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IMA Journal of Numerical Analysis
, Volume 38 (1) – Jan 1, 2018

29 pages

/lp/ou_press/an-a-posteriori-error-analysis-for-an-optimal-control-problem-pH0gI2jdl7

- Publisher
- Oxford University Press
- Copyright
- © The authors 2017. Published by Oxford University Press on behalf of the Institute of Mathematics and its Applications. All rights reserved.
- ISSN
- 0272-4979
- eISSN
- 1464-3642
- D.O.I.
- 10.1093/imanum/drx005
- Publisher site
- See Article on Publisher Site

Abstract In a previous work, we introduced a discretization scheme for a control-constrained optimal control problem involving the fractional Laplacian. For such a problem, we derived a near optimal a priori error estimate, for the approximation of the optimal control variable, that demands the convexity of the domain and some compatibility conditions on the data. To relax such restrictions, in this article, we introduce and analyse an efficient and, under certain assumptions, reliable a posteriori error estimator. We realize the fractional Laplacian as the Dirichlet-to-Neumann map for a nonuniformly elliptic problem posed on a semi infinite cylinder in one more spatial dimension. This extra dimension further motivates the design of an a posteriori error indicator. The latter is defined as the sum of three contributions that come from the discretization of the state and adjoint equations and the control variable. The indicator for the state and adjoint equations relies on an anisotropic error estimator in Muckenhoupt-weighted Sobolev spaces. We present an analysis that is valid in any dimension. On the basis of the devised a posteriori error estimator, we design a simple adaptive strategy that exhibits optimal experimental rates of convergence. 1. Introduction In this work, we shall be interested in the derivation and analysis of a computable, efficient and, under certain assumptions, reliable a posteriori error estimator for a control-constrained linear–quadratic optimal control problem involving the fractional powers of the Dirichlet Laplace operator. To the best of our knowledge, this is the first work that addresses this problem. To make matters precise, for $$n\ge1$$, we let $$\it \Omega$$ be an open and bounded polytopal domain of $$\mathbb{R}^n$$ with Lipschitz boundary $$\partial \it \Omega$$. Given $$s \in (0,1)$$, and a desired state $$\mathsf{u}_d: \it \Omega \rightarrow \mathbb{R}$$, we define the cost functional J(u,z)=12‖u−ud‖L2(Ω)2+μ2‖z‖L2(Ω)2, (1.1) where $$\mu > 0$$ is the so-called regularization parameter. With these ingredients at hand, we define the fractional optimal control problem as follows: find min J(u,z) (1.2) subject to the fractional state equation (−Δ)su=zinΩ,u=0on∂Ω (1.3) and the control constraints a(x′)≤z(x′)≤b(x′)a.e.x′∈Ω. (1.4) The operator $$(-\it \Delta)^s$$, with $$s \in (0,1)$$, denotes the fractional powers of the Dirichlet Laplace operator, which, for convenience, we will simply call the fractional Laplacian. The functions $$\mathsf{a}$$ and $$\mathsf{b}$$ both belong to $$L^2(\it \Omega)$$ and satisfy the property $$\mathsf{a}(x') \leq \mathsf{b}(x')$$ for almost every $$x' \in \it \Omega$$. A rather incomplete list of problems where fractional derivatives and fractional diffusion appear includes mechanics (Atanackovic et al, 2014) where they are used to model viscoelastic behavior (Debnath, 2003a), turbulence (del Castillo-Negrete et al, 2004; Chen, 2006) and the hereditary properties of materials (Gorenflo et al, 2002); diffusion processes (Nigmatullin, 1986; Abe & Thurner, 2005), in particular, processes in disordered media, where the disorder may change the laws of Brownian motion and thus lead to anomalous diffusion (Bouchaud & Georges, 1990; Barkai et al, 2000); nonlocal electrostatics (Ishizuka et al, 2008); finance (Levendorskiĭ, 2004); image processing (Gatto & Hesthaven, 2015); biophysics (Bueno-Orovio et al, 2014); chaotic dynamical systems (Saichev & Zaslavsky, 1997) and many others (Debnath, 2003b; Bucur & Valdinoci, 2015). Optimal control problems arise naturally in these applications and then it is essential to design numerical schemes to efficiently approximate them. The analysis of problems involving the fractional Laplacian is delicate and involves fine results in harmonic analysis (Stein, 1970; Landkof, 1972; Silvestre, 2007), one of the main difficulties being the nonlocality of the operator. This difficulty has been resolved to some extent by Caffarelli & Silvestre (2007), who have proposed a technique that turned out to be a breakthrough and has paved the way to study fractional Laplacians using local techniques. Namely, any power $$s \in (0,1)$$ of the fractional Laplacian in $$\mathbb{R}^n$$ can be realized as an operator that maps a Dirichlet boundary condition to a Neumann-type condition via an extension problem on the upper half-space $$\mathbb{R}_{+}^{n+1}$$. This result was later adapted in Stinga & Torrea (2010) and Capella et al (2011) to bounded domains $$\it \Omega$$, thus obtaining an extension problem posed on the semi infinite cylinder $$\mathscr{C} = \it \Omega \times \rm{(0,\infty)}$$. This extension corresponds to the following mixed boundary value problem: div(yα∇U)=0inC,U=0on∂LC,∂ναU=dszonΩ×{0}, (1.5) where $$\partial_L \mathscr{C}= \partial \it \Omega \times \rm [0,\infty)$$ is the lateral boundary of $$\mathscr{C}$$ and $$d_s = 2^{\alpha}\it \Gamma\rm(1-\it s)/\it \Gamma\rm (s)$$ is a positive normalization constant. The parameter $$\alpha$$ is defined as $$\alpha = 1-2s \in (-1,1)$$ and the conormal exterior derivative of $$\mathscr{U}$$ at $$\it \Omega \times \{ \rm 0 \}$$ is ∂ναU=−limy→0+yαUy. (1.6) We call $$y$$ the extended variable and call the dimension $$n+1$$ in $$\mathbb{R}_+^{n+1}$$ the extended dimension of problem (1.5). The limit in (1.6) must be understood in the distributional sense (see Caffarelli & Silvestre, 2007; Stinga & Torrea, 2010; Capella et al, 2011). With these elements at hand, we then write the fundamental result by Caffarelli & Silvestre (2007): the fractional Laplacian and the Dirichlet-to-Neumann map of problem (1.5) are related by $$ d_s (-\it \Delta)^s \mathsf{u} = \partial_{\nu^\alpha} \mathscr{U} $$ in $$\it \Omega$$. The use of the aforementioned localization techniques for the numerical treatment of problem (1.3) followed not so long after in Nochetto et al (2015). In this reference, the authors propose the following technique to solve problem (1.3): given $$\mathsf{z}$$, solve (1.5), thus obtaining a function $$\mathscr{U}$$; setting $$\mathsf{u}(x') = \mathscr{U} (x',0)$$, the solution to (1.3) is obtained. The implementation of this scheme uses standard components of finite element analysis, while its analysis combines asymptotic properties of Bessel functions (Abramowitz & Stegun, 1964), elements of harmonic analysis (Muckenhoupt, 1972; Duoandikoetxea, 2001) and a polynomial interpolation theory on weighted spaces (Durán & Lombardi, 2005; Nochetto et al, 2016b). The latter is valid for tensor product elements that exhibit a large aspect ratio in $$y$$ (anisotropy), which is necessary to fit the behavior of $$\mathscr{U}(x',y)$$ with $$x' \in \it \Omega$$ and $$y>0$$. The main advantage of this scheme is that it solves the local problem (1.5) instead of dealing with $$(-\it \Delta)^s$$ in (1.3). However, this comes at the expense of incorporating one more dimension to the problem, an issue that has been resolved to some extent with the design of fast solvers (Chen et al, 2016) and adaptive finite element methods (AFEMs) (Chen et al, 2015). Exploiting the ideas developed in Nochetto et al (2015), in the previous work Antil&Otárola (2015), we have proposed two numerical strategies to approximate the solution to (1.2)–(1.4). As a first step, we utilized the localization results of Caffarelli & Silvestre (2007), Stinga & Torrea (2010) and Capella et al (2011) and considered an equivalent optimal control problem: $$\min J(\mathscr{U}(\cdot,0),\mathsf{z})$$ subject to the linear state equation (1.5) and the control constraints (1.4). Since (1.5) is posed on the semi infinite cylinder $$\mathscr{C}$$, we have then introduced a truncated optimal control problem and analysed its approximation properties. On the basis of this, we have proposed two schemes based on the discretization of the state and adjoint equations with first-degree tensor product finite elements on anisotropic meshes: the variational approach by Hinze (2005) and a fully discrete scheme that discretizes the set of controls by piecewise constant functions (see Arada et al, 2002; Casas et al, 2005). For the latter scheme we derived a near optimal error estimate for the error in the control approximation: if $$\it \Omega$$ is convex, $$\mathsf{u}_d \in \mathbb{H}^{1-s}(\it \Omega)$$ and $$\mathsf{a}, \mathsf{b} \in \mathbb{R}$$ are such that $$\mathsf{a} < 0 < \mathsf{b}$$ for $$s \in (0,\tfrac{1}{2}]$$, then ‖z¯−Z¯‖L2(Ω)≲|logN|2sN−1/(n+1), (1.7) where $$\bar{\mathsf{z}}$$ denotes the optimal solution to the fractional optimal control problem, $$\bar{Z}$$ corresponds to the optimal solution of the discrete counterpart of (1.2)–(1.4) and $$N$$ denotes the number of the degrees of freedom of the underlying mesh. Since the aforementioned scheme incorporates one extra dimension, it raises the following question: how efficient is this method? The quest for an answer to this question motivates the study of AFEMs since it is known that they constitute an efficient class of numerical methods for approximating the solution to optimal control problems (Becker et al, 2000; Hintermüller et al, 2008; Kohls et al, 2014): they allow for their resolution with relatively modest computational resources. In addition, they can achieve optimal performance, measured as error vs. degrees of freedom, in situations when classical FEM cannot (see Nochetto et al, 2009; Nochetto & Veeser, 2012; Kohls et al, 2014). An essential ingredient of AFEMs is an a posteriori error estimator, which is a computable quantity that depends on the discrete solution and data, and provides information about the local quality of the approximate solution. For linear second-order elliptic boundary value problems, the theory has attained a mature understanding; see Verfürth (1996), Ainsworth & Oden (2000), Morin et al (2000), Nochetto et al (2009), Nochetto&Veeser (2012) for an up-to-date discussion including also the design of AFEMs, their convergence and optimal complexity. In contrast to this well-established theory, the a posteriori error analysis for a constrained optimal control problem has not been fully understood yet; the main source of difficulty is its inherent nonlinear feature. We refer the reader to Kohls et al (2014) for an up-to-date discussion. AFEMs for the fractional optimal control problem are also motivated by the fact that the a priori error estimate (1.7) requires $$\bar{\mathsf{z}} \in \mathbb{H}^{1-s}(\it \Omega)$$, which in turn demands $$\it \Omega$$ convex, $$\mathsf{u}_d \in \mathbb{H}^{1-s}(\it \Omega)$$ and $$\mathsf{a} < 0 < \mathsf{b}$$ for $$s \in (0,\tfrac{1}{2}]$$. If one of these conditions does not hold, the optimal control $$\bar{\mathsf{z}}$$ may have singularities in the $$x'$$-variables and thus exhibit fractional regularity. Consequently, quasi-uniform refinement of $$\it \Omega$$ would not result in an efficient solution technique; see Nochetto et al (2015, Section 6.3) for an illustration of this situation at the level of solving the state equation (1.5). The main contribution of this work is the design and analysis of a computable, efficient and, under certain assumptions, reliable a posteriori error estimator for the fractional optimal control problem (1.2)–(1.4). As highlighted before, there is undoubtedly a need for developing such an estimator, and this is the first work that provides a construction and analysis for it. Given a mesh $$\mathscr{T}$$ and corresponding approximations $$\mathscr{U}_{\mathscr{T}}$$, $$\mathscr{P}_{\mathscr{T}}$$ and $$\mathsf{z}_{\mathscr{T}}$$, the proposed error indicator is built on the basis of three contributions: Eocp=EU+EP+Ez, where $$\mathscr{E}_{\mathscr{U}}$$ and $$\mathscr{E}_{\mathscr{P}}$$ correspond to the anisotropic a posteriori error estimator on weighted Sobolev spaces of Chen et al (2015), for the state and adjoint equations, respectively. The error indicator $$\mathscr{E}_{\mathsf{z}}$$ is defined as the $$\ell^2$$-sum of the local contributions $$\mathscr{E}_{\mathsf{z}}(\mathsf{z}_{\mathscr{T}},\mathscr{P}_{\mathscr{T}}; T) = \| \mathsf{z}_{\mathscr{T}} - \Pi(-\mu^{-1} \mathscr{P}_{\mathscr{T}}(\cdot,0))\|_{L^2(\it \Omega)}$$, with $$T \in \mathscr{T}$$ and $$\it \Pi(v) = \min \{ \mathsf{b}, \max \{\mathsf{a},v\} \}$$. We present an analysis for $$\mathscr{E}_{\textrm{ocp}}$$: we prove its efficiency and, under certain assumptions, its reliability. We remark that the devised error estimator is able to deal with both the natural anisotropy of the mesh $$\mathscr{T}$$ in the extended variable and the degenerate coefficient $$y^{\alpha}$$. This approach is of value not only for the fractional optimal control problem but also in general for optimal control problems involving anisotropic meshes since rigorous anisotropic a posteriori error estimators are scarce in the literature. Our presentation is organized as follows. The notation is described in Section 2, where we also describe the definition of the fractional Laplacian and its localization via the Caffarelli–Silvestre extension. In Section 3, we review the a priori error analysis developed in Antil & Otárola (2015). Section 4 is the highlight of this contribution and is dedicated to the development and analysis of an error estimator for problem (1.2)–(1.4). As a first step, in Section 4.3 we introduce and analyse an ideal error estimator that is not computable but sets the stage for Section 4.4, where we devise a computable error estimator and show its equivalence, under suitable assumptions, to the error up to data oscillation terms. We conclude, in Section 5, with a numerical experiment that illustrates our theory. 2. Notation and preliminaries Throughout this work, $${\it {\Omega}}$$ is an open and bounded polytopal domain of $$\mathbb{R}^n$$ ($$n\geq1$$) with Lipschitz boundary $$\partial{\it {\Omega}}$$. We define the semi infinite cylinder with base $${\it {\Omega}}$$ and its lateral boundary, respectively, by $$\mathscr{C} = {\it {\Omega}} \times (0,\infty)$$ and $$ \partial_L \mathscr{C} = \partial {\it {\Omega}} \times [0,\infty). $$ Given , we define the truncated cylinder and accordingly. If $$x\in \mathbb{R}^{n+1}$$, we write $$ x = (x^1,\ldots,x^n, x^{n+1}) = (x', x^{n+1}) = (x',y), $$ with $$x^i \in \mathbb{R}$$ for $$i=1,\ldots,{n+1}$$, $$x' \in \mathbb{R}^n$$ and $$y\in\mathbb{R}$$; this notation distinguishes the extended dimension $$y$$. We denote by $$(-{\it {\Delta}})^s$$, $$s \in (0,1)$$ a fractional power of the Dirichlet Laplace operator $$(-{\it {\Delta}})$$. The parameter $$\alpha$$ belongs to $$(-1,1)$$ and is related to the power $$s$$ of the fractional Laplacian $$(-{\it {\Delta}})^s$$ by the formula $$\alpha = 1 -2s$$. Finally, the relation $$a \lesssim b$$ indicates that $$a \leq Cb$$, with a constant $$C$$ that does not depend on $$a$$ or $$b$$ nor-on the discretization parameters. The value of $$C$$ might change at each occurrence. 2.1 The fractional Laplace operator We adopt the spectral definition for the fractional powers of the Dirichlet Laplace operator (see Capella et al., 2011; Nochetto et al., 2015). The operator $$(-{\it {\Delta}})^{-1}:L^2({\it {\Omega}}) \rightarrow L^2({\it {\Omega}})$$ that solves $$-{\it {\Delta}} w = f$$ in $${\it {\Omega}}$$ and $$w = 0$$ on $$\partial {\it {\Omega}}$$, is compact, symmetric and positive, whence its spectrum $$\{ \lambda_k^{-1} \}_{k \in \mathbb{N}}$$ is discrete, real, positive and accumulates at zero. Moreover, the eigenfunctions {φk}k∈N:−Δφk=λkφk in Ω,φk=0 on Ω,k∈N form an orthonormal basis of $$L^2({\it {\Omega}})$$. Fractional powers of $$(-{\it {\Delta}})$$ can be defined by (−Δ)sw:=∑k=1∞λkswkφk,w∈C0∞(Ω),s∈(0,1), where $$w_k = \int_{{\it {\Omega}}} w \varphi_k $$. By density we extend this definition to Hs(Ω)={w=∑k=1∞wkφk:∑k=1∞λkswk2<∞}=[H01(Ω),L2(Ω)]1−s (see Nochetto et al., 2015 for details). For $$ s \in (0,1),$$ we denote by $$\mathbb{H}^{-s}({\it {\Omega}})$$ the dual space of $$\mathbb{H}^s({\it {\Omega}})$$. 