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Optimal error estimates for finite element discretization of elliptic optimal control problems with finitely many pointwise state constraints

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  • Dmitriy Leykekhman
  • Dominik Meidner
  • Boris Vexler

Abstract

In this paper we consider a model elliptic optimal control problem with finitely many state constraints in two and three dimensions. Such problems are challenging due to low regularity of the adjoint variable. For the discretization of the problem we consider continuous linear elements on quasi-uniform and graded meshes separately. Our main result establishes optimal a priori error estimates for the state, adjoint, and the Lagrange multiplier on the two types of meshes. In particular, in three dimensions the optimal second order convergence rate for all three variables is possible only on properly refined meshes. Numerical examples at the end of the paper support our theoretical results. Copyright Springer Science+Business Media New York 2013

Suggested Citation

  • Dmitriy Leykekhman & Dominik Meidner & Boris Vexler, 2013. "Optimal error estimates for finite element discretization of elliptic optimal control problems with finitely many pointwise state constraints," Computational Optimization and Applications, Springer, vol. 55(3), pages 769-802, July.
    • Handle: RePEc:spr:coopap:v:55:y:2013:i:3:p:769-802
    DOI: 10.1007/s10589-013-9537-8
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    References listed on IDEAS

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    1. Eduardo Casas & Mariano Mateos, 2012. "Numerical approximation of elliptic control problems with finitely many pointwise constraints," Computational Optimization and Applications, Springer, vol. 51(3), pages 1319-1343, April.
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    Cited by:

    1. Mariano Mateos & Ira Neitzel, 2016. "Dirichlet control of elliptic state constrained problems," Computational Optimization and Applications, Springer, vol. 63(3), pages 825-853, April.
    2. B. Jadamba & A. Khan & M. Sama, 2017. "Error estimates for integral constraint regularization of state-constrained elliptic control problems," Computational Optimization and Applications, Springer, vol. 67(1), pages 39-71, May.
    3. Francesco Ludovici & Ira Neitzel & Winnifried Wollner, 2018. "A Priori Error Estimates for State-Constrained Semilinear Parabolic Optimal Control Problems," Journal of Optimization Theory and Applications, Springer, vol. 178(2), pages 317-348, August.

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