Error estimates for the numerical approximation of Neumann control problems

Casas, Eduardo; Mateos, Mariano

doi:10.1007/s10589-007-9056-6

Cited by 59 publications

(58 citation statements)

References 10 publications

(19 reference statements)

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“…Indeed, to numerically solve a Neumann control problem, piecewise constant or piecewise linear functions are typically taken to approximate the controls. In both cases, the maximal order of the error estimates is h or h 3/2 , respectively; see [3]. A consequence of our estimate is that we also have error estimates of order h or h 3/2 , depending on the type of approximation used for the controls, for a fully discretized control problem using piecewise linear approximation of the states on a polygonal domain.…”

Section: Introductionmentioning

confidence: 78%

“…If Ω is a polygonal domain, then it is covered by the union of the triangles of the mesh, and Γ remains invariable. Then problems (NP) and (DP) are approximated by some discrete problems, and it is possible to estimate the differences ū −ū h L 2 (Γ) between the different solutions of (NP) and (DP) and the corresponding discrete approximations; see [3] or [4] for the Neumann case and [5] for the Dirichlet case. In the problems that we are considering here, the situation is more complicated because the numerical analysis with finite elements requires the approximation of Ω by a new (typically polygonal) domain Ω h , so that the comparison between the solutionsū andū h is more involved becauseū ∈ L 2 (Γ) andū h ∈ L 2 (Γ h ), where Γ h is the boundary of Ω h .…”

Section: Introductionmentioning

confidence: 99%

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Approximation of Boundary Control Problems on Curved Domains

Casas¹,

Sokołowski²

2010

SIAM J. Control Optim.

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Abstract. In this paper we consider boundary control problems associated to a semilinear elliptic equation defined in a curved domain Ω. The Dirichlet and Neumann cases are analyzed. To deal with the numerical analysis of these problems, the approximation of Ω by an appropriate domain Ω h (typically polygonal) is required. Here we do not consider the numerical approximation of the control problems. Instead, we formulate the corresponding infinite dimensional control problems in Ω h , and we study the influence of the replacement of Ω by Ω h on the solutions of the control problems. Our goal is to compare the optimal controls defined on Γ = ∂Ω with those defined on Γ h = ∂Ω h and to derive some error estimates. The use of a convenient parametrization of the boundary is needed for such estimates. 1. Introduction. In this paper we study boundary control problems defined on a curved domain Ω. We start with the Neumann problem (NP) and consider the Dirichlet problem (DP) afterward. To numerically solve these problems, usually it is convenient to approximate Ω by a polygonal domain Ω h , e.g., if finite elements are used for computations. Our goal is to analyze the effect of the domain change on the optimal controls. More precisely, two new optimal control problems (NP h ) and (DP h ) in Ω h are defined. The convergence of global or local solutions of problems (NP h ) and (DP h ) to the corresponding local or global solutions of (NP) and (DP), respectively, is investigated for the limit passage h → 0. The error estimates for the difference of optimal controls obtained for both problems in an appropriate norm are derived in a function of the parameter h. We restrict ourselves to the case of a convex domain Ω ⊂ R 2 approximated by a polygonal domain Ω h ; h is the maximal length of the edges of Ω h . A family of infinite dimensional control problems (NP h ) and (DP h ) defined in Ω h is considered, and the solutions of (NP h ) and (DP h ) are compared with the solutions of (NP) and (DP), respectively. In this way, the influence of small changes in the domain on the solutions of the control problems is analyzed.The numerical computation of the solution of (NP) and (DP) requires the discretization of the respective state equations, typically by using finite elements. If Ω is a polygonal domain, then it is covered by the union of the triangles of the mesh, and Γ

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Section: Introductionmentioning

confidence: 78%

Section: Introductionmentioning

confidence: 99%

Approximation of Boundary Control Problems on Curved Domains

Casas¹,

Sokołowski²

2010

SIAM J. Control Optim.

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“…The space of discretized controls is We denote the solutions of problem (P h ) and (P h ε ) byū h andū h ε respectively. We address the reader to [7] for the details about the optimization process, [4] for theory about continuous piecewise linear approximation of Neumann control problems, and [5] for theory about approximation of Dirichlet control problems.…”

Section: An Examplementioning

confidence: 99%

Penalization of Dirichlet optimal control problems

Casas¹,

Mateos²,

Raymond³

2008

ESAIM: COCV

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Abstract. We apply Robin penalization to Dirichlet optimal control problems governed by semilinear elliptic equations. Error estimates in terms of the penalization parameter are stated. The results are compared with some previous ones in the literature and are checked by a numerical experiment. A detailed study of the regularity of the solutions of the PDEs is carried out.Mathematics Subject Classification. 49M30, 35B30, 35B37.

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“…All the previous papers were devoted to distributed optimal control problems. In [7], the authors proved error estimates of order O(h 3/2 ) for Neumann control problems in a two-dimensional domain by assuming a similar assumption to the one introduced by Rösch. The use of piecewise linear approximations for a Dirichlet control problem was investigated in [8].…”

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confidence: 99%

Approximation of sparse controls in semilinear equations by piecewise linear functions

2012

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Abstract. Semilinear elliptic optimal control problems involving the L 1 norm of the control in the objective are considered. A priori finite element error estimates for piecewise linear discretizations for the control and the state are proved. These are obtained by a new technique based on an appropriate discretization of the objective function. Numerical experiments confirm the convergence rates.Key words. optimal control of partial differential equations, non-differentiable objective, sparse controls, finite element discretization, a priori error estimates, semilinear equations

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Error estimates for the numerical approximation of Neumann control problems

Cited by 59 publications

References 10 publications

Approximation of Boundary Control Problems on Curved Domains

Approximation of Boundary Control Problems on Curved Domains

Penalization of Dirichlet optimal control problems

Approximation of sparse controls in semilinear equations by piecewise linear functions

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