2.2 The Caffarelli–Silvestre extension problem In this section, we explore problem (1.5) and its relation with the nonlocal problem (1.3); we refer the reader to Caffarelli & Silvestre (2007), Stinga & Torrea (2010), Capella et al. (2011) and Nochetto et al. (2015) for details. Since $$\alpha \in (-1,1)$$, problem (1.5) is nonuniformly elliptic, and thus, it is necessary to introduce weighted Lebesgue and Sobolev spaces for its description. Let $$E$$ be an open set in $$\mathbb{R}^{n+1}$$. We define $$L^2(|y|^{\alpha},E)$$ as the Lebesgue space for the measure $$|y|^\alpha {\,{\text d}} x$$. We also define the weighted Sobolev space $$H^1(|y|^{\alpha},E) := \{ w \in L^2(|y|^{\alpha},E): | \nabla w | \in L^2(|y|^{\alpha},E) \}$$, which we endow with the norm ‖w‖H1(|y|α,E)=(‖w‖L2(|y|α,E)2+‖∇w‖L2(|y|α,E)2)1/2. (2.1) Since $$\alpha = 1-2s \in (-1,1)$$, the weight $$|y|^\alpha$$ belongs to the Muckenhoupt class $$A_2(\mathbb{R}^{n+1})$$ (see Turesson, 2000; Duoandikoetxea, 2001). Consequently, $$H^1(|y|^{\alpha},D)$$ is Hilbert and $$C^{\infty}(\mathscr{D}) \cap H^1(|y|^{\alpha},D)$$ is dense in $$H^1(|y|^{\alpha},D)$$ (Gol’dshtein & Ukhlov, 2009, Theorem 1; cf. Turesson, 2000, Proposition 2.1.2, Corollary 2.1.6). The natural space to seek a weak solution for problem (1.5) is HL1∘(yα,C):={w∈H1(yα,C):w=0 on ∂LC}. We recall the following weighted Poincaré inequality (Nochetto et al., 2015, inequality 2.21)): ‖w‖L2(yα,C)≲‖∇v‖L2(yα,C)∀w∈HL1∘(yα,C). This yields that the seminorm on $$\overset{\circ}{H_L^1}(y^{\alpha},\mathscr{C})$$ is equivalent to (2.1). For $$w \in H^1(y^{\alpha},\mathscr{C})$$, $$\text{tr}_{\it {\Omega}} w$$ denotes its trace onto $${\it {\Omega}} \times \{ 0 \}$$. We recall (Capella et al., 2011, Prop. 2.1; Nochetto et al., 2015, Prop. 2.5) trΩHL1∘(yα,C)=Hs(Ω),‖trΩw‖Hs(Ω)≤CtrΩ‖w‖HL1∘n(yα,C). (2.2) We must mention that $$C_{\text{tr}_{\it {\Omega}}} \leq d_s^{-{1}/{2}}$$ (Chen et al., 2015, Section 2.3), where $$d_s = 2^{\alpha}\Gamma(1-s)/\Gamma(s)$$. This estimate will be useful in the analysis of the proposed a posteriori error indicator. We conclude this section with the fundamental result by Caffarelli & Silvestre (2007) (see also Stinga & Torrea, 2010; Capella et al., 2011). If $$\mathsf{u} \in \mathbb{H}^s({\it {\Omega}})$$ and $$\mathscr{U} \in \overset{\circ}{H_L^1}(y^{\alpha},\mathscr{C})$$ solve (1.3) and (1.5), respectively, then ds(−Δ)su=∂ναU=−limy→0+yαUy in the sense of distributions. Here, $$s \in (0,1)$$ and $$\alpha = 1-2s \in (-1,1)$$. 3. A priori error estimates In an effort to make this work self-contained, in this section, we review the results of Antil & Otárola (2015), where an a priori error analysis for a fully discrete approximation of the fractional optimal control problem is investigated. This will also serve to make clear the limitations of this theory. 3.1 The extended optimal control problem We start by recalling an equivalent problem to (1.2)–(1.4): the extended optimal control problem. The main advantage of this problem is its local nature and it is based on the Cafarelli–Silvestre extension result. To describe it, we define the set of admissible controls as Zad={w∈L2(Ω):a(x′)≤w(x′)≤b(x′) a.e x′∈Ω}, (3.1) where $$\mathsf{a},\mathsf{b} \in L^2({\it {\Omega}})$$ and satisfy the property $$\mathsf{a}(x') \leq \mathsf{b}(x')$$ a.e. $$x' \in {\it {\Omega}}$$. The extended optimal control problem is then defined as follows: find $$\text{min } J(\text{tr}_{\it {\Omega}} \mathscr{U},\mathsf{z}),$$ subject to the linear state equation a(U,ϕ)=⟨z,trΩϕ⟩∀ϕ∈HL1∘(yα,C) (3.2) and the control constraints $$ \mathsf{z} \in \mathsf{Z}_{\textrm{ad}}. $$ The functional $$J$$ is defined by (1.2) with $$\mathsf{u}_d \in L^2({\it {\Omega}})$$ and $$\mu >0$$. For $$w, \phi \in \overset{\circ}{H_L^1}(y^{\alpha},\mathscr{C})$$, the bilinear form $$a$$ is defined by a(w,ϕ)=1ds∫Cyα∇w⋅∇ϕ and $$\langle \cdot, \cdot \rangle$$ denotes the duality pairing between $$\mathbb{H}^s({\it {\Omega}})$$ and $$\mathbb{H}^{-s}({\it {\Omega}})$$ which, as a consequence of (2.2), is well defined for $$\mathsf{z} \in \mathbb{H}^{-s}({\it {\Omega}})$$ and $$\phi \in \overset{\circ}{H_L^1}(y^{\alpha},\mathscr{C})$$. The extended optimal control problem has a unique optimal solution $$(\bar{\mathscr{U}},\bar{\mathsf{z}}) \in \overset{\circ}{H_L^1}(y^{\alpha},\mathscr{C}) \times \mathbb{H}^s({\it {\Omega}})$$Antil & Otárola, 2015, Theorem 3.11) and is equivalent to the fractional optimal control problem: $$\text{tr}_{\it {\Omega}} \bar{\mathscr{U}} = \bar{\mathsf{u}}$$ (Antil & Otárola, 2015, Theorem 3.12). 3.2 The truncated optimal control problem Since $$\mathscr{C}$$ is unbounded, problem (3.2) cannot be directly approximated with finite-element-like techniques. However, as Nochetto et al. (2015, Proposition 3.1) show, the solution $$\mathscr{U}$$ of problem (3.2) decays exponentially in the extended variable $$y$$. This suggests we consider a truncated optimal control problem, which is based on a truncation of the state equation (3.2). To describe it, we define and for all , the bilinear form $$ \ $$ (3.3) The truncated optimal control problem is then defined as follows: find $$ \text{min } J(\text{tr}_{\it {\Omega}} v,\mathsf{r}) $$ subject to the truncated state equation $$ \ $$ (3.4) and the control constraints $$ \mathsf{r} \in \mathsf{Z}_{\textrm{ad}}. $$ The existence and uniqueness of an optimal pair follows from Antil & Otárola (2015, Theorem 4.5). In addition, we have that the optimal control $$\bar{\mathsf{r}} \in \mathsf{Z}_{\textrm{ad}}$$ satisfies the variational inequality (trΩp¯+μr¯,r−r¯)L2(Ω)≥0∀r∈Zad, (3.5) where denotes the optimal adjoint state and solves $$ \ $$ (3.6) The following approximation properties follow from Antil & Otárola (2015, Lemma 4.6): if $$(\bar{\mathscr{U}},\bar{\mathsf{z}}) \in \overset{\circ}{H_L^1}(y^{\alpha},\mathscr{C}) \times \mathbb{H}^s({\it {\Omega}})$$ and solve the extended and truncated optimal control problems, respectively, then where $$\lambda_1$$ denotes the first eigenvalue of the operator $$-{\it {\Delta}}$$. 3.3 A fully discrete scheme In this section we recall the fully discrete scheme, proposed in Antil & Otárola (2015, Section 5.3), that approximates the solution to (1.2)–(1.4). We also review its a priori error analysis. To do so in this section, and this section only, we will assume the following regularity result, which is valid if, for instance, the domain $${\it {\Omega}}$$ is convex (Grisvard, 1985): ‖w‖H2(Ω)≲‖Δx′w‖L2(Ω)∀w∈H2(Ω)∩H01(Ω). (3.7) The analysis of the fully discrete scheme of Antil & Otárola (2015, Section 5.3) relies on the regularity properties of the optimal pairs $$(\bar{\mathscr{U}},\bar{\mathsf{z}})$$ and $$(\bar v, \bar{\mathsf{r}})$$ that solve the extended and truncated optimal control problems, respectively. We review such regularity properties in what follows. The results of Nochetto et al. (2015, Theorem 2.7) reveal that the second-order regularity of $$\mathscr{U}$$, solving (3.2), is much worse in the extended direction, namely ‖Δx′U‖L2(yα,C)+‖∂y∇x′U‖L2(yα,C) ≲‖z‖H1−s(Ω), (3.8) ‖Uyy‖L2(yβ,C) ≲‖z‖L2(Ω), (3.9) where $$\beta > 2\alpha + 1$$. These results are also valid for the solution $$v$$ of problem (3.4) (see Nochetto et al., 2016a, Remark 25). The estimates (3.8) and (3.9) have important consequences in the design of efficient numerical techniques to solve (3.2); they suggest that a graded mesh in the extended $$(n+1)$$ dimension must be used (Nochetto et al., 2015, Section 5). We recall the construction of the mesh over used in Antil & Otárola (2015) and Nochetto et al. (2015). First, we consider a graded partition of the interval with mesh points $$ \ $$ (3.10) and $$\gamma > 3/(1-\alpha)=3/(2s) > 1$$. Second, we consider $$\mathscr{T}_{{\it {\Omega}}} = \{ K \}$$ to be a conforming mesh of $${\it {\Omega}}$$, where $$K \subset \mathbb{R}^n$$ is an element that is isoparametrically equivalent either to the unit cube $$[0,1]^n$$ or the unit simplex in $$\mathbb{R}^n$$. We denote by $$\mathbb{T}_{{\it {\Omega}}}$$ the collections of all conforming refinements of an original mesh $$\mathscr{T}_{{\it {\Omega}}}^0$$. We assume that $$\mathbb{T}_{{\it {\Omega}}}$$ is shape regular (see Ciarlet, 2002). We then construct a mesh over as the tensor product triangulation of $$\mathscr{T}_{{\it {\Omega}}} \in \mathbb{T}_{{\it {\Omega}}}$$ and . We denote by $$\mathbb{T}$$ the set of all the meshes obtained with this procedure, and recall that $$\mathbb{T}$$ satisfies the following weak shape-regularity condition: if $$T_1 = K_1 \times I_1$$ and have nonempty intersection then there exists a positive constant such that $$ \ $$ (3.11) where $$h_I = |I|$$. This weak shape-regularity condition allows for anisotropy in the extended variable $$y$$ (Durán & Lombardi, 2005; Nochetto et al., 2015, 2016b). For , we define the finite element space $$ \ $$ (3.12) where is the Dirichlet boundary. The space $$\mathscr{P}_1(K)$$ is $$\mathbb{P}_1(K)$$—the space of polynomials of degree at most $$1$$, when the base $$K$$ of $$T = K \times I$$ is a simplex. If $$K$$ is a cube, $$\mathscr{P}_1(K)$$ stand for $$\mathbb{Q}_1(K)$$—the space of polynomials of degree not larger that $$1$$ in each variable. We also define the space , which is simply a $$\mathscr{P}_1$$ finite element space over the mesh $$\mathscr{T}_{\it {\Omega}}$$. Before describing the numerical scheme introduced and developed in Antil & Otárola (2015), we recall the regularity properties of the extended and truncated optimal controls $$\bar{\mathsf{z}}$$ and $$\bar{\mathsf{r}}$$, respectively. If $$\mathsf{u}_{d} \in \mathbb{H}^{1-s}({\it {\Omega}})$$ and $$\mathsf{a} \leq 0 \leq \mathsf{b}$$ for $$s \in (0,\tfrac{1}{2}]$$ then $$\bar{\mathsf{z}} \in H^1({\it {\Omega}}) \cap \mathbb{H}^{1-s}({\it {\Omega}})$$ (see Antil & Otárola, 