On the approximation of the solution of an optimal control problem governed by an elliptic equation

Geveci, Tunc

doi:10.1051/m2an/1979130403131

Cited by 164 publications

(110 citation statements)

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“…The case of linearquadratic elliptic control problems by finite elements was discussed in early papers by Falk [9], Geveci [10] and Malanowski [16], and Arnautu and Neittaanmäki [2], who already proved the optimal error estimate of order h in the L 2 -norm. In [16], also the case of piecewise linear control functions is addressed.…”

mentioning

confidence: 99%

Error Estimates for the Numerical Approximation of Boundary Semilinear Elliptic Control Problems

Casas

Mateos

Tröltzsch

2005

Comput Optim Applic

122

123

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We study the numerical approximation of boundary optimal control problems governed by semilinear elliptic partial differential equations with pointwise constraints on the control. The analysis of the approximate control problems is carried out. The uniform convergence of discretized controls to optimal controls is proven under natural assumptions by taking piecewise constant controls. Finally, error estimates are established. Abstract. We study the numerical approximation of boundary optimal control problems governed by semilinear elliptic partial differential equations with pointwise constraints on the control. The analysis of the approximate control problems is carried out. The uniform convergence of discretized controls to optimal controls is proven under natural assumptions by taking piecewise constant controls. Finally, error estimates are established.Key words. Boundary control, semilinear elliptic equation, numerical approximation, error estimates AMS subject classifications. 49J20, 49K20, 49M05, 65K101. Introduction. With this paper, we continue the discussion of error estimates for the numerical approximation of optimal control problems we have started for semilinear elliptic equations and distributed controls in [1]. The case of distributed control is the easiest one with respect to the mathematical analysis. In [1] it was shown that, roughly speaking, the distance between a locally optimal controlū and its numerical approximationū h has the order of the mesh size h in the L 2 -norm and in the L ∞ -norm. This estimate holds for a finite element approximation of the equation by standard piecewise linear elements and piecewise constant control functions.The analysis for boundary controls is more difficult, since the regularity of the state function is lower than that for distributed controls. Moreover, the internal approximation of the domain causes problems. In the general case, we have to approximate the boundary by a polygon. This requires the comparison of the original control that is located at the boundary Γ and the approximate control that is defined on the polygonal boundary Γ h . Moreover, the regularity of elliptic equations in domains with corners needs special care. To simplify the analysis, we assume here that Ω is a polygonal domain of R 2 . Though this makes the things easier, the lower regularity of states in polygonal domains complicates, together with the presence of nonlinearities, the analysis.Another novelty of our paper is the numerical confirmation of the predicted error estimates. We present two examples, where we know the exact solutions. The first one is of linear-quadratic type, while the second one is semilinear. We are able to verify our error estimates quite precisely.Let us mention some further papers related to this subject. The case of linearquadratic elliptic control problems by finite elements was discussed in early papers by Falk [9], Geveci [10] and Malanowski [16], and Arnautu and Neittaanmäki [2], who already proved the optimal error estimate of order h in the L 2 -...

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mentioning

confidence: 99%

Error Estimates for the Numerical Approximation of Boundary Semilinear Elliptic Control Problems

Casas

Mateos

Tröltzsch

2005

Comput Optim Applic

122

123

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“…China. 2 Key Laboratory for Nonlinear Science and System Structure, Chongqing Three Gorges University, Chongqing, 404000, P.R. China.…”

Section: Competing Interestsmentioning

confidence: 99%

New a posteriori error estimates for hp version of finite element methods of nonlinear parabolic optimal control problems

Liu

Hou

et al. 2016

J Inequal Appl

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In this paper, we investigate residual-based a posteriori error estimates for the hp version of the finite element approximation of nonlinear parabolic optimal control problems. By using the hp finite element approximation for both the state and the co-state variables and the hp discontinuous Galerkin finite element approximation for the control variable, we derive hp residual-based a posteriori error estimates for both the state and the control approximation. Such estimates, which are apparently not available in the literature, can be used to construct a reliable hp adaptive finite element approximation for the nonlinear parabolic optimal control problems. MSC: 49J20; 65N30Keywords: residual-based a posteriori error estimates; nonlinear parabolic optimal control problems; hp version of finite element method

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“…Convergence and super convergence for finite element approximation of linear or semi-linear elliptic optimization problems are investigated in [1,2] and [3,4], respectively.…”

Section: Introductionmentioning

confidence: 99%

Superconvergence of Variational Discretization for Bilinear Elliptic Optimization Problems

Tang

Hua

2017

Proceedings of the 2017 International Conference on Applied Mathematics, Modelling and Statistics Application (AMMSA 2017)

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Abstract-In this paper, variational discretization approximation of bilinear elliptic optimal control problems is considered. Firstly, we construct a variational discretization method for the bilinear elliptic optimal control problem with control constraints. Secondly, we derive the convergence of the approximation scheme. Thirdly, we analyze the superconvergence. Finally, we present a numerical example to conform our theoretical results.

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On the approximation of the solution of an optimal control problem governed by an elliptic equation

Cited by 164 publications

References 11 publications

Error Estimates for the Numerical Approximation of Boundary Semilinear Elliptic Control Problems

Error Estimates for the Numerical Approximation of Boundary Semilinear Elliptic Control Problems

New a posteriori error estimates for hp version of finite element methods of nonlinear parabolic optimal control problems

Superconvergence of Variational Discretization for Bilinear Elliptic Optimization Problems

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