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

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

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Article ID: iaor20133974
Volume: 55
Issue: 3
Start Page Number: 769
End Page Number: 802
Publication Date: Jul 2013
Journal: Computational Optimization and Applications
Authors: , ,
Keywords: error analysis, Lagrangian methods
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.

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