Dirichlet control of elliptic state constrained problems

Mateos, Mariano; Neitzel, Ira

doi:10.1007/s10589-015-9784-y

Cited by 15 publications

(24 citation statements)

References 25 publications

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“…We leave this to be considered elsewhere. For a 2D convex polygonal domain and f = 0, we use a recent regularity result of Mateos and Neitzel [32] below to give conditions on the domain and y d to guarantee the solution has the above regularity. For a higher dimensional convex polyhedral domain, the regularity theory is much more complicated, and we do not attempt to provide conditions to guarantee the above regularity in this work.…”

Section: Background: the Optimality System And Regularitymentioning

confidence: 99%

A superconvergent hybridizable discontinuous Galerkin method for Dirichlet boundary control of elliptic PDEs

Shen

Singler

et al. 2019

Numer. Math.

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We begin an investigation of hybridizable discontinuous Galerkin (HDG) methods for approximating the solution of Dirichlet boundary control problems governed by elliptic PDEs. These problems can involve atypical variational formulations, and often have solutions with low regularity on polyhedral domains. These issues can provide challenges for numerical methods and the associated numerical analysis. We propose an HDG method for a Dirichlet boundary control problem for the Poisson equation, and obtain optimal a priori error estimates for the control. Specifically, under certain assumptions, for a 2D convex polygonal domain we show the control converges at a superlinear rate. We present 2D and 3D numerical experiments to demonstrate our theoretical results.

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Section: Background: the Optimality System And Regularitymentioning

confidence: 99%

A superconvergent hybridizable discontinuous Galerkin method for Dirichlet boundary control of elliptic PDEs

Shen

Singler

et al. 2019

Numer. Math.

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“…Researchers have performed numerical analysis of computational methods for Dirichlet boundary control problems for over a decade. Many researchers considered the standard finite element method and obtained an error estimate for the optimal control of order h s for all s < min{1, π/2ω}, where ω is the largest angle of the boundary polygon; see, e.g., [8,36,37]. Apel et al in [1] considered special meshes and obtained an optimal convergence rate with s < min{3/2, π/ω − 1/2}.…”

Section: Introductionmentioning

confidence: 99%

An HDG Method for Dirichlet Boundary Control of Convection Dominated Diffusion PDEs

Chen¹,

Singler²,

Zhang³

2019

SIAM J. Numer. Anal.

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We first propose a hybridizable discontinuous Galerkin (HDG) method to approximate the solution of a convection dominated Dirichlet boundary control problem. Dirichlet boundary control problems and convection dominated problems are each very challenging numerically due to solutions with low regularity and sharp layers, respectively. Although there are some numerical analysis works in the literature on diffusion dominated convection diffusion Dirichlet boundary control problems, we are not aware of any existing numerical analysis works for convection dominated boundary control problems. Moreover, the existing numerical analysis techniques for convection dominated PDEs are not directly applicable for the Dirichlet boundary control problem because of the low regularity solutions. In this work, we obtain an optimal a priori error estimate for the control under some conditions on the domain and the desired state. We also present some numerical experiments to illustrate the performance of the HDG method for convection dominated Dirichlet boundary control problems.

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“…However, to the best of our knowledge, the argumentation is restricted to unconstrained problems. For the error of the controls in L 2 (Γ), the order shown in [9] is not improved.Nevertheless, the regularity of the control and the existing numerical experiments, see [18,17], suggested that for the control variable the order should be greater: h s for all s < min(1, π/ω 1 − 1/2) if one uses standard quasi-uniform meshes, and for all s < min(3/2, π/ω 1 − 1/2) if one uses certain quasi-uniform meshes which allow for superconvergence effects, see Definition 4.5. Our main results, Theorems 4.1 and 5.3, fully explain the observed orders of convergence in the literature for the control variable, improve existing results for the state variable in constrained linear-quadratic problems posed in convex domains, and provide the first available results in nonconvex domains.…”

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confidence: 99%

Error estimates for Dirichlet control problems in polygonal domains: Quasi-uniform meshes

Apel¹,

Mateos²,

Pfefferer³

et al. 2018

Mathematical Control &Amp; Related Fields

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The paper deals with finite element approximations of elliptic Dirichlet boundary control problems posed on two-dimensional polygonal domains. Error estimates are derived for the approximation of the control and the state variables. Special features of unconstrained and control constrained problems as well as general quasiuniform meshes and superconvergence meshes are carefully elaborated. Compared to existing results, the convergence rates for the control variable are not only improved but also fully explain the observed orders of convergence in the literature. Moreover, for the first time, results in non-convex domains are provided. of the adjoint state has a pole there, we getū ∈ H s (Γ) for all s < 3 2 . This regularity is determined by the larges convex angle and by the kinks due to the constraints.Unfortunately, this is not the whole truth. In exceptional cases, e. g. when the data enjoy certain symmetry, the leading singularity of type r 2/3 may not appear in the adjoint state. Instead, the solution may have a r 4/3 -singularity whose normal derivative has a r 1/3 -singularity which is not flattened by the projection Proj [a,b] . The control is less regular,ū ∈ H s (Γ) for all s < 5 6 . See Example 3.6 in [1]. Hence, dealing with these exceptional cases is not fun but necessary. If in the unconstrained case a stress intensity factor vanishes , i.e., the leading singularity does not occur, then the convergence result is still true, one may only see a better convergence in numerical tests. See Figure 3, right hand side and Remark 4.8. However, in the constrained case, the situation is the opposite. The exceptional case leads to the worstcase estimate. To deal with the "worst-case" and the "usual-case" in an unified way, we introduce in (2.4) some numbers related to the singular exponents.We distinguish two cases for the investigation of the discretization errors. After proving a general result in Section 3 we study the unconstrained case in Section 4 and the constrained case in Section 5. We focus on quasi-uniform meshes and distinguish general meshes and certain superconvergence meshes. In order not to overload the present paper, we postpone the study of graded meshes to [2]. The numerical tests in Section 6 confirm the theoretical results.The study of error estimates for Dirichlet control problems posed on polygonal domains can be traced back to [9], where a control constrained problem governed by a semilinear elliptic equation posed in a convex polygonal domain is studied. An order of convergence of h s is proved for all s < min(1, π/(2ω 1 )), where ω 1 is the largest interior angle, in both the control and the state variable. Later, in [18], it is proven that for unconstrained linear problems posed on convex domains, the state variable exhibits a better convergence property. The corresponding proof is based on a duality argument and estimates for the controls in weaker norms than L 2 (Γ). However, to the best of our knowledge, the argumentation is restricted to unconstrained problems. For the error of t...

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Dirichlet control of elliptic state constrained problems

Cited by 15 publications

References 25 publications

A superconvergent hybridizable discontinuous Galerkin method for Dirichlet boundary control of elliptic PDEs

A superconvergent hybridizable discontinuous Galerkin method for Dirichlet boundary control of elliptic PDEs

An HDG Method for Dirichlet Boundary Control of Convection Dominated Diffusion PDEs

Error estimates for Dirichlet control problems in polygonal domains: Quasi-uniform meshes

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