2015, Lemmas 3.5 and 5.9). Under the same framework, we have the same result for the truncated optimal control: $$\bar{\mathsf{r}} \in H^1({\it {\Omega}}) \cap \mathbb{H}^{1-s}({\it {\Omega}})$$ (see Otárola, 2016, Proposition 4.1). After all this preparation, we are ready to describe the fully discrete scheme of Antil & Otárola (2015) to approximate the fractional optimal control problem. The fully discrete optimal control problem reads as follows: $$\min J(\text{tr}_{\it {\Omega}} V, Z),$$ subject to the discrete state equation $$ \ $$ (3.13) and the discrete control constraints $$ Z \in \mathbb{Z}_{\rm ad}(\mathscr{T}_{{\it {\Omega}}}). $$ We recall that the functional $$J$$, the bilinear form and the discrete space are defined by (1.1), (3.3) and (3.12), respectively. The discrete and admissible set of controls is defined by Zad(TΩ)=Zad∩{Z∈L∞(Ω):Z|K∈P0(K)∀K∈TΩ}, i.e., the space of piecewise constant functions defined on the partition $$\mathscr{T}_{{\it {\Omega}}}$$ that verifies the control bounds, which we assume to be real constants. The existence and uniqueness of an optimal pair solving the aforementioned problem is standard (Antil & Otárola, 2015, Theorem 5.15). In addition, the optimal control $$\bar Z \in \mathbb{Z}_{\rm ad}(\mathscr{T}_{{\it {\Omega}}})$$ is uniquely characterized by the variational inequality (trΩP¯+μZ¯,Z−Z¯)L2(Ω)≥0∀Z∈Zad(TΩ), (3.14) where the optimal and discrete adjoint state solves $$ \ $$ (3.15) With the discrete solution at hand, we define U¯:=trΩV¯ (3.16) and thus obtain a fully discrete approximation $$(\bar{U},\bar{Z}) \in \mathbb{U}(\mathscr{T}_{{\it {\Omega}}}) \times \mathbb{Z}_{\rm ad}(\mathscr{T}_{{\it {\Omega}}})$$ of the optimal pair $$(\bar{\mathsf{u}},\bar{\mathsf{z}}) \in \mathbb{H}^s({\it {\Omega}}) \times \mathsf{Z}_{\textrm{ad}}$$ solving the fractional optimal control problem. To write the a priori error estimates for the fully discrete optimal control problem, we notice that and that $$\# \mathscr{T}_{\it {\Omega}} \approx M^n$$ implies . Consequently, if $$\mathscr{T}_{\it {\Omega}}$$ is quasi-uniform, we have that $$h_{\mathscr{T}_{{\it {\Omega}}}} \approx (\# \mathscr{T}_{{\it {\Omega}}})^{-1/n}$$. We then have the following result (Antil & Otárola, 2015, Corollary 5.17). Theorem 3.1 (Fractional control problem: error estimate) Let $$(\bar{V},\bar{Z}) $$ solve the fully discrete control problem and $$\bar{U} \in \mathbb{U}(\mathscr{T}_{{\it {\Omega}}})$$ be defined as in (3.16). If $${\it {\Omega}}$$ verifies (3.7), $$\mathsf{u}_d \in \mathbb{H}^{1-s}({\it {\Omega}})$$, $$\mathsf{a}, \mathsf{b} \in \mathbb{R}$$ and $$\mathsf{a} < 0 < \mathsf{b}$$ for $$s \in (0,\tfrac{1}{2}]$$ then we have $$ \ $$ (3.17) and $$ \ $$ (3.18) provided . Remark 3.2 (Domain and data regularity) The results of Theorem 3.1 are valid if and only if $${\it {\Omega}}$$ is such that (3.7) holds, $$\mathsf{u}_d \in \mathbb{H}^{1-s}({\it {\Omega}})$$, $$\mathsf{a}, \mathsf{b} \in \mathbb{R}$$ and $$\mathsf{a} < 0 < \mathsf{b}$$ for $$s \in (0,\tfrac{1}{2}]$$. 4. A posteriori error analysis The design and analysis of a posteriori error estimators for linear second-order elliptic boundary value problems on isotropic discretizations, i.e., meshes where the aspect ratio of all cells is bounded independently of the refinement level, has achieved a certain degree of maturity. Starting with the pioneering work of Babuška & Rheinboldt (1978), a great deal of work has been devoted to its study. We refer the reader to Verfürth (1996), Ainsworth & Oden (2000), Babuška & Strouboulis (2001), Morin et al. (2003), Nochetto et al. (2009) and Nochetto & Veeser (2012) for an up-to-date discussion including also the design of AFEMs, their convergence and optimal complexity. In contrast to this well-established theory, the a posteriori error estimation on anisotropic discretizations, i.e., meshes where the cells have disparate sizes in each direction, is still not completely understood. To the best of our knowledge, the first work that introduces an a posteriori error estimator on anisotropic meshes is Siebert (1996). The analysis provided in this work relies on certain assumptions on the mesh (Siebert, 1996, Section 2), on the exact solution (Siebert, 1996, Definition 3.1) and on the discrete solution (Siebert, 1996, Definition 5.2). However, no explicit examples of AFEMs satisfying these assumptions are provided and their construction is not evident. Afterward, the so-called matching function is introduced in Kunert (2000) and Kunert & Verfürth (2000) for deriving error indicators on anisotropic meshes. The presented analysis relies on the correct alignment of the grid with the exact solution. Indeed, the upper bound for the error involves the matching function, which depends on the error itself, and then it does not provide a real computable quantity (see Kunert, 2000, Theorem 2; Kunert & Verfürth, 2000, Theorem 5.1). The effect of approximating the matching function with a recovered-gradient-based technique is discussed in Kunert (2000) and Kunert & Verfürth (2000). To the best of our knowledge, the first article that attempts to deal with an anisotropic a posteriori error estimator for an optimal control problem is Picasso (2006). In this work, the author proposes, based on the the goal-oriented approach developed in Becker et al. (2000), an anisotropic error indicator for a parabolic optimal control problem involving the heat equation. However, the presented upper bound for the error (Picasso, 2006, Proposition 7) depends on the exact solution and therefore, it is not computable; see the discussion in Picasso (2006, Section 5). Later, Micheletti & Perotto (2011) present an anisotropic a posteriori error estimator for an optimal control problem of a scalar advection–reaction–diffusion equation. The analysis relies on the goal-oriented approach of Becker et al. (2000) and the a priori and a posteriori error analyses of Formaggia & Perotto (2001, 2003), respectively. The presented upper bound for the error depends on the exact optimal variables and therefore is not computable (Micheletti & Perotto, 2011,Proposition 3.5). This shortcoming is circumvented, computationally, by invoking a suitable recovery procedure. The main contribution of this work is the design and study of an a posteriori error indicator for the fractional optimal control problem (1.2)–(1.4). To accomplish this task, we invoke the a posteriori error indicator developed in Chen et al. (2015) that is based on the solution of local problems on stars; we remark that since problems (3.4) and (3.6) involve the coefficient $$y^{\alpha}$$$$(-1 < \alpha < 1)$$ that is not uniformly bounded, the usual residual estimator does not apply. The idea of working on stars goes back to Babuška & Miller (1987), who introduced local Dirichlet problems. Later, Carstensen&Funken (1999) and Morin et al. (2003) proposed solving local weighted problems on stars that deliver rather good effectivity indices. A convergence proof of AFEM driven by such error indicators is provided in Morin et al. (2003) for a Poisson problem and in Cascón & Nochetto (2012) for a general second-order elliptic partial differential equation (PDE); the latter also includes optimal complexity. We also refer the reader to Bank & Weiser (1985) for estimators based on solving Neumann problems on elements and their improvements via the so-called flux equilibration procedure (Ainsworth & Oden, 2000; see also Allendes et al., 2016) for applications of this procedure to a PDE-constrained optimization problem. Concerning the a posteriori error analysis for (1.2)–(1.4), we first propose and explore an ideal anisotropic error indicator that is constructed on the basis of solving local problems on cylindrical stars. This indicator is able to deal with both the coefficient $$y^{\alpha}$$ and the anisotropic mesh . Under a computationally implementable geometric condition imposed on the mesh, which does not depend on the exact optimal variables, we derive the equivalence between the ideal estimator and the error without oscillation terms. This ideal indicator sets the basis for defining a computable error estimator, which, under certain assumptions, is equivalent to the error up to data oscillations terms. 