mathcal{T}_{h}}\right\}\!, \\ M_{h}^{\partial} := & \left\{ \mu \in L^{2}(\partial{\mathcal{T}_{h}}): \mu|_{\partial K}\in \mathcal{P}_{k}(\partial K)\quad \forall \, K\in{\mathcal{T}_{h}}\right\}\!, \\ \mathcal{P}_{k}(\partial K) := & \left\{\mu \in L^{2}(\partial K): \mu |_{F} \in \mathcal{P}_{k}(F)\quad \forall \, F\in \mathcal{F}(K)\right\}\!. \end{align*} Moreover, if $$(\mathrm{L}^{h},\,\mathbf{u}^{h},\, p_{\perp }^{h}, \bar{p}^{h},\, {\widehat{\mathbf{u}}_{t}}^{h},\,{\widehat{\mathbf{u}}_{n}}^{h},\, \lambda ^{h})\in{\mathcal{G}}_{h}\times \boldsymbol{V}_{h}\times Q^{\perp }_{h} \times \overline{Q}_{h} \times{\boldsymbol{M}}^{t}_{h}(0)\times{\boldsymbol{M}}^{n}_{h}(0) \times M_{h}^{\partial }$$ is the numerical solution to the above equations, then $$(\mathrm{L}^{h},\mathbf{u}^{h},p_{\perp }^{h}+\bar{p}^{h},\,{\widehat{\mathbf{u}}_{t}}^{h})\in{\mathcal{G}}_{h}\times{\boldsymbol V}_{h}^{\textrm{div}}(0)\times \mathring{{Q}_{h}} \times{\boldsymbol{M}}^{t}_{h}(0)$$ is the only solution to (2.5). Note that $$\lambda ^{h}\in M_{h}^{\partial }$$ is a quantity that approximates $$0|_{\partial{\mathcal{T}_{h}}}$$. Proof. Let $$(\mathrm{L}^{h},\,\mathbf{u}^{h},\,p_{\perp }^{h},\, \bar{p}^{h},\,{\widehat{\mathbf{u}}_{t}}^{h},\,{\widehat{\mathbf{u}}_{n}}^{h},\, \lambda ^{h}) \in{\mathcal{G}}_{h}\times \boldsymbol{V}_{h}\times Q^{\perp }_{h} \times \overline{Q}_{h} \times{\boldsymbol{M}}^{t}_{h}(0)\times{\boldsymbol{M}}^{n}_{h}(0) \times M_{h}^{\partial }$$ be a numerical solution to equations (4.2). We prove such a numerical solution is unique and $$(\mathrm{L}^{h},\,\mathbf{u}^{h},\,p_{\perp }^{h} + \bar{p}^{h},\,\widehat{\mathbf{u}}_{t}^{h})$$ is the unique solution to equations (2.5). Since $$ \left(\mathbf{u}^{h} - \widehat{\mathbf{u}}_{n}^{h}\right)\cdot \mathbf{n} |_{\partial K} \in \mathcal{P}_{k}(\partial K)= M_{h}^{\partial}(K) \quad \forall \, K \in{\mathcal{T}_{h}}, $$ we have $$ \mathrm{t}\mathrm{r}_{n}^{F} (\mathbf{u}^{h}) = {\widehat{\mathbf{u}}_{n}}^{h}$$ on any facet $$F\in \mathcal{E}_{h}$$ by equations (4.2e). Hence, $$\mathbf{u}^{h}\in{\boldsymbol V}_{h}^{\textrm{div}}(0)$$. By equation (4.2f), we have $$p_{\perp }^{h} + \bar{p}^{h} \in \mathring{{Q}_{h}}$$. Then, taking $$\mathbf{v}^{h}\in{\boldsymbol V}_{h}^{\textrm{div}}(0)$$ in (4.2b), $$\widehat{\mathbf{v}}_{n}^{h} |_{F} = \mathrm{t}\mathrm{r}_{n}^{F} (\mathbf{v}^{h})$$ on any facet $$F\in \mathcal{E}_{h}$$ in (4.2d) and qh ∈ $$\mathring{{Q}_{h}}$$ in (4.2c), we have $$ \left(\mathrm{L}^{h},\mathbf{u}^{h},p_{\perp}^{h} + \bar{p}^{h},{\widehat{\mathbf{u}}_{t}}^{h}\right)\in{\mathcal{G}}_{h}\times{\boldsymbol V}_{h}^{\textrm{div}}(0)\times \mathring{{Q}_{h}} \times{\mathbf{M}}^{t}_{h}(0)$$ is the unique solution to equations (2.5). Now, we only need to show the uniqueness of λh. If there are two λh, then by equation (4.2b), their difference which we still call λh, satisfies $$ \langle \lambda^{h},\,\mathbf{v}^{h}\cdot \mathbf{n} \rangle_{\partial{{\mathcal{T}_{h}}}} = 0\quad \forall \, \mathbf{v}^{h} \in \mathbf{V}_{h}. $$ Since $$M_{h}^{\partial }(K) = \mathrm{t}\mathrm{r}_{n}(\mathbf{V}_{h}(K))$$ for any $$K \in{\mathcal{T}_{h}}$$, we have $$\lambda ^{h} = 0|_{\partial{\mathcal{T}_{h}}}$$. So, λh is also unique. This completes the proof. Then we identify local and global solvers. Because of the lack of uniqueness of pressure in the Brinkman equations, we will keep $$\bar{p}_{h}\in \overline{Q}_{h}$$ as a separate unknown. Given $$(\widehat{\mathbf{u}}_{t}, \widehat{\mathbf{u}}_{n})\in{\boldsymbol{M}}^{t}_{h}(0)\times{\boldsymbol{M}}^{n}_{h}(0)$$, $$\mathbf{f}\in L^{2}({\mathcal{T}_{h}})^{d}$$ and $$g\in L^{2}({\mathcal{T}_{h}})$$, we consider the solution to the set of local problems in each element $$K\in{\mathcal{T}_{h}}$$: find $$ \left(\mathrm{L}^{h},\mathbf{u}^{h},p_{\perp}^{h}, \lambda^{h}\right)\in{\mathcal{G}}(K)\times{\boldsymbol V}(K)\times{Q}^{\perp}(K) \times M_{h}^{\partial}(K) $$ such that \begin{align} ( \mathrm{L}^{h},\,\nu\,\mathrm{g}^{h} )_{K}+\left( \mathbf{u}^{h},\,{\boldsymbol{\nabla}} \cdot (\nu\,\mathrm{g}^{h}) \right)_{K} = \left\langle \widehat{\mathbf{u}}_{t}+ \widehat{\mathbf{u}}_{n},\, \nu\,\mathrm{g}^{h}\, \mathbf{n} \right\rangle_{\partial{K}}, \end{align} (4.3a) \begin{align} -\left( {\boldsymbol{\nabla}} \cdot (\nu\,\mathrm{L}^{h})-\nabla p_{\perp}^{h} - \gamma\, \mathbf{u}^{h},\,\mathbf{v}^{h} \right)_{K} - \langle \lambda^{h}\,\mathbf{n},\,\mathbf{v}^{h} \rangle_{\partial{K}} =(\mathbf{f},\mathbf{v}^{h})_{\mathcal{T}_{h}}, \end{align} (4.3b) \begin{align} \left( {\nabla\cdot}\mathbf{u}^{h},\,q_{\perp}^{h} \right)_{K} = (g, q_{\perp}^{h})_{\mathcal{T}_{h}}, \end{align} (4.3c) \begin{align} \left\langle (\mathbf{u}^{h} - \widehat{\mathbf{u}}_{n})\cdot \mathbf{n},\,\mu^{h} \right\rangle_{\partial{K}} = 0, \end{align} (4.3d) for all $$(\mathrm{g}^{h},\mathbf{v}^{h},q_{\perp }^{h}, \mu ^{h})\in{\mathcal{G}}(K)\times{\boldsymbol V}(K) \times{Q}_{\perp }(K) \times M_{h}^{\partial }(K)$$. The unique solvability of this problem is a simple consequence of the unique solvability of equations (4.2). The solution to (4.3) can be written $$ \left(\mathrm{L}^{h},\mathbf{u}^{h},p_{\perp}^{h}, \lambda^{h}\right) = \left(\mathrm{L}^{h}_{(\widehat{\mathbf{u}}_{t},\widehat{\mathbf{u}}_{n})},\,\mathbf{u}^{h}_{(\widehat{\mathbf{u}}_{t},\widehat{\mathbf{u}}_{n})},\, p^{h}_{\perp,(\widehat{\mathbf{u}}_{t},\widehat{\mathbf{u}}_{n})},\, \lambda^{h}_{(\widehat{\mathbf{u}}_{t},\widehat{\mathbf{u}}_{n})}\right) +\left(\mathrm{L}^{h}_{(\mathbf{f},g)},\,\mathbf{u}^{h}_{(\mathbf{f},g)},\,p^{h}_{\perp,(\mathbf{f},g)},\,\lambda^{h}_{(\mathbf{f}, g)}\right) $$ by considering separately the influence of $$(\widehat{\mathbf{u}}_{t},\widehat{\mathbf{u}}_{t})$$ and (f, g) in the solution. For example, $$\big(\mathrm{L}^{h}_{(\widehat{\mathbf{u}}_{t},\widehat{\mathbf{u}}_{n})},\,\mathbf{u}^{h}_{(\widehat{\mathbf{u}}_{t},\widehat{\mathbf{u}}_{n})}, \,p^{h}_{\perp ,(\widehat{\mathbf{u}}_{t},\widehat{\mathbf{u}}_{n})},\, \lambda ^{h}_{(\widehat{\mathbf{u}}_{t},\widehat{\mathbf{u}}_{n})}\big )$$ is the solution of (4.3) when (f, g) = (0, 0). According to equations (4.2c) (4.2d) (4.2f), the global (hybrid) problem is to find $$(\widehat{\mathbf{u}}_{t}^{h}, \widehat{\mathbf{u}}_{n}^{h}, \bar{p}^{h}) \in{\boldsymbol{M}}^{t}_{h}(0)\times{\boldsymbol{M}}^{n}_{h}(0) \times \overline{Q}_{h}$$ such that \begin{align} \left\langle \nu\,\mathrm{L}^{h}_{\left(\widehat{\mathbf{u}}_{t}^{h},\widehat{\mathbf{u}}_{n}^{h}\right)} \mathbf{n} - \left(p_{\perp, (\widehat{\mathbf{u}}_{t}^{h},\widehat{\mathbf{u}}_{n}^{h})}^{h}+\bar{p}^{h}\right) \mathbf{n} +\lambda^{h}_{\left(\widehat{\mathbf{u}}_{t}^{h},\widehat{\mathbf{u}}_{n}^{h}\right)}\mathbf{n},\,\widehat{\mathbf{v}}_{t}^{h}+\widehat{\mathbf{v}}_{n}^{h} \right\rangle_{\partial{{\mathcal{T}_{h}}}} \end{align} (4.4a) \begin{align} \nonumber &\qquad = \left\langle \nu\,\mathrm{L}^{h}_{(\mathbf{f}, g)} \mathbf{n} - p_{\perp, (\mathbf{f}, g)}^{h} \mathbf{n} +\lambda^{h}_{(\mathbf{f}, g)}\mathbf{n},\,\widehat{\mathbf{v}}_{t}^{h}+\widehat{\mathbf{v}}_{n}^{h} \right\rangle_{\partial{{\mathcal{T}_{h}}}}\!, \\ & \left( {\nabla\cdot}\left(\mathbf{u}^{h}_{(\widehat{\mathbf{u}}_{t}^{h},\widehat{\mathbf{u}}_{n}^{h})} + \mathbf{u}^{h}_{(\mathbf{f}, g)}\right)\!,\,\bar{q}^{h} \right)_{{\mathcal{T}_{h}}} = (g, \bar{q}^{h})_{\mathcal{T}_{h}}, \end{align} (4.4b) \begin{align} (\bar{p}^{h}, 1)_{\mathcal{T}_{h}} =0, \end{align} (4.4c) for all $$(\widehat{\mathbf{v}}_{t}^{h},\, \widehat{\mathbf{v}}_{n}^{h},\, \bar{q}^{h}) \in{\mathbf{M}}^{t}_{h}(0)\times{\mathbf{M}}^{n}_{h}(0) \times \overline{Q}_{h}$$. Again, the unique solvability of this problem is a simple consequence of that for equations (4.2). Moreover, we have the following characterization of equations (4.4). Its proof is trivial; see, e.g., Cockburn & Sayas (2014). Proposition 4.2 Equations (4.4) can be rewritten \begin{align*} A_{h}\left(\widehat{\mathbf{u}}_{t}^{h},\,\widehat{\mathbf{u}}_{n}^{h};\,\widehat{\mathbf{v}}_{t}^{h},\,\widehat{\mathbf{v}}_{n}^{h}\right) + B_{h}\left(\widehat{\mathbf{v}}_{n}^{h};\, \bar{p}^{h}\right) &= F_{h} \left(\widehat{\mathbf{v}}_{t}^{h},\,\widehat{\mathbf{v}}_{n}^{h}\right)\!,\\ B_{h}\left(\widehat{\mathbf{u}}_{n}^{h}; \bar{q}^{h}\right) &= 0,\\ (\bar{p}^{h}, 1)_{\mathcal{T}_{h}} &= 0, \end{align*} where \begin{align} A_{h}\left(\widehat{\mathbf{u}}_{t}^{h},\,\widehat{\mathbf{u}}_{n}^{h};\,\widehat{\mathbf{v}}_{t}^{h},\,\widehat{\mathbf{v}}_{n}^{h}\right) := \left( \nu \mathrm{L}^{h}_{\left(\widehat{\mathbf{u}}_{t}^{h},\widehat{\mathbf{u}}_{n}^{h}\right)},\,\mathrm{L}^{h}_{\left(\widehat{\mathbf{v}}_{t},\widehat{\mathbf{v}}_{n}\right)} \right)_{{\mathcal{T}_{h}}} + \left( \gamma \mathbf{u}^{h}_{\left(\widehat{\mathbf{u}}_{t}^{h},\widehat{\mathbf{u}}_{n}^{h}\right)} ,\,\mathbf{u}^{h}_{\left(\widehat{\mathbf{v}}_{t}^{h},\widehat{\mathbf{v}}_{n}^{h}\right)} \right)_{{\mathcal{T}_{h}}}\!, \end{align} (4.5a) \begin{align} B_{h}\left(\widehat{\mathbf{v}}_{n}^{h};\, \bar{p}^{h}\right) := - \left\langle \bar{p}^{h},\,\widehat{\mathbf{v}}_{n}^{h}\cdot \mathbf{n} \right\rangle_{\partial{{\mathcal{T}_{h}}}}\!, \end{align} (4.5b) \begin{align}\!\!\!\!\!\!\!\!\!\!\!\!\! F_{h} \left(\widehat{\mathbf{v}}_{t}^{h},\,\widehat{\mathbf{v}}_{n}^{h}\right) := & \left(\mathbf{f}, \mathbf{u}^{h}_{\left(\widehat{\mathbf{v}}_{t}^{h},\widehat{\mathbf{v}}_{n}^{h}\right)}\right)_{{\mathcal{T}_{h}}} - \left( \nu \mathrm{L}^{h}_{(\mathbf{f},g)},\,\mathrm{L}^{h}_{\left(\widehat{\mathbf{v}}_{t},\widehat{\mathbf{v}}_{n}\right)} \right)_{{\mathcal{T}_{h}}} \\ \nonumber & - \left( \gamma \mathbf{u}^{h}_{(\mathbf{f},g)} ,\,\mathbf{u}^{h}_{\left(\widehat{\mathbf{v}}_{t}^{h},\widehat{\mathbf{v}}_{n}^{h}\right)} \right)_{{\mathcal{T}_{h}}}\!. \end{align} (4.5c) 5. Numerical results In this section, we present two-dimensional numerical studies on both rectangular and triangular meshes to validate the theoretical results in Section 2. We use the deal.II (Bangerth et al., 2016) software to implement the HDG method (2.5) on rectangular meshes, and NGSolve (Schöberl, 1997; 2016) on triangular meshes. Recall that our approximation spaces are given in Table 1. The implementation on rectangular meshes uses the hybridization discussed in Section 4, while the implementation on triangular meshes uses NGSolve’s built-in static condensation approach; see Schöberl (2016). We present four numerical tests with a manufactured solution to validate our theoretical results in Section 2. For all the tests, the body forces f and g are chosen such that the exact solution (u, p) takes the form \begin{align*} \mathbf{u} =&\; \left(\sin(2\,\pi x)\sin(2\,\pi y), \sin(2\,\pi x)\sin(2\,\pi y)\right)^{\textrm{T}}\!,\\ p =&\; \sin(m\,\pi x)\sin(m\,\pi y),\ \textrm{where}\ m\ \textrm{is a fixed number.} \end{align*} We take ν = 1, γ = 1 and m = 2 for the first test, ν = 1, γ = 1 and m = 20 for the second test, ν = 0.0001, γ = 1 and m = 2 for the third test and ν = 1 e−8, γ = 1 and m = 2 for the last test. The first two tests are in the Stokes-dominated regime, while the last two tests are in the Darcy-dominated regime. The second test examines the effect of pressure regularity on the convergence of the velocity field. Table 2 History of convergence for the H(div)-conforming HDG method on square meshes Mesh D.O.F. $$\|\mathrm{L} - \mathrm{L}^{h} \|_{{\mathcal{T}_{h}}}$$ $$\|\mathbf{u}-\mathbf{u}^{h}\|_{{\mathcal{T}_{h}}}$$ $$\|p - p^{h} \|_{{\mathcal{T}_{h}}}$$ $$\|\mathbf{u}-\mathbf{u}^{*,h}\|_{{\mathcal{T}_{h}}}$$ k Nele Nglobal Nlocal Error Order Error Order Error Order Error Order First test: ν = 1, γ = 1, m = 2. 0 64 288 704 2.393e+00 — 1.622e-01 — 4.133e-01 — 5.398e-02 — 256 1088 2816 1.224e+00 0.97 8.043e-02 1.01 1.300e-01 1.67 1.337e-02 2.01 1024 4224 11264 6.157e-01 0.99 4.011e-02 1.00 4.782e-02 1.44 3.335e-03 2.00 4096 16640 45056 3.083e-01 1.00 2.004e-02 1.00 2.108e-02 1.18 8.331e-04 2.00 1 64 576 1856 4.951e-01 — 1.829e-02 — 1.178e-01 — 6.955e-03 — 256 2176 7424 1.286e-01 1.94 4.211e-03 2.12 1.559e-02 2.92 7.790e-04 3.16 1024 8448 29696 3.245e-02 1.99 1.026e-03 2.04 2.518e-03 2.63 9.367e-05 3.06 4096 33280 118784 8.131e-03 2.00 2.546e-04 2.01 5.171e-04 2.28 1.159e-05 3.02 2 64 864 3328 5.810e-02 — 1.399e-03 — 1.281e-02 — 7.069e-04 — 256 3264 13312 7.352e-03 2.98 1.481e-04 3.24 9.173e-04 3.80 4.129e-05 4.10 1024 12672 53248 9.223e-04 2.99 1.731e-05 3.10 7.743e-05 3.57 2.533e-06 4.03 4096 49920 212992 1.154e-04 3.00 2.122e-06 3.03 8.097e-06 3.26 1.575e-07 4.01 3 64 1152 5248 5.598e-03 — 9.147e-05 — 1.740e-03 — 6.264e-05 — 256 4352 20992 3.600e-04 3.96 4.127e-06 4.47 9.163e-05 4.25 2.049e-06 4.93 1024 16896 83968 2.272e-05 3.99 2.222e-07 4.21 5.203e-06 4.14 6.492e-08 4.98 4096 66560 335872 1.424e-06 4.00 1.325e-08 4.07 3.112e-07 4.06 2.036e-09 5.00 Second test: ν = 1, γ = 1, m = 20. 0 64 288 704 2.393e+00 — 1.622e-01 — 6.293e-01 — 5.398e-02 — 256 1088 2816 1.224e+00 0.97 8.043e-02 1.01 4.983e-01 0.34 1.337e-02 2.01 1024 4224 11264 6.157e-01 0.99 4.011e-02 1.00 3.494e-01 0.51 3.335e-03 2.00 4096 16640 45056 3.083e-01 1.00 2.004e-02 1.00 1.934e-01 0.85 8.331e-04 2.00 1 64 576 1856 4.951e-01 — 1.829e-02 — 5.117e-01 — 6.955e-03 — 256 2176 7424 1.286e-01 1.94 4.211e-03 2.12 4.186e-01 0.29 7.790e-04 3.16 1024 8448 29696 3.245e-02 1.99 1.026e-03 2.04 1.631e-01 1.36 9.367e-05 3.06 4096 33280 118784 8.131e-03 2.00 2.546e-04 2.01 4.573e-02 1.83 1.159e-05 3.02 2 64 864 3328 5.810e-02 — 1.399e-03 — 4.917e-01 — 7.069e-04 — 256 3264 13312 7.352e-03 2.98 1.481e-04 3.24 2.722e-01 0.85 4.129e-05 4.10 1024 12672 53248 9.223e-04 2.99 1.731e-05 3.10 5.209e-02 2.39 2.533e-06 4.03 4096 49920 212992 1.154e-04 3.00 2.122e-06 3.03 7.240e-03 2.85 1.575e-07 4.01 3 64 1152 5248 5.598e-03 — 9.147e-05 — 4.744e-01 — 6.264e-05 — 256 4352 20992 3.600e-04 3.96 4.127e-06 4.47 1.362e-01 1.80 2.049e-06 4.93 1024 16896 83968 2.272e-05 3.99 2.222e-07 4.21 1.252e-02 3.44 6.492e-08 4.98 4096 66560 335872 1.424e-06 4.00 1.325e-08 4.07 8.610e-04 3.86 2.036e-09 5.00 Mesh D.O.F. $$\|\mathrm{L} - \mathrm{L}^{h} \|_{{\mathcal{T}_{h}}}$$ $$\|\mathbf{u}-\mathbf{u}^{h}\|_{{\mathcal{T}_{h}}}$$ $$\|p - p^{h} \|_{{\mathcal{T}_{h}}}$$ $$\|\mathbf{u}-\mathbf{u}^{*,h}\|_{{\mathcal{T}_{h}}}$$ k Nele Nglobal Nlocal Error Order Error Order Error Order Error Order First test: ν = 1, γ = 1, m = 2. 