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

Authors:
Dmitriy Leykekhman;Dominik Meidner;Boris Vexler
Affiliations:
Department of Mathematics, University of Connecticut, Storrs, USA 06269-3009;Lehrstuhl für Mathematische Optimierung, Fakultät für Mathematik, Technische Universität München, Garching b. München, Germany 85748;Lehrstuhl für Mathematische Optimierung, Fakultät für Mathematik, Technische Universität München, Garching b. München, Germany 85748
Venue:
Computational Optimization and Applications
Year:
2013

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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.