4.1 Preliminaries Let us begin the discussion on a posteriori error estimation with some terminology and notation that follows from Chen et al. (2015). We recall that the mesh over is obtained as the tensor product triangulation of $$\mathscr{T}_{{\it {\Omega}}} \in \mathbb{T}_{{\it {\Omega}}}$$ and and that $$ \mathscr{T}_{{\it {\Omega}}} = \{ K \}$$, where $$K$$ is isoparametrically equivalent to $$[0,1]^n$$ or the unit simplex in $$\mathbb{R}^n$$. Given a node $$z$$ on the mesh , we write $$z = (z',z''),$$ where $$z'$$ and $$z''$$ are nodes on the meshes $$\mathscr{T}_{{\it {\Omega}}}$$ and , respectively. Given a cell $$K \in \mathscr{T}_{{\it {\Omega}}}$$, we denote by the set of nodes of $$K$$ and by the set of interior nodes, i.e., . With this notation at hand, we define and Given , we define , and then and accordingly. Given , we define the star around $$z'$$ as Sz′=⋃{K∈TΩ: K∋z′}⊂Ω and the cylindrical star around $$z'$$ as $$ \ $$ (4.1) Given $$K \in \mathscr{T}_{{\it {\Omega}}}$$, we define its patch as $$ S_K := \bigcup_{z' \in K} S_{z'}. $$ For , its patch $$S_T$$ is defined similarly. Given we define its cylindrical patch as $$ \ $$ For each , we set $$h_{z'} := \min\{h_{K}: K \ni z' \}$$. 4.2 Local weighted Sobolev spaces To define the a posteriori error estimator proposed in this work, we need to introduce some local weighted Sobolev spaces. Definition 4.1 (Local spaces) Given , we define W(Cz′)={w∈H1(yα,Cz′):w=0 on ∂Cz′∖Ω×{0}}, (4.2) where $$\mathscr{C}_{z'}$$ denotes the cylindrical star around $$z'$$ defined in (4.1). Since $$y^\alpha$$ belongs to the class $$A_2(\mathbb{R}^{n+1})$$ (see Muckenhoupt, 1972; Duoandikoetxea, 2001), the space $$\mathbb{W}(\mathscr{C}_{z'})$$ is Hilbert. In addition, we have the following weighted Poincaré-type inequality (Chen et al., 2015, Proposition 5.8): if $$w \in \mathbb{W}(\mathscr{C}_{z'})$$ then $$ \ $$ (4.3) where denotes the truncation parameter introduced in Section 3.2. We also have the following trace inequality that follows from Capella et al. (2011, Proposition 2.1): if $$w \in \mathbb{W}(\mathscr{C}_{z'})$$ then ‖trΩw‖L2(Sz′)≤CtrΩ‖∇w‖L2(yα,Cz′). (4.4) We notice that the same arguments as Chen et al. (2015, Section 2.3) yield $$C_{\text{tr}_{\it {\Omega}}} \leq d_s^{-{1}/{2}}$$. 4.3 An ideal a posteriori error estimator On the basis of the notation introduced in Sections 4.1 and 4.2, we propose and analyse an ideal a posteriori error estimator for the fractional optimal control problem (1.2)–(1.4). The proposed error indicator is ideal because it is not computable: it is based on the resolution of local problems on infinite- dimensional spaces. However, it provides the intuition required to define a discrete and computable error indicator, as is explained in Section 4.4. The construction of this ideal indicator allows for the anisotropic meshes defined in Section 3 and the nonuniformly coefficient $$y^{\alpha}$$ of problem (3.2). We prove that it is equivalent to the error without oscillation terms. The ideal error indicator is defined as the sum of three contributions: $$ \ $$ (4.5) where corresponds to the anisotropic mesh constructed in Section 3.3 and $$\bar{V}$$, $$\bar{P}$$ and $$\bar{Z}$$ denote the optimal variables solving the fully discrete optimal control problem described in Section 3.3. We now proceed to describe each contribution in (4.5) separately. To accomplish this task, we introduce, for $$w,\psi \in \mathbb{W}(\mathscr{C}_{z'})$$, the bilinear form az′(w,ψ)=1ds∫Cz′yα∇w⋅∇ψ. (4.6) Then, the first contribution in (4.5) is defined on the basis of the indicator developed in Chen et al. (2015, Section 5.3). We define $$\zeta_{z'} \in \mathbb{W}(\mathscr{C}_{z'})$$ as the solution to az′(ζz′,ψ)=⟨Z¯,trΩψ⟩−az′(V¯,ψ)∀ψ∈W(Cz′), (4.7) where we recall that the space $$\mathbb{W}(\mathscr{C}_{z'})$$ is defined in (4.2). With this definition at hand, we then define the local error estimator EV(V¯,Z¯;Cz′):=‖∇ζz′‖L2(yα,Cz′) (4.8) and the global error estimator We now describe the second contribution in (4.5). To accomplish this task, we define $$\chi_{z'} \in \mathbb{W}(\mathscr{C}_{z'})$$ as the solution to the local problem az′(χz′,ψ)=⟨trΩV¯−ud,trΩψ⟩−az′(P¯,ψ)∀ψ∈W(Cz′). (4.9) We then define the local error indicator EP(P¯,V¯;Cz′):=‖∇χz′‖L2(yα,Cz′) (4.10) and the global error indicator . Finally, we define a global error estimator for the optimal control as follows: EZ(Z¯,P¯;TΩ):=(∑K∈TΩEZ2(Z¯,P¯;K))1/2 (4.11) with the local error indicators EZ(Z¯,P¯;K):=‖Z¯−Π(−1μtrΩP¯)‖L2(K). (4.12) In (4.12), $${\it \Pi}: L^2({\it {\Omega}}) \rightarrow \mathsf{Z}_{\textrm{ad}}$$ denotes the nonlinear projection operator defined by Π(w)=min{b,max{a,w}}, (4.13) where $$\mathsf{a}$$ and $$\mathsf{b}$$ denote the control bounds defining the set $$\mathsf{Z}_{\textrm{ad}}$$ in (3.1). To invoke the results of Chen et al. (2015, Section 5.3), we introduce an implementable geometric condition that will allow us to consider graded meshes in $${\it {\Omega}}$$ while preserving the anisotropy in the extended direction $$y$$ that is necessary to retain optimal orders of approximation. The flexibility of having graded meshes in $${\it {\Omega}}$$ is essential for compensating some possible singularities in the $$x'$$-variables. We thus assume the following condition over the family of triangulations $$\mathbb{T}$$: there exists a positive constant $$C_{\mathbb{T}}$$ such that, for every mesh , we have $$ \ $$ (4.14) for all interior nodes $$z'$$ of $$\mathscr{T}_{{\it {\Omega}}}$$. Here, denotes the largest size in the $$y$$-direction. We remark that this condition is fully implementable. We now derive an estimate of the energy error in terms of the total error estimator $$\mathscr{E}_{\textrm{ocp}}$$ defined in (4.5) (reliability). Theorem 4.2 (Global upper bound) Let be the solution to the optimality system associated with the truncated optimal control problem defined in Section 3.2 and its numerical approximation defined in Section 3.3. If (4.14) holds then $$ \ $$ (4.15) where the hidden constant is independent of the continuous and discrete optimal variables and the size of the elements in the meshes $$\mathscr{T}_{{\it {\Omega}}}$$ and . Proof The proof involves six steps. Step 1. With the definition (4.12) of the local error indicator $$\mathscr{E}_{Z}$$ in mind, we define the auxiliary control $$\tilde{\mathsf{r}} = {\it \Pi} (-\frac{1}{\mu} \text{tr}_{\it {\Omega}} \bar{P})$$ and notice that it verifies (trΩP¯+μr~,r−r~)L2(Ω)≥0∀r∈Zad. (4.16) Then, an application of the triangle inequality yields ‖r¯−Z¯‖L2(Ω)≤‖r¯−r~‖L2(Ω)+‖r~−Z¯‖L2(Ω). (4.17) We notice that the second term on the right-hand side of the previous inequality corresponds to the definition of the global indicator (4.11). Thus, it suffices to bound the first term, i.e., $$\| \bar{\mathsf{r}} - \tilde{\mathsf{r}} \|_{L^2({\it {\Omega}})}$$. Step 2. Set $${\mathsf{r}} = \tilde {\mathsf{r}}$$ in (3.5) and $${\mathsf{r}} = \bar{{\mathsf{r}}}$$ in (4.16). Adding the obtained inequalities we arrive at μ‖r¯−r~‖L2(Ω)2≤(trΩ(p¯−P¯),r~−r¯)L2(Ω), (4.18) where $$\bar{p}$$ and $$\bar{P}$$ solve (3.6) and (3.15), respectively. To control the right-hand side of this expression, we introduce the auxiliary adjoint state $$q$$ as follows: $$ \ $$ (4.19) By writing $$\bar{p} - \bar{P} = (\bar{p}-q) + (q - \bar{P})$$, the estimate (4.18) immediately yields μ‖r¯−r~‖L2(Ω)2≤(trΩ(p¯−q),r~−r¯)L2(Ω)+(trΩ(q−P¯),r~−r¯)L2(Ω). (4.20) We conclude this step by noticing that, by construction, the solution to problem (3.15) corresponds to the Galerkin approximation of the solution to (4.19). Then, Chen et al. (2015, Proposition 5.14) yields $$ \ $$ (4.21) where in the first inequality we used the trace estimate (2.2) and in last one we used Young’s inequality; we notice that $$\mathfrak{C} = C^2 C_{\text{tr}_{\it {\Omega}}}^2/ \mu $$, where $$C$$ denotes the constant that appears in the estimate of Chen et al. (2015, Proposition 5.14) and $$C_{\text{tr}_{\it {\Omega}}}$$ is the one in (2.2). Step 3. The goal of this step is to bound the term $$\mathrm{I}:= (\text{tr}_{\it {\Omega}}(\bar{p} - q), \tilde{{\mathsf{r}}} - \bar{{\mathsf{r}}})_{L^2({\it {\Omega}})}$$. To accomplish this task, we introduce another auxiliary adjoint state, $$ \ $$ (4.22) where $$\tilde v$$ is defined as the unique solution to $$ \ $$ (4.23) and $$\tilde{{\mathsf{r}}} = {\it \Pi} (-\frac{1}{\mu} \text{tr}_{\it {\Omega}} \bar{P})$$. We then write $$\bar{p} - q = (\bar{p} - w) + (w - q)$$ and bound each contribution to the term $$\mathrm{I}$$ separately. To do this, we observe that $$\bar{v} - \tilde{v}$$ solves the problem $$ \ $$ On the other hand, for all , $$\bar{p} - w$$ solves . Combining these two problems, we arrive at $$ \ $$ (4.24) We now estimate the term $$\mathrm{I}_2:= (\text{tr}_{\it {\Omega}}(w - q), \tilde{{\mathsf{r}}} - \bar{{\mathsf{r}}})_{L^2({\it {\Omega}})}$$, where $$w$$ and $$q$$ solve problems (4.22) and (4.19), respectively. We observe that the difference $$w - q$$ solves for all . Thus, the trace estimate (2.2) and the stability of problem (4.19) yield $$ \ $$ (4.25) It suffices to bound the term $$\| \text{tr}_{\it {\Omega}} (\tilde{v}-\bar{V}) \|_{L^2({\it {\Omega}})}$$. To accomplish this task, we invoke the triangle inequality and obtain the estimate $$\| \text{tr}_{\it {\Omega}} (\tilde{v}-\bar{V}) \|_{L^2({\it {\Omega}})} \leq \| \text{tr}_{\it {\Omega}} (\tilde{v}-v^*) \|_{L^2({\it {\Omega}})}+ \| \text{tr}_{\it {\Omega}} (v^*-\bar{V}) \|_{L^2({\it {\Omega}})}$$, where $$v^*$$ denotes the unique solution to the following problem: $$ \ $$ (4.26) Now, we invoke (2.2) and the stability of (4.26) to derive that $$\| \text{tr}_{\it {\Omega}} (\tilde{v}-v^*) \|_{L^2({\it {\Omega}})} \lesssim \| \tilde {\mathsf{r}} - \bar{Z}\|_{L^2({\it {\Omega}})}$$. This, in view of the definition of $$\mathscr{E}_{Z}$$, given by (4.11)–(4.12), yields ‖trΩ(v~−v∗)‖L2(Ω)≲EZ(Z¯,P¯;TΩ). (4.27) To control the remainder term, we observe that problem (3.13) corresponds to the Galerkin discretization of (4.26). Consequently, (2.2) and Chen et al. (2015, Proposition 5.14) yield $$ \ $$ (4.28) In view of (4.25), the collection of estimates (4.27) and (4.28) allows us to obtain where $$C$$ denotes a positive constant. Since (4.24) tells us that $$\mathrm{I}_1 \leq 0$$, we obtain a similar estimate for the term $$\mathrm{I} = \mathrm{I}_1 + \mathrm{I}_2$$. This estimate implies, on the basis of (4.20) and (4.21), the bound which, invoking (4.17), provides an estimate for the error in the control approximation: $$ \ $$ (4.29) Step 4. The goal of this step is to bound the seminorm of $$\nabla(\bar{v}-\bar{V})$$ in terms of the ideal error indicator (4.5). We employ similar arguments to the ones developed in step 2. We write $$\bar{v}-\bar{V} = (\bar{v}- v^*) + (v^*- \bar{V})$$, where $$v^*$$ is defined in (4.26). The stability of problem (4.26) and the estimate (4.29) immediately provide the bound This estimate, combined with (4.28), allows us to conclude that $$ \ $$ (4.30) Step 5. We bound the term . To accomplish this task, we invoke the triangle inequality and write where $$q$$ is defined as in (4.19). Applying the stability of problem (4.19), the trace estimate (2.2) and (4.30), we arrive at On the other hand, since $$\bar P$$, solution to (3.15), corresponds to the Galerkin approximation of $$q$$, the solution to (4.19), we invoke Chen et al. (2015, Proposition 5.14) and conclude that Collecting the derived estimates, we obtain that $$ \ $$ (4.31) Step 6. Finally, the desired estimate (4.15) follows from a simple collection of the estimates (4.29), (4.30) and (4.31). □ We now derive a local lower bound that measures the quality of $$\mathscr{E}_{\textrm{ocp}}$$ (efficiency). To achieve this, we define C(ds,μ)=max{2ds−1,ds−1/2(μ−1+ds−1/2),1+ds−1/2}. (4.32) Theorem 4.3 (Local lower bound) Let be the solution to the optimality system associated with the truncated optimal control problem defined in Section 3.2 and its numerical approximation defined in Section 3.3. Then, EV(V¯,Z¯;Cz′)+EP(P¯,V¯;Cz′)+EZ(Z¯,P¯;Sz′) ≤C(ds,μ)(‖∇(v¯−V¯)‖L2(yα,Cz′)+‖∇(p¯−P¯)‖L2(yα,Cz′)+‖r¯−Z¯‖L2(Sz′)), (4.33) where $$C(d_s,\mu)$$ depends only on $$d_s$$ and the parameter $$\mu$$ and is defined in (4.32). Proof. We proceed in three steps. Step 1. We begin by analysing the efficiency properties of the indicator $$\mathscr{E}_{V}$$ defined, locally, by (4.8). Let . We invoke the fact that $$\zeta_{z'}$$ solves the local problem (4.7) to conclude that EV2(V¯,Z¯;Cz′)=az′(ζz′,ζz′)=⟨r¯,trΩζz′⟩−az′(V¯,ζz′)+⟨Z¯−r¯,trΩζz′⟩. Now, since $$\zeta_{z'}$$ is supported on $$\mathscr{C}_{z'}$$ we can extend it by zero to . As a consequence, $$\zeta_{z'}$$ is a valid test function in (3.4) and thus we have . This, in view of the previous expression, yields EV2(V¯,Z¯;Cz′)=az′(v¯−V¯,ζz′)+⟨Z¯−r¯,trΩζz′⟩. Define $$e_V = \bar{v} - \bar{V}$$, where $$\bar v$$ solves (3.4). Invoking (4.4) with $$C_{\text{tr}_{\it {\Omega}}} \leq d_s^{-{1}/{2}}$$ and a simple application of the Cauchy–Schwarz inequality, we arrive at EV2(V¯,Z¯;Cz′) ≤ds−1‖∇eV‖L2(yα,Cz′)‖∇ζz′‖L2(yα,Cz′)+‖r¯−Z¯‖L2(Sz′)‖trΩζz′‖L2(Sz′) ≤(ds−1‖∇eV‖L2(yα,Cz′)+ds−1/2‖r¯−Z¯‖L2(Sz′))‖∇ζz′‖L2(yα,Cz′). This, in view of definition (4.8), implies the efficiency of $$\mathscr{E}_{V}$$: EV(V¯,Z¯;Cz′)≤ds−1‖∇eV‖L2(yα,Cz′)+ds−1/2‖r¯−Z¯‖L2(Sz′). (4.34) Step 2. In this step, we elucidate the efficiency properties of the indicator $$\mathscr{E}_{P}$$ defined in (4.10). Following the arguments elaborated in step 1, we write EP2(P¯,V¯;Cz′)=⟨trΩ(V¯−v¯),trΩχz′⟩+az′(eP,χz′), where $$\chi_{z'} \in \mathbb{W}(\mathscr{C}_{z'})$$ solves (4.9) and $$e_P := \bar p - \bar P$$. An application of (4.4) with $$C_{\text{tr}_{\it {\Omega}}} \leq d_s^{-{1}/{2}}$$ and the Cauchy–Schwarz inequality yield EP(V¯,Z¯;Cz′)≤ds−1‖∇eV‖L2(yα,Cz′)+ds−1‖∇eP‖L2(yα,Cz′). (4.35) Step 3. The goal of this step is to analyse the efficiency properties of the indicator $$\mathscr{E}_{Z}$$ defined by (4.11)–(4.12). A trivial application of the triangle inequality yields EZ(Z¯,P¯;Sz′)≤‖Z¯−Π(−1μtrΩp¯)‖L2(Sz′)+‖Π(−1μtrΩp¯)−Π(−1μtrΩP¯)‖L2(Sz′), where $${\it \Pi}$$ denotes the nonlinear projector defined by (4.13). Now, in view of the local Lipschitz continuity of $${\it \Pi}$$, the fact that $$\bar{{\mathsf{r}}} = {\it \Pi} (-\tfrac{1}{\mu} \text{tr}_{\it {\Omega}} \bar p)$$ and the trace estimate (4.4) imply that EZ(Z¯,P¯;Sz′)≤‖r¯−Z¯‖L2(Sz′)+ds−1/2μ‖∇eP‖L2(yα,Cz′). (4.36) Step 4. The desired estimate (4.33) follows from a collection of the estimates (4.34), (4.35) and (4.36). This concludes the proof. □ Remark 4.4 (Local efficiency) Examining the proof of Theorem 4.3, we realize that the error indicators $$\mathscr{E}_{V}$$, $$\mathscr{E}_{P}$$ and $$\mathscr{E}_{Z}$$ are locally efficient; see inequalities (4.34), (4.35) and (4.36), respectively. In addition, in all these inequalities, the involved constants are known and depend only on the parameters $$s$$, through the constant $$d_s$$, and $$\mu$$. The key ingredients to derive the local efficiency property of the error estimator $$\mathscr{E}_{Z}$$ are the local Lipschitz continuity of $${\it \Pi}$$ and the trace estimate (4.4). We comment that obtaining local a posteriori error bounds for the discretization of an optimal control problem is not always possible. We refer the reader to Kohls et al. (2014, Remark 3.3) for a thorough discussion on this matter. 4.4 A computable a posteriori error estimator The a posteriori error estimator proposed and analysed in Section 4.3 has an obvious drawback: given a node $$z'$$, its construction requires knowledge of the functions $$\zeta_{z'}$$ and $$\chi_{z'}$$ that solve exactly the infinite-dimensional problems (4.7) and (4.9), respectively. However, it provides intuition and sets the mathematical framework under which we will define a computable and anisotropic a posteriori error estimator. To describe it, we define the following discrete local spaces. Definition 4.5 (Discrete local spaces) For , we define where, if $$K$$ is a quadrilateral, $$\mathscr{P}_2(K)$$ stands for $$\mathbb{Q}_2(K)$$—the space of polynomials of degree not larger than $$2$$ in each variable. If $$K$$ is a simplex, $$\mathscr{P}_2(K)$$ corresponds to $$\mathbb{P}_2(K) \oplus \mathbb{B}(K)$$ where $$\mathbb{P}_2(K)$$ stands for the space of polynomials of total degree at most $$2$$, and $$\mathbb{B}(K)$$ is the space spanned by a local cubic bubble function. With these discrete spaces at hand, we proceed to define the computable counterpart of the error indicator $$\mathscr{E}_{\mathrm{ocp}}$$ given by (4.5). This indicator is defined as $$ \ $$ (4.37) where is the anisotropic mesh defined in Section 3.3 and $$\bar{V}$$, $$\bar{P}$$ and $$\bar{Z}$$ denote the optimal variables solving the fully discrete optimal control problem. To describe the first contribution in (4.37), we define $$\eta_{z'} \in \mathscr{W}(\mathscr{C}_{z'})$$ as the solution to az′(ηz′,W)=⟨Z¯,trΩW⟩−az′(V¯,W)∀W∈W(Cz′). (4.38) We then define the local and computable error estimator, associated to the state equation, as EV(V¯,Z¯;Cz′):=‖∇ηz′‖L2(yα,Cz′) (4.39) and the global error estimator The second contribution in (4.37) is defined on the basis of the discrete object $$\theta_{z'} \in \mathscr{W}(\mathscr{C}_{z'})$$ that solves the following local problem: az′(θz′,W)=⟨trΩV¯−ud,trΩW⟩−az′(P¯,W)∀W∈W(Cz′). (4.40) We thus define the local and computable error indicator EP(P¯,V¯;Cz′):=‖∇θz′‖L2(yα,Cz′) (4.41) and the global error indicator . The third contribution in (4.37), i.e., the error indicator associated with the optimal control $$E_{Z}$$, is defined by (4.11)–(4.12). We now explore the connection between the error estimator $$E_{\textrm{ocp}}$$ and the error. We first obtain a lower bound that does not involve any oscillation term. Theorem 4.6 (Local lower bound) Let be the solution to the optimality system associated with the truncated optimal control problem defined in Section 3.2 and its numerical approximation defined in Section 3.3. Then, EV(V¯,Z¯;Cz′)+EP(P¯,V¯;Cz′)+EZ(Z¯,P¯;Sz′) ≤C(ds,μ)(‖∇(v¯−V¯)‖L2(yα,Cz′)+‖∇(p¯−P¯)‖L2(yα,Cz′)+‖r¯−Z¯‖L2(Sz′)), (4.42) where $$C(d_s,\mu)$$ depends only on $$d_s$$ and the parameter $$\mu$$ and is defined in (4.32). Proof. The proof of the estimate (4.42) repeats the arguments developed in the proof of Theorem 4.3. We analyse the local efficiency of the indicator $$E_V$$ defined in (4.39). To do this, we let . Employing the fact that $$\eta_{z'}$$ solves problem (4.38) and recalling that $$\bar{{\mathsf{r}}}$$ denotes the continuous optimal control, we arrive at EV2(V¯,Z¯;Cz′)=az′(ηz′,ηz′)=⟨r¯,trΩηz′⟩+⟨Z¯−r¯,trΩηz′⟩−az′(V¯,ηz′). Invoking the trace estimate (4.4) with $$C_{\text{tr}_{\it {\Omega}}} \leq d_s^{-{1}/{2}}$$, the fact that $$\bar v$$ solves problem (3.4) and the Cauchy–Schwarz inequality, we obtain EV2(V¯,Z¯;Cz′)≤(ds−1‖∇(v¯−V¯)‖L2(yα,Cz′)+ds−1/2‖r¯−Z¯‖L2(Sz′))‖∇ηz′‖L2(yα,Cz′), which, in light of (4.39), immediately yields the desired result EV(V¯,Z¯;Cz′)≤ds−1‖∇(v¯−V¯)‖L2(yα,Cz′)+ds−1/2‖r¯−Z¯‖L2(Sz′). The efficiency analyses for the contributions $$E_{P}$$ and $$E_{Z}$$ follow similar arguments. We skip details for brevity. □ Remark 4.7 (Strong efficiency) We remark that that the lower bound (4.42) implies a strong concept of efficiency: it is free of any oscillation term and the involved constant $$C(d_s,\mu)$$ is known and given by (4.32). The relative size of the local error indicator dictates mesh refinement regardless of fine structure of the data. The analysis is valid for the family of anisotropic meshes and allows the nonuniformly coefficients involved in problems (3.4) and (3.6). We now proceed to analyse the reliability properties of the anisotropic and computable error indicator $$E_{\textrm{ocp}}$$ defined in (4.37). To achieve this, we introduce the so-called data oscillation. Given a function $$f \in L^2({\it {\Omega}})$$ and , we define the local oscillation of the function $$f$$ as osc(f;Sz′):=hz′s‖f−fz′‖L2(Sz′), (4.43) where $$h_{z'} = \min\{h_{K}: K \ni z' \}$$ and $$f_{z'}|_K \in \mathbb{R}$$ is the average of $$f$$ over $$K$$, i.e., fz′|K:=⨏Kf. (4.44) The global data oscillation is then defined as $$ \ $$ (4.45) To present our results in a concise manner, we define $$D = (\mathsf{u}_d, \text{tr}_{\it {\Omega}} \bar V)$$ and osc(D;Sz′):=osc(ud;Sz′)+osc(trΩV¯;Sz′), (4.46) where $$\mathsf{osc}(\mathsf{u}_d ;S_{z'})$$ and $$\mathsf{osc}(\text{tr}_{\it {\Omega}} \bar V ;S_{z'})$$ are defined in view of (4.43). In fact, osc(ud;Sz′)=hz′s‖ud−udz′‖L2(Sz′),osc(trΩV¯;Sz′)=hz′s‖trΩV¯−trΩV¯z′‖L2(Sz′), where $$\mathsf{u}_{d_{z'}}$$ and $$\text{tr}_{\it {\Omega}} \bar V_{z'}$$ are defined in light of (4.44). We also define the total error indicator $$ \ $$ (4.47) This indicator will be used to mark elements for refinement in the AFEM proposed in Section 5. The following remark is then necessary. Remark 4.8 (Marking) We comment that, in contrast to Cascón et al. (2008), the proposed AFEM will utilize the total error indicator, namely the sum of energy error and oscillation, for marking. This could be avoided if $$E_{\textrm{ocp}}(\bar V, \bar P, \bar Z; \mathscr{C}_{z'}) \geq C \mathsf{osc}(D; S_{z'})$$ for $$C > 0$$. While this property is trivial for the residual estimator with $$C = 1$$, it is in general false for other families of estimators such as the one we are proposing in this work. We refer to Cascón & Nochetto (2012) for a thorough discussion on this matter. Let and, for any $$\mathscr{M} \subset \mathscr{K}_{\mathscr{T}_{{\it {\Omega}}}}$$, we set and $$ \ $$ (4.48) where, we recall that . With these ingredients at hand, we present the following result. Theorem 4.9 (Global upper bound) Let be the solution to the optimality system associated with the truncated optimal control problem defined in Section 3.2 and its numerical approximation defined in Section 3.3. If (4.14) and Chen et al. (2015, Conjecture 5.28) hold then $$ \ $$ (4.49) where the hidden constant is independent of the continuous and discrete optimal variables and the sizes of the elements in the meshes $$\mathscr{T}_{{\it {\Omega}}}$$ and . Proof. The proof of the estimate (4.49) follows closely the arguments developed in the proof of Theorem 4.2; the difference being the use of the computable error indicator $$E_{\textrm{ocp}}$$ instead of the ideal estimator $$\mathscr{E}_{\textrm{ocp}}$$. We start by bounding the error in the control approximation. Defining $$\tilde {\mathsf{r}} = {\it \Pi} (-\frac{1}{\mu} \text{tr}_{\it {\Omega}} \bar P)$$, estimate (4.17) implies that ‖r¯−Z¯‖L2(Ω)≤‖r¯−r~‖L2(Ω)+EZ(Z¯,P¯;TΩ). (4.50) To control the remainder term, we invoke (4.20) with $$q$$ defined by (4.19) and write μ‖r¯−r~‖L2(Ω)2≤(trΩ(p¯−q),r~−r¯)L2(Ω)+(trΩ(q−P¯),r~−r¯)L2(Ω)=I+II. (4.51) To control the term $$\textrm{II}$$, we invoke the fact that $$\bar P$$, the solution to problem (3.15), corresponds to the Galerkin approximation of $$q$$, the solution to problem (4.19). This, in view of Chen et al. (2015, Theorem 5.37), yields $$ \ $$ (4.52) where $$C$$ denotes a positive constant and $$\mathsf{osc}$$ is defined by (4.43) and (4.45). To control the term $$\mathrm{I}$$, we write $$\mathrm{I} = \mathrm{I}_1 + \mathrm{I}_2 := (\text{tr}_{\it {\Omega}}(\bar{p} - w), \tilde {\mathsf{r}} - \bar{{\mathsf{r}}})_{L^2({\it {\Omega}})} + (\text{tr}_{\it {\Omega}}(w - q), \tilde {\mathsf{r}} - \bar{{\mathsf{r}}})_{L^2({\it {\Omega}})}$$, where $$w$$ is defined as in (4.22). Step 3 in the proof of Theorem 4.2 implies that $$\mathrm{I}_1 \leq 0$$. To control the term $$\mathrm{I}_2$$, we invoke (4.25) and write $$ \ $$ (4.53) We now write $$\tilde{v}-\bar{V} = (\tilde{v}-v^*) - (v^* - \bar{V})$$, where $$v^*$$ is defined as in (4.26), and estimate each contribution separately. First, stability of (4.26) yields ‖trΩ(v~−v∗)‖L2(Ω)≲EZ(Z¯,P¯;TΩ). (4.54) Second, since $$\bar V$$ corresponds to the Galerkin approximation of $$v^*$$, Chen et al. (2015, Theorem 5.37) implies the estimate $$ \ $$ (4.55) This, in view of (4.53) and (4.54), implies that where $$C$$ denotes a positive constant. Since $$\mathrm{I}_1 \leq 0$$, a similar estimate holds for $$\mathrm{I} = \mathrm{I}_1 + \mathrm{I}_2$$. This estimate, in conjunction with the previous bound, and the estimates (4.50), (4.51) and (4.52) implies that The estimates for the terms and follow similar arguments to the ones elaborated on in steps 4 and 5 of the proof of Theorem 4.2. For brevity, we skip the rest of the details. □ Remark 4.10 (Chen et al., 2015, Conjecture 5.28) Examining the proof of Theorem 4.9, we realize that the key steps where Chen et al. (2015, Theorem 5.37) is invoked are (4.52) and (4.55). The results of Chen et al. (2015, Theorem 5.37) are valid under the assumption of the existence of an operator $$\mathscr{M}_{z'}$$ that verifies the conditions stipulated in Chen et al. (2015, Conjecture 5.28). The construction of the operator $$\mathscr{M}_{z'}$$ is an open problem. The numerical experiments of Chen et al. (2015, Section 6) provide consistent computational evidence of the existence of $$\mathscr{M}_{z'}$$ with the requisite properties. 5. Numerical experiments In this section we describe a numerical example that illustrates the performance of the proposed error estimator. To accomplish this task, we formulate an AFEM based on the following iterative loop: SOLVE→ESTIMATE→MARK→REFINE. (5.1) 5.1 Design of AFEM We proceed to describe the four modules in (5.1). SOLVE: Given , we compute , the solution to the fully discrete optimal control problem defined in Section 3.3: To solve the minimization problem, we have used the projected Broyden–Fletcher–Goldfarb–Shanno (BFGS) method with Armijo line search (see Kelley, 1999). The optimization algorithm is terminated when the $$\ell^2$$ norm of the projected gradient is less than or equal to $$10^{-5}$$. ESTIMATE: Once a discrete solution is obtained, we compute, for each , the local error indicator $$E_{\textrm{ocp}}$$, which is defined by Eocp(V¯,P¯,Z¯;Cz′)=EV(V¯,Z¯;Cz′)+EP(P¯,V¯;Cz′)+EZ(Z¯,P¯;Sz′), where the indicators $$E_{V}$$, $$E_{P}$$ and $$E_{Z}$$ are defined by (4.39), (4.41) and (4.12), respectively. We then compute the oscillation term (4.46) and construct the total error indicator (4.47): where . For notational convenience, and in view of the fact that we replaced $$\mathscr{C}_{z'}$$ by $$S_{z'}$$ in the previous formula. MARK: Using the so-called Dörfler marking strategy (Dörfler, 1996) (bulk chasing strategy) with parameter $$\theta$$ with $$\theta \in (0,1]$$, we select a set M=MARK({E(V¯,P¯,Z¯;Sz′)}Sz′∈KTΩ,(V¯,P¯,Z¯))⊂KTΩ of minimal cardinality that satisfies E((V¯,P¯,Z¯),M)≥θE((V¯,P¯,Z¯),KTΩ). REFINEMENT:We generate a new mesh $$\mathscr{T}_{\it {\Omega}}'$$ by bisecting all the elements $$K \in \mathscr{T}_{{\it {\Omega}}}$$ contained in $$\mathscr{M}$$ based on the newest vertex bisection method (see Nochetto et al., 2009; Nochetto & Veeser, 2012). We choose the truncation parameter as to balance the approximation and truncation errors (Nochetto et al., 2015, Remark 5.5). The mesh is constructed by the rule (3.10), with the number of degrees of freedom $$M$$ sufficiently large so that (4.14) holds. This is attained by first creating a partition with $$M \approx (\# \mathscr{T}_{\it {\Omega}}')^{1/n}$$ and checking (4.14). If this condition is violated, we increase the number of points until we get the desired result. The new mesh is obtained as the tensor product of $$\mathscr{T}_{\it {\Omega}}'$$ and . 