0 64 288 704 2.393e+00 — 1.622e-01 — 4.133e-01 — 5.398e-02 — 256 1088 2816 1.224e+00 0.97 8.043e-02 1.01 1.300e-01 1.67 1.337e-02 2.01 1024 4224 11264 6.157e-01 0.99 4.011e-02 1.00 4.782e-02 1.44 3.335e-03 2.00 4096 16640 45056 3.083e-01 1.00 2.004e-02 1.00 2.108e-02 1.18 8.331e-04 2.00 1 64 576 1856 4.951e-01 — 1.829e-02 — 1.178e-01 — 6.955e-03 — 256 2176 7424 1.286e-01 1.94 4.211e-03 2.12 1.559e-02 2.92 7.790e-04 3.16 1024 8448 29696 3.245e-02 1.99 1.026e-03 2.04 2.518e-03 2.63 9.367e-05 3.06 4096 33280 118784 8.131e-03 2.00 2.546e-04 2.01 5.171e-04 2.28 1.159e-05 3.02 2 64 864 3328 5.810e-02 — 1.399e-03 — 1.281e-02 — 7.069e-04 — 256 3264 13312 7.352e-03 2.98 1.481e-04 3.24 9.173e-04 3.80 4.129e-05 4.10 1024 12672 53248 9.223e-04 2.99 1.731e-05 3.10 7.743e-05 3.57 2.533e-06 4.03 4096 49920 212992 1.154e-04 3.00 2.122e-06 3.03 8.097e-06 3.26 1.575e-07 4.01 3 64 1152 5248 5.598e-03 — 9.147e-05 — 1.740e-03 — 6.264e-05 — 256 4352 20992 3.600e-04 3.96 4.127e-06 4.47 9.163e-05 4.25 2.049e-06 4.93 1024 16896 83968 2.272e-05 3.99 2.222e-07 4.21 5.203e-06 4.14 6.492e-08 4.98 4096 66560 335872 1.424e-06 4.00 1.325e-08 4.07 3.112e-07 4.06 2.036e-09 5.00 Second test: ν = 1, γ = 1, m = 20. 0 64 288 704 2.393e+00 — 1.622e-01 — 6.293e-01 — 5.398e-02 — 256 1088 2816 1.224e+00 0.97 8.043e-02 1.01 4.983e-01 0.34 1.337e-02 2.01 1024 4224 11264 6.157e-01 0.99 4.011e-02 1.00 3.494e-01 0.51 3.335e-03 2.00 4096 16640 45056 3.083e-01 1.00 2.004e-02 1.00 1.934e-01 0.85 8.331e-04 2.00 1 64 576 1856 4.951e-01 — 1.829e-02 — 5.117e-01 — 6.955e-03 — 256 2176 7424 1.286e-01 1.94 4.211e-03 2.12 4.186e-01 0.29 7.790e-04 3.16 1024 8448 29696 3.245e-02 1.99 1.026e-03 2.04 1.631e-01 1.36 9.367e-05 3.06 4096 33280 118784 8.131e-03 2.00 2.546e-04 2.01 4.573e-02 1.83 1.159e-05 3.02 2 64 864 3328 5.810e-02 — 1.399e-03 — 4.917e-01 — 7.069e-04 — 256 3264 13312 7.352e-03 2.98 1.481e-04 3.24 2.722e-01 0.85 4.129e-05 4.10 1024 12672 53248 9.223e-04 2.99 1.731e-05 3.10 5.209e-02 2.39 2.533e-06 4.03 4096 49920 212992 1.154e-04 3.00 2.122e-06 3.03 7.240e-03 2.85 1.575e-07 4.01 3 64 1152 5248 5.598e-03 — 9.147e-05 — 4.744e-01 — 6.264e-05 — 256 4352 20992 3.600e-04 3.96 4.127e-06 4.47 1.362e-01 1.80 2.049e-06 4.93 1024 16896 83968 2.272e-05 3.99 2.222e-07 4.21 1.252e-02 3.44 6.492e-08 4.98 4096 66560 335872 1.424e-06 4.00 1.325e-08 4.07 8.610e-04 3.86 2.036e-09 5.00 Table 2 History of convergence for the H(div)-conforming HDG method on square meshes Mesh D.O.F. $$\|\mathrm{L} - \mathrm{L}^{h} \|_{{\mathcal{T}_{h}}}$$ $$\|\mathbf{u}-\mathbf{u}^{h}\|_{{\mathcal{T}_{h}}}$$ $$\|p - p^{h} \|_{{\mathcal{T}_{h}}}$$ $$\|\mathbf{u}-\mathbf{u}^{*,h}\|_{{\mathcal{T}_{h}}}$$ k Nele Nglobal Nlocal Error Order Error Order Error Order Error Order First test: ν = 1, γ = 1, m = 2. 0 64 288 704 2.393e+00 — 1.622e-01 — 4.133e-01 — 5.398e-02 — 256 1088 2816 1.224e+00 0.97 8.043e-02 1.01 1.300e-01 1.67 1.337e-02 2.01 1024 4224 11264 6.157e-01 0.99 4.011e-02 1.00 4.782e-02 1.44 3.335e-03 2.00 4096 16640 45056 3.083e-01 1.00 2.004e-02 1.00 2.108e-02 1.18 8.331e-04 2.00 1 64 576 1856 4.951e-01 — 1.829e-02 — 1.178e-01 — 6.955e-03 — 256 2176 7424 1.286e-01 1.94 4.211e-03 2.12 1.559e-02 2.92 7.790e-04 3.16 1024 8448 29696 3.245e-02 1.99 1.026e-03 2.04 2.518e-03 2.63 9.367e-05 3.06 4096 33280 118784 8.131e-03 2.00 2.546e-04 2.01 5.171e-04 2.28 1.159e-05 3.02 2 64 864 3328 5.810e-02 — 1.399e-03 — 1.281e-02 — 7.069e-04 — 256 3264 13312 7.352e-03 2.98 1.481e-04 3.24 9.173e-04 3.80 4.129e-05 4.10 1024 12672 53248 9.223e-04 2.99 1.731e-05 3.10 7.743e-05 3.57 2.533e-06 4.03 4096 49920 212992 1.154e-04 3.00 2.122e-06 3.03 8.097e-06 3.26 1.575e-07 4.01 3 64 1152 5248 5.598e-03 — 9.147e-05 — 1.740e-03 — 6.264e-05 — 256 4352 20992 3.600e-04 3.96 4.127e-06 4.47 9.163e-05 4.25 2.049e-06 4.93 1024 16896 83968 2.272e-05 3.99 2.222e-07 4.21 5.203e-06 4.14 6.492e-08 4.98 4096 66560 335872 1.424e-06 4.00 1.325e-08 4.07 3.112e-07 4.06 2.036e-09 5.00 Second test: ν = 1, γ = 1, m = 20. 0 64 288 704 2.393e+00 — 1.622e-01 — 6.293e-01 — 5.398e-02 — 256 1088 2816 1.224e+00 0.97 8.043e-02 1.01 4.983e-01 0.34 1.337e-02 2.01 1024 4224 11264 6.157e-01 0.99 4.011e-02 1.00 3.494e-01 0.51 3.335e-03 2.00 4096 16640 45056 3.083e-01 1.00 2.004e-02 1.00 1.934e-01 0.85 8.331e-04 2.00 1 64 576 1856 4.951e-01 — 1.829e-02 — 5.117e-01 — 6.955e-03 — 256 2176 7424 1.286e-01 1.94 4.211e-03 2.12 4.186e-01 0.29 7.790e-04 3.16 1024 8448 29696 3.245e-02 1.99 1.026e-03 2.04 1.631e-01 1.36 9.367e-05 3.06 4096 33280 118784 8.131e-03 2.00 2.546e-04 2.01 4.573e-02 1.83 1.159e-05 3.02 2 64 864 3328 5.810e-02 — 1.399e-03 — 4.917e-01 — 7.069e-04 — 256 3264 13312 7.352e-03 2.98 1.481e-04 3.24 2.722e-01 0.85 4.129e-05 4.10 1024 12672 53248 9.223e-04 2.99 1.731e-05 3.10 5.209e-02 2.39 2.533e-06 4.03 4096 49920 212992 1.154e-04 3.00 2.122e-06 3.03 7.240e-03 2.85 1.575e-07 4.01 3 64 1152 5248 5.598e-03 — 9.147e-05 — 4.744e-01 — 6.264e-05 — 256 4352 20992 3.600e-04 3.96 4.127e-06 4.47 1.362e-01 1.80 2.049e-06 4.93 1024 16896 83968 2.272e-05 3.99 2.222e-07 4.21 1.252e-02 3.44 6.492e-08 4.98 4096 66560 335872 1.424e-06 4.00 1.325e-08 4.07 8.610e-04 3.86 2.036e-09 5.00 Mesh D.O.F. $$\|\mathrm{L} - \mathrm{L}^{h} \|_{{\mathcal{T}_{h}}}$$ $$\|\mathbf{u}-\mathbf{u}^{h}\|_{{\mathcal{T}_{h}}}$$ $$\|p - p^{h} \|_{{\mathcal{T}_{h}}}$$ $$\|\mathbf{u}-\mathbf{u}^{*,h}\|_{{\mathcal{T}_{h}}}$$ k Nele Nglobal Nlocal Error Order Error Order Error Order Error Order First test: ν = 1, γ = 1, m = 2. 0 64 288 704 2.393e+00 — 1.622e-01 — 4.133e-01 — 5.398e-02 — 256 1088 2816 1.224e+00 0.97 8.043e-02 1.01 1.300e-01 1.67 1.337e-02 2.01 1024 4224 11264 6.157e-01 0.99 4.011e-02 1.00 4.782e-02 1.44 3.335e-03 2.00 4096 16640 45056 3.083e-01 1.00 2.004e-02 1.00 2.108e-02 1.18 8.331e-04 2.00 1 64 576 1856 4.951e-01 — 1.829e-02 — 1.178e-01 — 6.955e-03 — 256 2176 7424 1.286e-01 1.94 4.211e-03 2.12 1.559e-02 2.92 7.790e-04 3.16 1024 8448 29696 3.245e-02 1.99 1.026e-03 2.04 2.518e-03 2.63 9.367e-05 3.06 4096 33280 118784 8.131e-03 2.00 2.546e-04 2.01 5.171e-04 2.28 1.159e-05 3.02 2 64 864 3328 5.810e-02 — 1.399e-03 — 1.281e-02 — 7.069e-04 — 256 3264 13312 7.352e-03 2.98 1.481e-04 3.24 9.173e-04 3.80 4.129e-05 4.10 1024 12672 53248 9.223e-04 2.99 1.731e-05 3.10 7.743e-05 3.57 2.533e-06 4.03 4096 49920 212992 1.154e-04 3.00 2.122e-06 3.03 8.097e-06 3.26 1.575e-07 4.01 3 64 1152 5248 5.598e-03 — 9.147e-05 — 1.740e-03 — 6.264e-05 — 256 4352 20992 3.600e-04 3.96 4.127e-06 4.47 9.163e-05 4.25 2.049e-06 4.93 1024 16896 83968 2.272e-05 3.99 2.222e-07 4.21 5.203e-06 4.14 6.492e-08 4.98 4096 66560 335872 1.424e-06 4.00 1.325e-08 4.07 3.112e-07 4.06 2.036e-09 5.00 Second test: ν = 1, γ = 1, m = 20. 0 64 288 704 2.393e+00 — 1.622e-01 — 6.293e-01 — 5.398e-02 — 256 1088 2816 1.224e+00 0.97 8.043e-02 1.01 4.983e-01 0.34 1.337e-02 2.01 1024 4224 11264 6.157e-01 0.99 4.011e-02 1.00 3.494e-01 0.51 3.335e-03 2.00 4096 16640 45056 3.083e-01 1.00 2.004e-02 1.00 1.934e-01 0.85 8.331e-04 2.00 1 64 576 1856 4.951e-01 — 1.829e-02 — 5.117e-01 — 6.955e-03 — 256 2176 7424 1.286e-01 1.94 4.211e-03 2.12 4.186e-01 0.29 7.790e-04 3.16 1024 8448 29696 3.245e-02 1.99 1.026e-03 2.04 1.631e-01 1.36 9.367e-05 3.06 4096 33280 118784 8.131e-03 2.00 2.546e-04 2.01 4.573e-02 1.83 1.159e-05 3.02 2 64 864 3328 5.810e-02 — 1.399e-03 — 4.917e-01 — 7.069e-04 — 256 3264 13312 7.352e-03 2.98 1.481e-04 3.24 2.722e-01 0.85 4.129e-05 4.10 1024 12672 53248 9.223e-04 2.99 1.731e-05 3.10 5.209e-02 2.39 2.533e-06 4.03 4096 49920 212992 1.154e-04 3.00 2.122e-06 3.03 7.240e-03 2.85 1.575e-07 4.01 3 64 1152 5248 5.598e-03 — 9.147e-05 — 4.744e-01 — 6.264e-05 — 256 4352 20992 3.600e-04 3.96 4.127e-06 4.47 1.362e-01 1.80 2.049e-06 4.93 1024 16896 83968 2.272e-05 3.99 2.222e-07 4.21 1.252e-02 3.44 6.492e-08 4.98 4096 66560 335872 1.424e-06 4.00 1.325e-08 4.07 8.610e-04 3.86 2.036e-09 5.00 Table 2 Continued Mesh D.O.F. $$\|\mathrm{L} - \mathrm{L}^{h} \|_{{\mathcal{T}_{h}}}$$ $$\|\mathbf{u}-\mathbf{u}^{h}\|_{{\mathcal{T}_{h}}}$$ $$\|p - p^{h} \|_{{\mathcal{T}_{h}}}$$ $$\|\mathbf{u}-\mathbf{u}^{*,h}\|_{{\mathcal{T}_{h}}}$$ k Nele Nglobal Nlocal Error Order Error Order Error Order Error Order Third test: ν = 0.0001, γ = 1, m = 2. 0 64 288 704 2.399e+00 — 1.621e-01 — 1.567e-01 — 5.329e-02 — 256 1088 2816 1.226e+00 0.97 8.039e-02 1.01 7.970e-02 0.98 1.313e-02 2.02 1024 4224 11264 6.160e-01 0.99 4.011e-02 1.00 4.002e-02 0.99 3.268e-03 2.01 4096 16640 45056 3.083e-01 1.00 2.004e-02 1.00 2.003e-02 1.00 8.164e-04 2.00 1 64 576 1856 3.779e-01 — 1.679e-02 — 2.967e-02 — 6.192e-03 — 256 2176 7424 9.967e-02 1.92 4.096e-03 2.04 7.556e-03 1.97 7.509e-04 3.04 1024 8448 29696 2.761e-02 1.85 1.020e-03 2.01 1.898e-03 1.99 9.297e-05 3.01 4096 33280 118784 7.630e-03 1.86 2.544e-04 2.00 4.750e-04 2.00 1.157e-05 3.01 2 64 864 3328 4.844e-02 — 1.223e-03 — 3.755e-03 — 6.990e-04 — 256 3264 13312 6.177e-03 2.97 1.399e-04 3.13 4.773e-04 2.98 4.215e-05 4.05 1024 12672 53248 8.198e-04 2.91 1.708e-05 3.03 5.992e-05 2.99 2.571e-06 4.04 4096 49920 212992 1.099e-04 2.90 2.118e-06 3.01 7.498e-06 3.00 1.584e-07 4.02 3 64 1152 5248 4.973e-03 — 7.545e-05 — 3.567e-04 — 6.160e-05 — 256 4352 20992 3.248e-04 3.94 3.766e-06 4.32 2.264e-05 3.98 2.038e-06 4.92 1024 16896 83968 2.136e-05 3.93 2.173e-07 4.12 1.420e-06 3.99 6.486e-08 4.97 4096 66560 335872 1.390e-06 3.94 1.322e-08 4.04 8.885e-08 4.00 2.035e-09 4.99 Fourth test: ν = 1e−8, γ = 1, m = 2. 0 64 288 704 2.400e+00 — 1.621e-01 — 1.567e-01 — 5.329e-02 — 256 1088 2816 1.226e+00 0.97 8.039e-02 1.01 7.970e-02 0.98 1.313e-02 2.02 1024 4224 11264 6.162e-01 0.99 4.011e-02 1.00 4.002e-02 0.99 3.268e-03 2.01 4096 16640 45056 3.084e-01 1.00 2.004e-02 1.00 2.003e-02 1.00 8.156e-04 2.00 1 64 576 1856 3.717e-01 — 1.678e-02 — 2.967e-02 — 6.166e-03 — 256 2176 7424 9.358e-02 1.99 4.091e-03 2.04 7.556e-03 1.97 7.472e-04 3.04 1024 8448 29696 2.344e-02 2.00 1.018e-03 2.01 1.898e-03 1.99 9.261e-05 3.01 4096 33280 118784 5.863e-03 2.00 2.541e-04 2.00 4.750e-04 2.00 1.155e-05 3.00 2 64 864 3328 4.775e-02 — 1.222e-03 — 3.755e-03 — 6.963e-04 — 256 3264 13312 5.940e-03 3.01 1.396e-04 3.13 4.773e-04 2.98 4.237e-05 4.04 1024 12672 53248 7.406e-04 3.00 1.701e-05 3.04 5.992e-05 2.99 2.629e-06 4.01 4096 49920 212992 9.251e-05 3.00 2.112e-06 3.01 7.498e-06 3.00 1.640e-07 4.00 3 64 1152 5248 4.919e-03 — 7.526e-05 — 3.567e-04 — 6.159e-05 — 256 4352 20992 3.114e-04 3.98 3.737e-06 4.33 2.264e-05 3.98 2.032e-06 4.92 1024 16896 83968 1.955e-05 3.99 2.151e-07 4.12 1.420e-06 3.99 6.451e-08 4.98 4096 66560 335872 1.223e-06 4.00 1.314e-08 4.03 8.885e-08 4.00 2.024e-09 4.99 Mesh D.O.F. $$\|\mathrm{L} - \mathrm{L}^{h} \|_{{\mathcal{T}_{h}}}$$ $$\|\mathbf{u}-\mathbf{u}^{h}\|_{{\mathcal{T}_{h}}}$$ $$\|p - p^{h} \|_{{\mathcal{T}_{h}}}$$ $$\|\mathbf{u}-\mathbf{u}^{*,h}\|_{{\mathcal{T}_{h}}}$$ k Nele Nglobal Nlocal Error Order Error Order Error Order Error Order Third test: ν = 0.0001, γ = 1, m = 2. 0 64 288 704 2.399e+00 — 1.621e-01 — 1.567e-01 — 5.329e-02 — 256 1088 2816 1.226e+00 0.97 8.039e-02 1.01 7.970e-02 0.98 1.313e-02 2.02 1024 4224 11264 6.160e-01 0.99 4.011e-02 1.00 4.002e-02 0.99 3.268e-03 2.01 4096 16640 45056 3.083e-01 1.00 2.004e-02 1.00 2.003e-02 1.00 8.164e-04 2.00 1 64 576 1856 3.779e-01 — 1.679e-02 — 2.967e-02 — 6.192e-03 — 256 2176 7424 9.967e-02 1.92 4.096e-03 2.04 7.556e-03 1.97 7.509e-04 3.04 1024 8448 29696 2.761e-02 1.85 1.020e-03 2.01 1.898e-03 1.99 9.297e-05 3.01 4096 33280 118784 7.630e-03 1.86 2.544e-04 2.00 4.750e-04 2.00 1.157e-05 3.01 2 64 864 3328 4.844e-02 — 1.223e-03 — 3.755e-03 — 6.990e-04 — 256 3264 13312 6.177e-03 2.97 1.399e-04 3.13 4.773e-04 2.98 4.215e-05 4.05 1024 12672 53248 8.198e-04 2.91 1.708e-05 3.03 5.992e-05 2.99 2.571e-06 4.04 4096 49920 212992 1.099e-04 2.90 2.118e-06 3.01 7.498e-06 3.00 1.584e-07 4.02 3 64 1152 5248 4.973e-03 — 7.545e-05 — 3.567e-04 — 6.160e-05 — 256 4352 20992 3.248e-04 3.94 3.766e-06 4.32 2.264e-05 3.98 2.038e-06 4.92 1024 16896 83968 2.136e-05 3.93 2.173e-07 4.12 1.420e-06 3.99 6.486e-08 4.97 4096 66560 335872 1.390e-06 3.94 1.322e-08 4.04 8.885e-08 4.00 2.035e-09 4.99 Fourth test: ν = 1e−8, γ = 1, m = 2. 0 64 288 704 2.400e+00 — 1.621e-01 — 1.567e-01 — 5.329e-02 — 256 1088 2816 1.226e+00 0.97 8.039e-02 1.01 7.970e-02 0.98 1.313e-02 2.02 1024 4224 11264 6.162e-01 0.99 4.011e-02 1.00 4.002e-02 0.99 3.268e-03 2.01 4096 16640 45056 3.084e-01 1.00 2.004e-02 1.00 2.003e-02 1.00 8.156e-04 2.00 1 64 576 1856 3.717e-01 — 1.678e-02 — 2.967e-02 — 6.166e-03 — 256 2176 7424 9.358e-02 1.99 4.091e-03 2.04 7.556e-03 1.97 7.472e-04 3.04 1024 8448 29696 2.344e-02 2.00 1.018e-03 2.01 1.898e-03 1.99 9.261e-05 3.01 4096 33280 118784 5.863e-03 2.00 2.541e-04 2.00 4.750e-04 2.00 1.155e-05 3.00 2 64 864 3328 4.775e-02 — 1.222e-03 — 3.755e-03 — 6.963e-04 — 256 3264 13312 5.940e-03 3.01 1.396e-04 3.13 4.773e-04 2.98 4.237e-05 4.04 1024 12672 53248 7.406e-04 3.00 1.701e-05 3.04 5.992e-05 2.99 2.629e-06 4.01 4096 49920 212992 9.251e-05 3.00 2.112e-06 3.01 7.498e-06 3.00 1.640e-07 4.00 3 64 1152 5248 4.919e-03 — 7.526e-05 — 3.567e-04 — 6.159e-05 — 256 4352 20992 3.114e-04 3.98 3.737e-06 4.33 2.264e-05 3.98 2.032e-06 4.92 1024 16896 83968 1.955e-05 3.99 2.151e-07 4.12 1.420e-06 3.99 6.451e-08 4.98 4096 66560 335872 1.223e-06 4.00 1.314e-08 4.03 8.885e-08 4.00 2.024e-09 4.99 Table 2 Continued Mesh D.O.F. $$\|\mathrm{L} - \mathrm{L}^{h} \|_{{\mathcal{T}_{h}}}$$ $$\|\mathbf{u}-\mathbf{u}^{h}\|_{{\mathcal{T}_{h}}}$$ $$\|p - p^{h} \|_{{\mathcal{T}_{h}}}$$ $$\|\mathbf{u}-\mathbf{u}^{*,h}\|_{{\mathcal{T}_{h}}}$$ k Nele Nglobal Nlocal Error Order Error Order Error Order Error Order Third test: ν = 0.0001, γ = 1, m = 2. 0 64 288 704 2.399e+00 — 1.621e-01 — 1.567e-01 — 5.329e-02 — 256 1088 2816 1.226e+00 0.97 8.039e-02 1.01 7.970e-02 0.98 1.313e-02 2.02 1024 4224 11264 6.160e-01 0.99 4.011e-02 1.00 4.002e-02 0.99 3.268e-03 2.01 4096 16640 45056 3.083e-01 1.00 2.004e-02 1.00 2.003e-02 1.00 8.164e-04 2.00 1 64 576 1856 3.779e-01 — 1.679e-02 — 2.967e-02 — 6.192e-03 — 256 2176 7424 9.967e-02 1.92 4.096e-03 2.04 7.556e-03 1.97 7.509e-04 3.04 1024 8448 29696 2.761e-02 1.85 1.020e-03 2.01 1.898e-03 1.99 9.297e-05 3.01 4096 33280 118784 7.630e-03 1.86 2.544e-04 2.00 4.750e-04 2.00 1.157e-05 3.01 2 64 864 3328 4.844e-02 — 1.223e-03 — 3.755e-03 — 6.990e-04 — 256 3264 13312 6.177e-03 2.97 1.399e-04 3.13 4.773e-04 2.98 4.215e-05 4.05 1024 12672 53248 8.198e-04 2.91 1.708e-05 3.03 5.992e-05 2.99 2.571e-06 4.04 4096 49920 212992 1.099e-04 2.90 2.118e-06 3.01 7.498e-06 3.00 1.584e-07 4.02 3 64 1152 5248 4.973e-03 — 7.545e-05 — 3.567e-04 — 6.160e-05 — 256 4352 20992 3.248e-04 3.94 3.766e-06 4.32 2.264e-05 3.98 2.038e-06 4.92 1024 16896 83968 2.136e-05 3.93 2.173e-07 4.12 1.420e-06 3.99 6.486e-08 4.97 4096 66560 335872 1.390e-06 3.94 1.322e-08 4.04 8.885e-08 4.00 2.035e-09 4.99 Fourth test: ν = 1e−8, γ = 1, m = 2. 