5.2 Implementation The AFEM (5.1) is implemented within the MATLAB software library iFEM (Chen, 2009). All matrices have been assembled exactly. The right-hand sides are computed by a quadrature formula that is exact for polynomials of degree $$4$$. All linear systems were solved using the multigrid method with line smoother introduced and analysed in Chen et al. (2016). To compute the solution $$\eta_{z'}$$ to the discrete local problem (4.38), we proceed as follows: we loop around each node , collect data about the cylindrical star $$\mathscr{C}_{z'}$$ and assemble the small linear system (4.38). This linear system is solved by the built-in direct solver of MATLAB. To compute the solution $$\theta_{z'}$$ to the discrete local problem (4.40) we proceed similarly. All integrals involving only the weight and discrete functions are computed exactly, whereas those also involving data functions are computed elementwise by a quadrature formula that is exact for polynomials of degree 7. For convenience, in the MARK step we change the estimator from starwise to elementwise. To accomplish this task, we first scale the nodalwise estimator as $$E_{\textrm{ocp}}^2(\bar V, \bar P, \bar Z; \mathscr{C}_{z'}) / (\# S_{z'})$$ and then, for each element $$K \in \mathscr{T}_{{\it {\Omega}}}$$, we compute where . The scaling is introduced so that The cellwise data oscillation is now defined as osc(f;K)2:=hK2s‖f−f¯K‖L2(K)2, where $$\bar f_K$$ denotes the average of $$f$$ over the element $$K$$. This quantity is computed using a quadrature formula that is exact for polynomials of degree 7. 5.3 L-shaped domain with incompatible data For our numerical example, we consider the worst possible scenario: (D1) $$\mathsf{a} = 0.1$$, $$\mathsf{b} = 0.3$$. This implies that the optimal control $$\bar{\mathsf{z}} \not \in \mathbb{H}^{1-s}({\it {\Omega}})$$ when $$s \leq \frac{1}{2}$$. We will refer to the optimal control $$\bar{\mathsf{z}}$$ as an incompatible datum for problem (3.4). (D2) $$\mathsf{u}_d = 1$$. This element does not belong to $$\mathbb{H}^{1-s}({\it {\Omega}})$$ when $$s \leq \frac{1}{2}$$. Therefore, for $$s \leq\frac{1}{2}$$, $$\mathsf{u}_d$$ is an incompatible datum for problem (3.6). (D3) $${\it {\Omega}} = (-1,1)^2 \setminus (0,1) \times (-1,0)$$, i.e., an L-shaped domain; see Fig. 2. In view of (D1) and (D2), we conclude that the right-hand sides of the state and adjoint equations, problems (3.4) and (3.6), respectively, are incompatible for $$s \leq \frac{1}{2}$$. As discussed in Nochetto et al. (2015, Section 6.3), at the level of the state equation, this results in lower rates of convergence when quasi-uniform refinement of $${\it {\Omega}}$$ is employed. In addition, we consider a situation where the domain $${\it {\Omega}}$$ is noncovex. As a result, the hypothesis of Theorem 3.17 does not hold and then it cannot be applied. We set $$\mu = 1$$, and we comment that we do not explicitly enforce the mesh restriction (4.14), which shows that this is nothing but an artifact in our theory. As Fig. 1 illustrates, using our proposed AFEM driven by the error indicator (4.47), we can recover the optimal rates of convergence (3.17)–(3.18) for all values of $$s$$ considered: $$s = 0.2, 0.4, 0.6$$ and $$s = 0.8$$. We remark, again, that we are operating under the conditions (D1)–(D3) and then Theorem 3.17 cannot be applied. Since, for $$s\leq\frac{1}{2}$$, the data are incompatible (D1)–(D2), the optimal and adjoint states exhibit boundary layers. To capture them, our AFEM refines near the boundary; see Fig. 2 (middle). In contrast, when $$s > \frac{1}{2}$$, such incompatibilities do not occur and then our AFEM focuses on resolving the reentrant corner; see Fig. 2 (right). The left panel in Fig. 2 shows the initial mesh. We comment that the middle and the right panels are obtained with 17 AFEM cycles. Fig. 1. View large Download slide Computational rate of convergence for our anisotropic AFEM with incompatible right-hand sides for both the state equation and the adjoint equation over an L-shaped domain (nonconvex domain). We consider n = 2. Since the exact solution is not known for this problem, we present the total error estimator with respect to the number of degrees of freedom. In all cases, we recover the optimal rate of convergence . Fig. 1. View large Download slide Computational rate of convergence for our anisotropic AFEM with incompatible right-hand sides for both the state equation and the adjoint equation over an L-shaped domain (nonconvex domain). We consider n = 2. Since the exact solution is not known for this problem, we present the total error estimator with respect to the number of degrees of freedom. In all cases, we recover the optimal rate of convergence . Fig. 2. View largeDownload slide The left panel shows the initial grid. The middle and right panels shows adaptive grids, obtained after 17 refinements, for $$s = 0.2$$ and $$s = 0.8$$, respectively. We consider an L-shaped domain with incompatible right-hand side for the state and adjoint equations. As expected, when $$s \leq 1/2$$ the incompatible data ($$\bar{\mathsf{z}}, \mathsf{u}_d \notin \mathbb{H}^{1-s}({\it {\Omega}})$$) result in boundary layers for both the state and the adjoint state. In order to capture them, our AFEM refines near the boundary. In contrast, when $$s > 1/2$$ the refinement is more pronounced near the reentrant corner; the data $$\bar{\mathsf{z}}$$ and $$\mathsf{u}_d$$ are compatible in this case. Fig. 2. View largeDownload slide The left panel shows the initial grid. The middle and right panels shows adaptive grids, obtained after 17 refinements, for $$s = 0.2$$ and $$s = 0.8$$, respectively. We consider an L-shaped domain with incompatible right-hand side for the state and adjoint equations. As expected, when $$s \leq 1/2$$ the incompatible data ($$\bar{\mathsf{z}}, \mathsf{u}_d \notin \mathbb{H}^{1-s}({\it {\Omega}})$$) result in boundary layers for both the state and the adjoint state. In order to capture them, our AFEM refines near the boundary. In contrast, when $$s > 1/2$$ the refinement is more pronounced near the reentrant corner; the data $$\bar{\mathsf{z}}$$ and $$\mathsf{u}_d$$ are compatible in this case. Remark 5.1 (Extensions) We discuss a few extensions of this work. Cost functional. Let us consider the cost functional J(u,z)=ψ(u)+μ2‖z‖L2(Ω)2; (5.2) see Kohls et al. (2014) for suitable assumptions that $$\psi$$ must satisfy. We comment that it is possible to extend the results of this article to the scenario where (5.2) replaces (1.1). We refer the reader to Kohls et al. (2014) for an a posteriori error analysis for an optimal control problem with (5.2) as a cost functional and local PDEs as constraints. We also mention that we are currently analysing optimal control problems involving fractional diffusion and an $$L^1$$-control cost. The latter term leads to sparsely supported optimal controls, which are desirable, for instance, in actuator placement problems. General operators. Let us consider the second-order, symmetric and uniformly elliptic operator $$\mathscr{L}$$, supplemented with homogeneous Dirichlet boundary conditions: Lw=−divx′(A∇x′w)+cw. (5.3) Denote by $$\mathscr{L}^s$$ a fractional power of $$\mathscr{L}$$ supplemented with Dirichlet boundary conditions; $$s \in (0,1)$$. Thus, the results of this article can be extended to obtain an a posteriori error analysis for the fractional optimal control problem, where the state equation (1.3) is replaced by $$\mathscr{L}^s \mathsf{u} = \mathsf{z}$$ in $${\it {\Omega}}$$. Semilinear equations. Semilinear problems appear in many practical situations, for instance, superconductivity (Tröltzsch, 2010) and fluid dynamics (Gunzburger, 2003). It is thus in our interest to study an optimal control problem involving the fractional semilinear equation (−Δ)su+f(x′,u)=z in Ω,u=0 on ∂Ω. (5.4) We refer the reader to Alibaud & Imbert (2009), Capella et al. (2011) and Antil et al. 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IMA Journal of Numerical Analysis – Oxford University Press

**Published: ** Jan 1, 2018

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