0 64 288 704 2.400e+00 — 1.621e-01 — 1.567e-01 — 5.329e-02 — 256 1088 2816 1.226e+00 0.97 8.039e-02 1.01 7.970e-02 0.98 1.313e-02 2.02 1024 4224 11264 6.162e-01 0.99 4.011e-02 1.00 4.002e-02 0.99 3.268e-03 2.01 4096 16640 45056 3.084e-01 1.00 2.004e-02 1.00 2.003e-02 1.00 8.156e-04 2.00 1 64 576 1856 3.717e-01 — 1.678e-02 — 2.967e-02 — 6.166e-03 — 256 2176 7424 9.358e-02 1.99 4.091e-03 2.04 7.556e-03 1.97 7.472e-04 3.04 1024 8448 29696 2.344e-02 2.00 1.018e-03 2.01 1.898e-03 1.99 9.261e-05 3.01 4096 33280 118784 5.863e-03 2.00 2.541e-04 2.00 4.750e-04 2.00 1.155e-05 3.00 2 64 864 3328 4.775e-02 — 1.222e-03 — 3.755e-03 — 6.963e-04 — 256 3264 13312 5.940e-03 3.01 1.396e-04 3.13 4.773e-04 2.98 4.237e-05 4.04 1024 12672 53248 7.406e-04 3.00 1.701e-05 3.04 5.992e-05 2.99 2.629e-06 4.01 4096 49920 212992 9.251e-05 3.00 2.112e-06 3.01 7.498e-06 3.00 1.640e-07 4.00 3 64 1152 5248 4.919e-03 — 7.526e-05 — 3.567e-04 — 6.159e-05 — 256 4352 20992 3.114e-04 3.98 3.737e-06 4.33 2.264e-05 3.98 2.032e-06 4.92 1024 16896 83968 1.955e-05 3.99 2.151e-07 4.12 1.420e-06 3.99 6.451e-08 4.98 4096 66560 335872 1.223e-06 4.00 1.314e-08 4.03 8.885e-08 4.00 2.024e-09 4.99 Mesh D.O.F. $$\|\mathrm{L} - \mathrm{L}^{h} \|_{{\mathcal{T}_{h}}}$$ $$\|\mathbf{u}-\mathbf{u}^{h}\|_{{\mathcal{T}_{h}}}$$ $$\|p - p^{h} \|_{{\mathcal{T}_{h}}}$$ $$\|\mathbf{u}-\mathbf{u}^{*,h}\|_{{\mathcal{T}_{h}}}$$ k Nele Nglobal Nlocal Error Order Error Order Error Order Error Order Third test: ν = 0.0001, γ = 1, m = 2. 0 64 288 704 2.399e+00 — 1.621e-01 — 1.567e-01 — 5.329e-02 — 256 1088 2816 1.226e+00 0.97 8.039e-02 1.01 7.970e-02 0.98 1.313e-02 2.02 1024 4224 11264 6.160e-01 0.99 4.011e-02 1.00 4.002e-02 0.99 3.268e-03 2.01 4096 16640 45056 3.083e-01 1.00 2.004e-02 1.00 2.003e-02 1.00 8.164e-04 2.00 1 64 576 1856 3.779e-01 — 1.679e-02 — 2.967e-02 — 6.192e-03 — 256 2176 7424 9.967e-02 1.92 4.096e-03 2.04 7.556e-03 1.97 7.509e-04 3.04 1024 8448 29696 2.761e-02 1.85 1.020e-03 2.01 1.898e-03 1.99 9.297e-05 3.01 4096 33280 118784 7.630e-03 1.86 2.544e-04 2.00 4.750e-04 2.00 1.157e-05 3.01 2 64 864 3328 4.844e-02 — 1.223e-03 — 3.755e-03 — 6.990e-04 — 256 3264 13312 6.177e-03 2.97 1.399e-04 3.13 4.773e-04 2.98 4.215e-05 4.05 1024 12672 53248 8.198e-04 2.91 1.708e-05 3.03 5.992e-05 2.99 2.571e-06 4.04 4096 49920 212992 1.099e-04 2.90 2.118e-06 3.01 7.498e-06 3.00 1.584e-07 4.02 3 64 1152 5248 4.973e-03 — 7.545e-05 — 3.567e-04 — 6.160e-05 — 256 4352 20992 3.248e-04 3.94 3.766e-06 4.32 2.264e-05 3.98 2.038e-06 4.92 1024 16896 83968 2.136e-05 3.93 2.173e-07 4.12 1.420e-06 3.99 6.486e-08 4.97 4096 66560 335872 1.390e-06 3.94 1.322e-08 4.04 8.885e-08 4.00 2.035e-09 4.99 Fourth test: ν = 1e−8, γ = 1, m = 2. 0 64 288 704 2.400e+00 — 1.621e-01 — 1.567e-01 — 5.329e-02 — 256 1088 2816 1.226e+00 0.97 8.039e-02 1.01 7.970e-02 0.98 1.313e-02 2.02 1024 4224 11264 6.162e-01 0.99 4.011e-02 1.00 4.002e-02 0.99 3.268e-03 2.01 4096 16640 45056 3.084e-01 1.00 2.004e-02 1.00 2.003e-02 1.00 8.156e-04 2.00 1 64 576 1856 3.717e-01 — 1.678e-02 — 2.967e-02 — 6.166e-03 — 256 2176 7424 9.358e-02 1.99 4.091e-03 2.04 7.556e-03 1.97 7.472e-04 3.04 1024 8448 29696 2.344e-02 2.00 1.018e-03 2.01 1.898e-03 1.99 9.261e-05 3.01 4096 33280 118784 5.863e-03 2.00 2.541e-04 2.00 4.750e-04 2.00 1.155e-05 3.00 2 64 864 3328 4.775e-02 — 1.222e-03 — 3.755e-03 — 6.963e-04 — 256 3264 13312 5.940e-03 3.01 1.396e-04 3.13 4.773e-04 2.98 4.237e-05 4.04 1024 12672 53248 7.406e-04 3.00 1.701e-05 3.04 5.992e-05 2.99 2.629e-06 4.01 4096 49920 212992 9.251e-05 3.00 2.112e-06 3.01 7.498e-06 3.00 1.640e-07 4.00 3 64 1152 5248 4.919e-03 — 7.526e-05 — 3.567e-04 — 6.159e-05 — 256 4352 20992 3.114e-04 3.98 3.737e-06 4.33 2.264e-05 3.98 2.032e-06 4.92 1024 16896 83968 1.955e-05 3.99 2.151e-07 4.12 1.420e-06 3.99 6.451e-08 4.98 4096 66560 335872 1.223e-06 4.00 1.314e-08 4.03 8.885e-08 4.00 2.024e-09 4.99 In Table 2, we present the L2-convergence rates for Lh, uh, ph and u*, h for the HDG method (2.5) with polynomial degree varying from k = 0 to k = 3 on rectangular meshes. The first-level mesh consists of 8 × 8 congruent squares, and the consequent meshes are obtained by uniform refinements. Here, the local postprocessing u*, h ∈ Pk+1(K) is defined elementwise by the following set of equations: \begin{align*} ({\boldsymbol{\nabla}} \mathbf{u}^{*,h},{\boldsymbol{\nabla}} \mathbf{v})_{K} = &\; (\mathrm{L}^{h}, {\boldsymbol{\nabla}} \mathbf{v})_{K}\quad \forall \, \mathbf{v}\in\boldsymbol{\mathcal{P}}_{k+1}(K),\\ (\mathbf{u}^{*,h},\mathbf{w})_{K} = &\; (\mathbf{u}^{h}, \mathbf{w})_{K}\quad \forall \, \mathbf{w}\in\boldsymbol{\mathcal{P}}_{0}(K). \end{align*} It is quite easy to show (cf. Stenberg, 1991; Cockburn et al., 2010) that u*, h convergence with an order of k + 2 − δ0, k, where $$\delta _{0,k}=\begin{cases} 1 & if\, k=0\\ 0 & if\, k>0\\ \end{cases} $$ is the Kronecker delta, if u has enough regularity. Table 3 History of convergence for the H(div)-conforming HDG method on triangular meshes Mesh D.O.F. $$\|\mathrm{L} - \mathrm{L}^{h} \|_{{\mathcal{T}_{h}}}$$ $$\|\mathbf{u}-\mathbf{u}^{h}\|_{{\mathcal{T}_{h}}}$$ $$\|p - p^{h} \|_{{\mathcal{T}_{h}}}$$ $$\|\mathbf{u}-\mathbf{u}^{*,h}\|_{{\mathcal{T}_{h}}}$$ k Nele Nglobal Nlocal Error Order Error Order Error Order Error Order First test: ν = 1, γ = 1, m = 2. 1 32 256 555 1.567e+00 — 8.253e-02 — 5.144e-01 — 5.985e-02 — 128 960 2203 3.378e-01 2.21 3.220e-02 1.36 1.158e-01 2.15 6.449e-03 3.21 512 3712 8763 8.757e-02 1.95 8.073e-03 2.00 2.712e-02 2.09 8.455e-04 2.93 2048 14592 34939 2.213e-02 1.98 2.018e-03 2.00 6.559e-03 2.05 1.073e-04 2.98 8192 57856 139515 5.550e-03 2.00 5.045e-04 2.00 1.615e-03 2.02 1.348e-05 2.99 2 32 368 1163 9.679e-02 — 3.553e-02 — 4.949e-02 — 2.407e-03 — 128 1376 4635 3.471e-02 1.48 3.432e-03 3.37 1.183e-02 2.07 4.712e-04 2.35 512 5312 18491 4.381e-03 2.99 4.359e-04 2.98 1.488e-03 2.99 2.964e-05 3.99 2048 20864 73851 5.488e-04 3.00 5.472e-05 2.99 1.862e-04 3.00 1.854e-06 4.00 8192 82688 295163 6.864e-05 3.00 6.847e-06 3.00 2.325e-05 3.00 1.159e-07 4.00 3 32 480 1995 3.551e-02 — 1.557e-03 — 2.159e-02 — 1.760e-03 — 128 1792 7963 1.815e-03 4.29 .245e-04 2.79 9.946e-04 4.44 4.237e-05 5.38 512 6912 31803 1.172e-04 3.95 1.418e-05 3.99 6.099e-05 4.03 1.356e-06 4.97 2048 27136 127099 7.398e-06 3.99 8.883e-07 4.00 3.774e-06 4.01 4.266e-08 4.99 8192 107520 508155 4.638e-07 4.00 5.555e-08 4.00 2.348e-07 4.01 1.336e-09 5.00 Second test: ν = 1, γ = 1, m = 20. 1 32 256 555 1.582e+00 — 8.376e-02 — 1.022e+00 — 6.085e-02 — 128 960 2203 3.652e-01 2.12 3.256e-02 1.36 6.395e-01 0.68 7.492e-03 3.02 512 3712 8763 8.758e-02 2.06 8.073e-03 2.01 3.010e-01 1.09 8.457e-04 3.15 2048 14592 34939 2.213e-02 1.98 2.018e-03 2.00 1.091e-01 1.46 1.073e-04 2.98 8192 57856 139515 5.550e-03 2.00 5.045e-04 2.00 3.012e-02 1.86 1.348e-05 2.99 2 32 368 1163 1.813e-01 — 3.599e-02 — 6.015e-01 — 5.208e-03 — 128 1376 4635 3.471e-02 2.39 3.432e-03 3.39 4.004e-01 0.59 4.715e-04 3.47 512 5312 18491 4.381e-03 2.99 4.359e-04 2.98 1.741e-01 1.20 2.964e-05 3.99 2048 20864 73851 5.488e-04 3.00 5.472e-05 2.99 3.076e-02 2.50 1.854e-06 4.00 8192 82688 295163 6.864e-05 3.00 6.847e-06 3.00 4.192e-03 2.88 1.159e-07 4.00 3 32 480 1995 4.193e-02 — 1.793e-03 — 5.592e-01 — 1.907e-03 — 128 1792 7963 1.815e-03 4.53 2.245e-04 3.00 3.068e-01 0.87 4.237e-05 5.49 512 6912 31803 1.172e-04 3.95 1.418e-05 3.99 3.201e-02 3.26 1.356e-06 4.97 2048 27136 127099 7.398e-06 3.99 8.883e-07 4.00 1.507e-03 4.41 4.266e-08 4.99 8192 107520 508155 4.638e-07 4.00 5.555e-08 4.00 6.589e-05 4.52 1.336e-09 5.00 Mesh D.O.F. $$\|\mathrm{L} - \mathrm{L}^{h} \|_{{\mathcal{T}_{h}}}$$ $$\|\mathbf{u}-\mathbf{u}^{h}\|_{{\mathcal{T}_{h}}}$$ $$\|p - p^{h} \|_{{\mathcal{T}_{h}}}$$ $$\|\mathbf{u}-\mathbf{u}^{*,h}\|_{{\mathcal{T}_{h}}}$$ k Nele Nglobal Nlocal Error Order Error Order Error Order Error Order First test: ν = 1, γ = 1, m = 2. 1 32 256 555 1.567e+00 — 8.253e-02 — 5.144e-01 — 5.985e-02 — 128 960 2203 3.378e-01 2.21 3.220e-02 1.36 1.158e-01 2.15 6.449e-03 3.21 512 3712 8763 8.757e-02 1.95 8.073e-03 2.00 2.712e-02 2.09 8.455e-04 2.93 2048 14592 34939 2.213e-02 1.98 2.018e-03 2.00 6.559e-03 2.05 1.073e-04 2.98 8192 57856 139515 5.550e-03 2.00 5.045e-04 2.00 1.615e-03 2.02 1.348e-05 2.99 2 32 368 1163 9.679e-02 — 3.553e-02 — 4.949e-02 — 2.407e-03 — 128 1376 4635 3.471e-02 1.48 3.432e-03 3.37 1.183e-02 2.07 4.712e-04 2.35 512 5312 18491 4.381e-03 2.99 4.359e-04 2.98 1.488e-03 2.99 2.964e-05 3.99 2048 20864 73851 5.488e-04 3.00 5.472e-05 2.99 1.862e-04 3.00 1.854e-06 4.00 8192 82688 295163 6.864e-05 3.00 6.847e-06 3.00 2.325e-05 3.00 1.159e-07 4.00 3 32 480 1995 3.551e-02 — 1.557e-03 — 2.159e-02 — 1.760e-03 — 128 1792 7963 1.815e-03 4.29 .245e-04 2.79 9.946e-04 4.44 4.237e-05 5.38 512 6912 31803 1.172e-04 3.95 1.418e-05 3.99 6.099e-05 4.03 1.356e-06 4.97 2048 27136 127099 7.398e-06 3.99 8.883e-07 4.00 3.774e-06 4.01 4.266e-08 4.99 8192 107520 508155 4.638e-07 4.00 5.555e-08 4.00 2.348e-07 4.01 1.336e-09 5.00 Second test: ν = 1, γ = 1, m = 20. 1 32 256 555 1.582e+00 — 8.376e-02 — 1.022e+00 — 6.085e-02 — 128 960 2203 3.652e-01 2.12 3.256e-02 1.36 6.395e-01 0.68 7.492e-03 3.02 512 3712 8763 8.758e-02 2.06 8.073e-03 2.01 3.010e-01 1.09 8.457e-04 3.15 2048 14592 34939 2.213e-02 1.98 2.018e-03 2.00 1.091e-01 1.46 1.073e-04 2.98 8192 57856 139515 5.550e-03 2.00 5.045e-04 2.00 3.012e-02 1.86 1.348e-05 2.99 2 32 368 1163 1.813e-01 — 3.599e-02 — 6.015e-01 — 5.208e-03 — 128 1376 4635 3.471e-02 2.39 3.432e-03 3.39 4.004e-01 0.59 4.715e-04 3.47 512 5312 18491 4.381e-03 2.99 4.359e-04 2.98 1.741e-01 1.20 2.964e-05 3.99 2048 20864 73851 5.488e-04 3.00 5.472e-05 2.99 3.076e-02 2.50 1.854e-06 4.00 8192 82688 295163 6.864e-05 3.00 6.847e-06 3.00 4.192e-03 2.88 1.159e-07 4.00 3 32 480 1995 4.193e-02 — 1.793e-03 — 5.592e-01 — 1.907e-03 — 128 1792 7963 1.815e-03 4.53 2.245e-04 3.00 3.068e-01 0.87 4.237e-05 5.49 512 6912 31803 1.172e-04 3.95 1.418e-05 3.99 3.201e-02 3.26 1.356e-06 4.97 2048 27136 127099 7.398e-06 3.99 8.883e-07 4.00 1.507e-03 4.41 4.266e-08 4.99 8192 107520 508155 4.638e-07 4.00 5.555e-08 4.00 6.589e-05 4.52 1.336e-09 5.00 Table 3 History of convergence for the H(div)-conforming HDG method on triangular meshes Mesh D.O.F. $$\|\mathrm{L} - \mathrm{L}^{h} \|_{{\mathcal{T}_{h}}}$$ $$\|\mathbf{u}-\mathbf{u}^{h}\|_{{\mathcal{T}_{h}}}$$ $$\|p - p^{h} \|_{{\mathcal{T}_{h}}}$$ $$\|\mathbf{u}-\mathbf{u}^{*,h}\|_{{\mathcal{T}_{h}}}$$ k Nele Nglobal Nlocal Error Order Error Order Error Order Error Order First test: ν = 1, γ = 1, m = 2. 1 32 256 555 1.567e+00 — 8.253e-02 — 5.144e-01 — 5.985e-02 — 128 960 2203 3.378e-01 2.21 3.220e-02 1.36 1.158e-01 2.15 6.449e-03 3.21 512 3712 8763 8.757e-02 1.95 8.073e-03 2.00 2.712e-02 2.09 8.455e-04 2.93 2048 14592 34939 2.213e-02 1.98 2.018e-03 2.00 6.559e-03 2.05 1.073e-04 2.98 8192 57856 139515 5.550e-03 2.00 5.045e-04 2.00 1.615e-03 2.02 1.348e-05 2.99 2 32 368 1163 9.679e-02 — 3.553e-02 — 4.949e-02 — 2.407e-03 — 128 1376 4635 3.471e-02 1.48 3.432e-03 3.37 1.183e-02 2.07 4.712e-04 2.35 512 5312 18491 4.381e-03 2.99 4.359e-04 2.98 1.488e-03 2.99 2.964e-05 3.99 2048 20864 73851 5.488e-04 3.00 5.472e-05 2.99 1.862e-04 3.00 1.854e-06 4.00 8192 82688 295163 6.864e-05 3.00 6.847e-06 3.00 2.325e-05 3.00 1.159e-07 4.00 3 32 480 1995 3.551e-02 — 1.557e-03 — 2.159e-02 — 1.760e-03 — 128 1792 7963 1.815e-03 4.29 .245e-04 2.79 9.946e-04 4.44 4.237e-05 5.38 512 6912 31803 1.172e-04 3.95 1.418e-05 3.99 6.099e-05 4.03 1.356e-06 4.97 2048 27136 127099 7.398e-06 3.99 8.883e-07 4.00 3.774e-06 4.01 4.266e-08 4.99 8192 107520 508155 4.638e-07 4.00 5.555e-08 4.00 2.348e-07 4.01 1.336e-09 5.00 Second test: ν = 1, γ = 1, m = 20. 1 32 256 555 1.582e+00 — 8.376e-02 — 1.022e+00 — 6.085e-02 — 128 960 2203 3.652e-01 2.12 3.256e-02 1.36 6.395e-01 0.68 7.492e-03 3.02 512 3712 8763 8.758e-02 2.06 8.073e-03 2.01 3.010e-01 1.09 8.457e-04 3.15 2048 14592 34939 2.213e-02 1.98 2.018e-03 2.00 1.091e-01 1.46 1.073e-04 2.98 8192 57856 139515 5.550e-03 2.00 5.045e-04 2.00 3.012e-02 1.86 1.348e-05 2.99 2 32 368 1163 1.813e-01 — 3.599e-02 — 6.015e-01 — 5.208e-03 — 128 1376 4635 3.471e-02 2.39 3.432e-03 3.39 4.004e-01 0.59 4.715e-04 3.47 512 5312 18491 4.381e-03 2.99 4.359e-04 2.98 1.741e-01 1.20 2.964e-05 3.99 2048 20864 73851 5.488e-04 3.00 5.472e-05 2.99 3.076e-02 2.50 1.854e-06 4.00 8192 82688 295163 6.864e-05 3.00 6.847e-06 3.00 4.192e-03 2.88 1.159e-07 4.00 3 32 480 1995 4.193e-02 — 1.793e-03 — 5.592e-01 — 1.907e-03 — 128 1792 7963 1.815e-03 4.53 2.245e-04 3.00 3.068e-01 0.87 4.237e-05 5.49 512 6912 31803 1.172e-04 3.95 1.418e-05 3.99 3.201e-02 3.26 1.356e-06 4.97 2048 27136 127099 7.398e-06 3.99 8.883e-07 4.00 1.507e-03 4.41 4.266e-08 4.99 8192 107520 508155 4.638e-07 4.00 5.555e-08 4.00 6.589e-05 4.52 1.336e-09 5.00 Mesh D.O.F. $$\|\mathrm{L} - \mathrm{L}^{h} \|_{{\mathcal{T}_{h}}}$$ $$\|\mathbf{u}-\mathbf{u}^{h}\|_{{\mathcal{T}_{h}}}$$ $$\|p - p^{h} \|_{{\mathcal{T}_{h}}}$$ $$\|\mathbf{u}-\mathbf{u}^{*,h}\|_{{\mathcal{T}_{h}}}$$ k Nele Nglobal Nlocal Error Order Error Order Error Order Error Order First test: ν = 1, γ = 1, m = 2. 1 32 256 555 1.567e+00 — 8.253e-02 — 5.144e-01 — 5.985e-02 — 128 960 2203 3.378e-01 2.21 3.220e-02 1.36 1.158e-01 2.15 6.449e-03 3.21 512 3712 8763 8.757e-02 1.95 8.073e-03 2.00 2.712e-02 2.09 8.455e-04 2.93 2048 14592 34939 2.213e-02 1.98 2.018e-03 2.00 6.559e-03 2.05 1.073e-04 2.98 8192 57856 139515 5.550e-03 2.00 5.045e-04 2.00 1.615e-03 2.02 1.348e-05 2.99 2 32 368 1163 9.679e-02 — 3.553e-02 — 4.949e-02 — 2.407e-03 — 128 1376 4635 3.471e-02 1.48 3.432e-03 3.37 1.183e-02 2.07 4.712e-04 2.35 512 5312 18491 4.381e-03 2.99 4.359e-04 2.98 1.488e-03 2.99 2.964e-05 3.99 2048 20864 73851 5.488e-04 3.00 5.472e-05 2.99 1.862e-04 3.00 1.854e-06 4.00 8192 82688 295163 6.864e-05 3.00 6.847e-06 3.00 2.325e-05 3.00 1.159e-07 4.00 3 32 480 1995 3.551e-02 — 1.557e-03 — 2.159e-02 — 1.760e-03 — 128 1792 7963 1.815e-03 4.29 .245e-04 2.79 9.946e-04 4.44 4.237e-05 5.38 512 6912 31803 1.172e-04 3.95 1.418e-05 3.99 6.099e-05 4.03 1.356e-06 4.97 2048 27136 127099 7.398e-06 3.99 8.883e-07 4.00 3.774e-06 4.01 4.266e-08 4.99 8192 107520 508155 4.638e-07 4.00 5.555e-08 4.00 2.348e-07 4.01 1.336e-09 5.00 Second test: ν = 1, γ = 1, m = 20. 1 32 256 555 1.582e+00 — 8.376e-02 — 1.022e+00 — 6.085e-02 — 128 960 2203 3.652e-01 2.12 3.256e-02 1.36 6.395e-01 0.68 7.492e-03 3.02 512 3712 8763 8.758e-02 2.06 8.073e-03 2.01 3.010e-01 1.09 8.457e-04 3.15 2048 14592 34939 2.213e-02 1.98 2.018e-03 2.00 1.091e-01 1.46 1.073e-04 2.98 8192 57856 139515 5.550e-03 2.00 5.045e-04 2.00 3.012e-02 1.86 1.348e-05 2.99 2 32 368 1163 1.813e-01 — 3.599e-02 — 6.015e-01 — 5.208e-03 — 128 1376 4635 3.471e-02 2.39 3.432e-03 3.39 4.004e-01 0.59 4.715e-04 3.47 512 5312 18491 4.381e-03 2.99 4.359e-04 2.98 1.741e-01 1.20 2.964e-05 3.99 2048 20864 73851 5.488e-04 3.00 5.472e-05 2.99 3.076e-02 2.50 1.854e-06 4.00 8192 82688 295163 6.864e-05 3.00 6.847e-06 3.00 4.192e-03 2.88 1.159e-07 4.00 3 32 480 1995 4.193e-02 — 1.793e-03 — 5.592e-01 — 1.907e-03 — 128 1792 7963 1.815e-03 4.53 2.245e-04 3.00 3.068e-01 0.87 4.237e-05 5.49 512 6912 31803 1.172e-04 3.95 1.418e-05 3.99 3.201e-02 3.26 1.356e-06 4.97 2048 27136 127099 7.398e-06 3.99 8.883e-07 4.00 1.507e-03 4.41 4.266e-08 4.99 8192 107520 508155 4.638e-07 4.00 5.555e-08 4.00 6.589e-05 4.52 1.336e-09 5.00 Table 3 Continued Mesh D.O.F. $$\|\mathrm{L} - \mathrm{L}^{h} \|_{{\mathcal{T}_{h}}}$$ $$\|\mathbf{u}-\mathbf{u}^{h}\|_{{\mathcal{T}_{h}}}$$ $$\|p - p^{h} \|_{{\mathcal{T}_{h}}}$$ $$\|\mathbf{u}-\mathbf{u}^{*,h}\|_{{\mathcal{T}_{h}}}$$ k Nele Nglobal Nlocal Error Order Error Order Error Order Error Order Third test: ν = 0.0001, γ = 1, m = 2. 1 32 256 555 1.436e+00 — 7.825e-02 — 3.891e-02 — 5.242e-02 — 128 960 2203 3.932e-01 1.87 3.013e-02 1.38 1.949e-02 1.00 8.254e-03 2.67 512 3712 8763 1.241e-01 1.66 7.719e-03 1.96 4.951e-03 1.98 1.497e-03 2.46 2048 14592 34939 3.414e-02 1.86 1.962e-03 1.98 1.243e-03 1.99 2.153e-04 2.80 8192 57856 139515 7.387e-03 2.21 4.987e-04 1.98 3.110e-04 2.00 2.215e-05 3.28 2 32 368 1163 3.442e-01 — 3.288e-02 — 2.266e-02 — 1.065e-02 — 128 1376 4635 6.978e-02 2.30 3.132e-03 3.39 2.169e-03 3.39 9.514e-04 3.48 512 5312 18491 1.042e-02 2.74 4.049e-04 2.95 2.748e-04 2.98 7.304e-05 3.70 2048 20864 73851 1.085e-03 3.26 5.285e-05 2.94 3.447e-05 3.00 4.209e-06 4.12 8192 82688 295163 9.842e-05 3.46 6.770e-06 2.96 4.313e-06 3.00 1.938e-07 4.44 3 32 480 1995 3.231e-02 — 1.545e-03 — 6.370e-04 — 1.717e-03 — 128 1792 7963 6.311e-03 2.36 1.928e-04 3.00 2.490e-05 4.68 6.757e-05 4.67 512 6912 31803 3.905e-04 4.01 1.295e-05 3.90 1.348e-06 4.21 2.318e-06 4.87 2048 27136 127099 1.768e-05 4.46 8.583e-07 3.91 8.055e-08 4.07 6.261e-08 5.21 8192 107520 508155 7.284e-07 4.60 5.500e-08 3.96 4.974e-09 4.02 2.111e-09 4.89 Fourth test: ν = 1e −8, γ = 1, m = 2. 1 32 256 555 1.434e+00 — 7.825e-02 — 3.891e-02 — 5.233e-02 — 128 960 2203 4.167e-01 1.78 3.011e-02 1.38 1.949e-02 1.00 8.738e-03 2.58 512 3712 8763 1.581e-01 1.40 7.687e-03 1.97 4.951e-03 1.98 1.919e-03 2.19 2048 14592 34939 7.048e-02 1.17 1.932e-03 1.99 1.243e-03 1.99 4.633e-04 2.05 8192 57856 139515 3.411e-02 1.05 4.836e-04 2.00 3.110e-04 2.00 1.151e-04 2.01 2 32 368 1163 3.778e-01 — 3.286e-02 — 2.266e-02 — 1.139e-02 — 128 1376 4635 9.041e-02 2.06 3.110e-03 3.40 2.169e-03 3.39 1.172e-03 3.28 512 5312 18491 2.349e-02 1.94 3.895e-04 3.00 2.748e-04 2.98 1.440e-04 3.02 2048 20864 73851 5.980e-03 1.97 4.859e-05 3.00 3.447e-05 3.00 1.805e-05 3.00 8192 82688 295163 1.504e-03 1.99 6.065e-06 3.00 4.313e-06 3.00 2.261e-06 3.00 3 32 480 1995 3.228e-02 — 1.551e-03 — 6.370e-04 — 1.711e-03 — 128 1792 7963 1.094e-02 1.56 1.882e-04 3.04 2.489e-05 4.68 9.261e-05 4.21 512 6912 31803 1.314e-03 3.06 1.191e-05 3.98 1.348e-06 4.21 5.506e-06 4.07 2048 27136 127099 1.612e-04 3.03 7.466e-07 4.00 8.055e-08 4.06 3.403e-07 4.02 8192 107520 508155 1.995e-05 3.01 4.669e-08 4.00 4.974e-09 4.02 2.122e-08 4.00 Mesh D.O.F. $$\|\mathrm{L} - \mathrm{L}^{h} \|_{{\mathcal{T}_{h}}}$$ $$\|\mathbf{u}-\mathbf{u}^{h}\|_{{\mathcal{T}_{h}}}$$ $$\|p - p^{h} \|_{{\mathcal{T}_{h}}}$$ $$\|\mathbf{u}-\mathbf{u}^{*,h}\|_{{\mathcal{T}_{h}}}$$ k Nele Nglobal Nlocal Error Order Error Order Error Order Error Order Third test: ν = 0.0001, γ = 1, m = 2. 1 32 256 555 1.436e+00 — 7.825e-02 — 3.891e-02 — 5.242e-02 — 128 960 2203 3.932e-01 1.87 3.013e-02 1.38 1.949e-02 1.00 8.254e-03 2.67 512 3712 8763 1.241e-01 1.66 7.719e-03 1.96 4.951e-03 1.98 1.497e-03 2.46 2048 14592 34939 3.414e-02 1.86 1.962e-03 1.98 1.243e-03 1.99 2.153e-04 2.80 8192 57856 139515 7.387e-03 2.21 4.987e-04 1.98 3.110e-04 2.00 2.215e-05 3.28 2 32 368 1163 3.442e-01 — 3.288e-02 — 2.266e-02 — 1.065e-02 — 128 1376 4635 6.978e-02 2.30 3.132e-03 3.39 2.169e-03 3.39 9.514e-04 3.48 512 5312 18491 1.042e-02 2.74 4.049e-04 2.95 2.748e-04 2.98 7.304e-05 3.70 2048 20864 73851 1.085e-03 3.26 5.285e-05 2.94 3.447e-05 3.00 4.209e-06 4.12 8192 82688 295163 9.842e-05 3.46 6.770e-06 2.96 4.313e-06 3.00 1.938e-07 4.44 3 32 480 1995 3.231e-02 — 1.545e-03 — 6.370e-04 — 1.717e-03 — 128 1792 7963 6.311e-03 2.36 1.928e-04 3.00 2.490e-05 4.68 6.757e-05 4.67 512 6912 31803 3.905e-04 4.01 1.295e-05 3.90 1.348e-06 4.21 2.318e-06 4.87 2048 27136 127099 1.768e-05 4.46 8.583e-07 3.91 8.055e-08 4.07 6.261e-08 5.21 8192 107520 508155 7.284e-07 4.60 5.500e-08 3.96 4.974e-09 4.02 2.111e-09 4.89 Fourth test: ν = 1e −8, γ = 1, m = 2. 1 32 256 555 1.434e+00 — 7.825e-02 — 3.891e-02 — 5.233e-02 — 128 960 2203 4.167e-01 1.78 3.011e-02 1.38 1.949e-02 1.00 8.738e-03 2.58 512 3712 8763 1.581e-01 1.40 7.687e-03 1.97 4.951e-03 1.98 1.919e-03 2.19 2048 14592 34939 7.048e-02 1.17 1.932e-03 1.99 1.243e-03 1.99 4.633e-04 2.05 8192 57856 139515 3.411e-02 1.05 4.836e-04 2.00 3.110e-04 2.00 1.151e-04 2.01 2 32 368 1163 3.778e-01 — 3.286e-02 — 2.266e-02 — 1.139e-02 — 128 1376 4635 9.041e-02 2.06 3.110e-03 3.40 2.169e-03 3.39 1.172e-03 3.28 512 5312 18491 2.349e-02 1.94 3.895e-04 3.00 2.748e-04 2.98 1.440e-04 3.02 2048 20864 73851 5.980e-03 1.97 4.859e-05 3.00 3.447e-05 3.00 1.805e-05 3.00 8192 82688 295163 1.504e-03 1.99 6.065e-06 3.00 4.313e-06 3.00 2.261e-06 3.00 3 32 480 1995 3.228e-02 — 1.551e-03 — 6.370e-04 — 1.711e-03 — 128 1792 7963 1.094e-02 1.56 1.882e-04 3.04 2.489e-05 4.68 9.261e-05 4.21 512 6912 31803 1.314e-03 3.06 1.191e-05 3.98 1.348e-06 4.21 5.506e-06 4.07 2048 27136 127099 1.612e-04 3.03 7.466e-07 4.00 8.055e-08 4.06 3.403e-07 4.02 8192 107520 508155 1.995e-05 3.01 4.669e-08 4.00 4.974e-09 4.02 2.122e-08 4.00 Table 3 Continued Mesh D.O.F. $$\|\mathrm{L} - \mathrm{L}^{h} \|_{{\mathcal{T}_{h}}}$$ $$\|\mathbf{u}-\mathbf{u}^{h}\|_{{\mathcal{T}_{h}}}$$ $$\|p - p^{h} \|_{{\mathcal{T}_{h}}}$$ $$\|\mathbf{u}-\mathbf{u}^{*,h}\|_{{\mathcal{T}_{h}}}$$ k Nele Nglobal Nlocal Error Order Error Order Error Order Error Order Third test: ν = 0.0001, γ = 1, m = 2. 1 32 256 555 1.436e+00 — 7.825e-02 — 3.891e-02 — 5.242e-02 — 128 960 2203 3.932e-01 1.87 3.013e-02 1.38 1.949e-02 1.00 8.254e-03 2.67 512 3712 8763 1.241e-01 1.66 7.719e-03 1.96 4.951e-03 1.98 1.497e-03 2.46 2048 14592 34939 3.414e-02 1.86 1.962e-03 1.98 1.243e-03 1.99 2.153e-04 2.80 8192 57856 139515 7.387e-03 2.21 4.987e-04 1.98 3.110e-04 2.00 2.215e-05 3.28 2 32 368 1163 3.442e-01 — 3.288e-02 — 2.266e-02 — 1.065e-02 — 128 1376 4635 6.978e-02 2.30 3.132e-03 3.39 2.169e-03 3.39 9.514e-04 3.48 512 5312 18491 1.042e-02 2.74 4.049e-04 2.95 2.748e-04 2.98 7.304e-05 3.70 2048 20864 73851 1.085e-03 3.26 5.285e-05 2.94 3.447e-05 3.00 4.209e-06 4.12 8192 82688 295163 9.842e-05 3.46 6.770e-06 2.96 4.313e-06 3.00 1.938e-07 4.44 3 32 480 1995 3.231e-02 — 1.545e-03 — 6.370e-04 — 1.717e-03 — 128 1792 7963 6.311e-03 2.36 1.928e-04 3.00 2.490e-05 4.68 6.757e-05 4.67 512 6912 31803 3.905e-04 4.01 1.295e-05 3.90 1.348e-06 4.21 2.318e-06 4.87 2048 27136 127099 1.768e-05 4.46 8.583e-07 3.91 8.055e-08 4.07 6.261e-08 5.21 8192 107520 508155 7.284e-07 4.60 5.500e-08 3.96 4.974e-09 4.02 2.111e-09 4.89 Fourth test: ν = 1e −8, γ = 1, m = 2. 1 32 256 555 1.434e+00 — 7.825e-02 — 3.891e-02 — 5.233e-02 — 128 960 2203 4.167e-01 1.78 3.011e-02 1.38 1.949e-02 1.00 8.738e-03 2.58 512 3712 8763 1.581e-01 1.40 7.687e-03 1.97 4.951e-03 1.98 1.919e-03 2.19 2048 14592 34939 7.048e-02 1.17 1.932e-03 1.99 1.243e-03 1.99 4.633e-04 2.05 8192 57856 139515 3.411e-02 1.05 4.836e-04 2.00 3.110e-04 2.00 1.151e-04 2.01 2 32 368 1163 3.778e-01 — 3.286e-02 — 2.266e-02 — 1.139e-02 — 128 1376 4635 9.041e-02 2.06 3.110e-03 3.40 2.169e-03 3.39 1.172e-03 3.28 512 5312 18491 2.349e-02 1.94 3.895e-04 3.00 2.748e-04 2.98 1.440e-04 3.02 2048 20864 73851 5.980e-03 1.97 4.859e-05 3.00 3.447e-05 3.00 1.805e-05 3.00 8192 82688 295163 1.504e-03 1.99 6.065e-06 3.00 4.313e-06 3.00 2.261e-06 3.00 3 32 480 1995 3.228e-02 — 1.551e-03 — 6.370e-04 — 1.711e-03 — 128 1792 7963 1.094e-02 1.56 1.882e-04 3.04 2.489e-05 4.68 9.261e-05 4.21 512 6912 31803 1.314e-03 3.06 1.191e-05 3.98 1.348e-06 4.21 5.506e-06 4.07 2048 27136 127099 1.612e-04 3.03 7.466e-07 4.00 8.055e-08 4.06 3.403e-07 4.02 8192 107520 508155 1.995e-05 3.01 4.669e-08 4.00 4.974e-09 4.02 2.122e-08 4.00 Mesh D.O.F. $$\|\mathrm{L} - \mathrm{L}^{h} \|_{{\mathcal{T}_{h}}}$$ $$\|\mathbf{u}-\mathbf{u}^{h}\|_{{\mathcal{T}_{h}}}$$ $$\|p - p^{h} \|_{{\mathcal{T}_{h}}}$$ $$\|\mathbf{u}-\mathbf{u}^{*,h}\|_{{\mathcal{T}_{h}}}$$ k Nele Nglobal Nlocal Error Order Error Order Error Order Error Order Third test: ν = 0.0001, γ = 1, m = 2. 1 32 256 555 1.436e+00 — 7.825e-02 — 3.891e-02 — 5.242e-02 — 128 960 2203 3.932e-01 1.87 3.013e-02 1.38 1.949e-02 1.00 8.254e-03 2.67 512 3712 8763 1.241e-01 1.66 7.719e-03 1.96 4.951e-03 1.98 1.497e-03 2.46 2048 14592 34939 3.414e-02 1.86 1.962e-03 1.98 1.243e-03 1.99 2.153e-04 2.80 8192 57856 139515 7.387e-03 2.21 4.987e-04 1.98 3.110e-04 2.00 2.215e-05 3.28 2 32 368 1163 3.442e-01 — 3.288e-02 — 2.266e-02 — 1.065e-02 — 128 1376 4635 6.978e-02 2.30 3.132e-03 3.39 2.169e-03 3.39 9.514e-04 3.48 512 5312 18491 1.042e-02 2.74 4.049e-04 2.95 2.748e-04 2.98 7.304e-05 3.70 2048 20864 73851 1.085e-03 3.26 5.285e-05 2.94 3.447e-05 3.00 4.209e-06 4.12 8192 82688 295163 9.842e-05 3.46 6.770e-06 2.96 4.313e-06 3.00 1.938e-07 4.44 3 32 480 1995 3.231e-02 — 1.545e-03 — 6.370e-04 — 1.717e-03 — 128 1792 7963 6.311e-03 2.36 1.928e-04 3.00 2.490e-05 4.68 6.757e-05 4.67 512 6912 31803 3.905e-04 4.01 1.295e-05 3.90 1.348e-06 4.21 2.318e-06 4.87 2048 27136 127099 1.768e-05 4.46 8.583e-07 3.91 8.055e-08 4.07 6.261e-08 5.21 8192 107520 508155 7.284e-07 4.60 5.500e-08 3.96 4.974e-09 4.02 2.111e-09 4.89 Fourth test: ν = 1e −8, γ = 1, m = 2. 1 32 256 555 1.434e+00 — 7.825e-02 — 3.891e-02 — 5.233e-02 — 128 960 2203 4.167e-01 1.78 3.011e-02 1.38 1.949e-02 1.00 8.738e-03 2.58 512 3712 8763 1.581e-01 1.40 7.687e-03 1.97 4.951e-03 1.98 1.919e-03 2.19 2048 14592 34939 7.048e-02 1.17 1.932e-03 1.99 1.243e-03 1.99 4.633e-04 2.05 8192 57856 139515 3.411e-02 1.05 4.836e-04 2.00 3.110e-04 2.00 1.151e-04 2.01 2 32 368 1163 3.778e-01 — 3.286e-02 — 2.266e-02 — 1.139e-02 — 128 1376 4635 9.041e-02 2.06 3.110e-03 3.40 2.169e-03 3.39 1.172e-03 3.28 512 5312 18491 2.349e-02 1.94 3.895e-04 3.00 2.748e-04 2.98 1.440e-04 3.02 2048 20864 73851 5.980e-03 1.97 4.859e-05 3.00 3.447e-05 3.00 1.805e-05 3.00 8192 82688 295163 1.504e-03 1.99 6.065e-06 3.00 4.313e-06 3.00 2.261e-06 3.00 3 32 480 1995 3.228e-02 — 1.551e-03 — 6.370e-04 — 1.711e-03 — 128 1792 7963 1.094e-02 1.56 1.882e-04 3.04 2.489e-05 4.68 9.261e-05 4.21 512 6912 31803 1.314e-03 3.06 1.191e-05 3.98 1.348e-06 4.21 5.506e-06 4.07 2048 27136 127099 1.612e-04 3.03 7.466e-07 4.00 8.055e-08 4.06 3.403e-07 4.02 8192 107520 508155 1.995e-05 3.01 4.669e-08 4.00 4.974e-09 4.02 2.122e-08 4.00 In Table 3, we present the same convergence study with polynomial degree varying from k = 1 to k = 3 on triangular meshes. The first level mesh consists of 2 × 4 × 4 congruent triangles, and the consequent meshes are obtained by uniform refinements. In both tables, Nele denotes the number of elements, Nglobal denotes the number of globally coupled degrees of freedom and Nlocal denotes the number of local (static-condensed) degrees of freedom. From the results for the first test in Tables 2 and 3, we observe an optimal convergence order of k + 1 for all the three variables Lh, uh and ph, and a superconvergence order of k + 2 for the postprocessing u*, h. The convergence results for Lh, uh and ph are in full agreement with the theoretical predictions in Corollary 2.5 and Theorem 2.6. The superconvergence for u*, h is in agreement with the theoretical predictions in Theorem 2.6 for k ≥ 1, while the superconvergence of u*, h for k = 0 on rectangular meshes is not covered by our analysis in Theorem 2.6. From the results for the second test in Tables 2 and 3, we observe the same L2-errors in Lh, uh and u*, h as the corresponding ones in the first test. This indicates that velocity error is independent of the pressure, in full agreement with the estimates in Corollary 2.5. We also observe that the L2-error for ph is significantly larger than that for the first test. It is clear that, in this test, convergence for pressure is not in the asymptotic regime yet. From the results for the Darcy-dominated regimes in the third and fourth tests in Tables 2 and 3, we observe a similar L2-error in the velocity. This indicates the uniform stability of the proposed HDG method since the L2-error of the velocity does not depend on the ratio γ/ν, which is in full agreement with the velocity estimate in Corollary 2.5. We also observe a suboptimal convergence of one order less for the velocity gradient Lh, and the loss of superconvergence of u*, h for the fourth test on triangular meshes in Table 2. This is expected since in the limiting case when $$\nu \rightarrow 0$$, the regularity constant Cr for the dual problem (2.12) will blow up, and the Brinkman equations will collapse to the Darcy equations, in which the control of velocity gradient in Corollary 2.5 and that of the velocity error in Theorem 2.6 will be lost. However, such losses of convergence do not appear in Table 2 on rectangular meshes, which are better than our analysis in Section 3 indicates. 6. Conclusion We present and analyse a class of parameter-free superconvergent H(div)-conforming HDG methods on both simplicial and rectangular meshes for the Brinkman equations. Numerical results in two dimensions are presented to validate the theoretical findings. Funding Research Grants Council of the Hong Kong Special Administrative Region, China (Project No. City U 11304017) to W.Q. Acknowledgements G. 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Published: Feb 20, 2018

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