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Computational Optimization and Applications

, Volume 55, Issue 3, pp 769–802 | Cite as

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

  • Dmitriy Leykekhman
  • Dominik Meidner
  • Boris Vexler
Article

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.

Keywords

Optimal control Finite elements Error estimates State constraints 

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Copyright information

© Springer Science+Business Media New York 2013

Authors and Affiliations

  • Dmitriy Leykekhman
    • 1
  • Dominik Meidner
    • 2
  • Boris Vexler
    • 2
  1. 1.Department of MathematicsUniversity of ConnecticutStorrsUSA
  2. 2.Lehrstuhl für Mathematische Optimierung, Fakultät für MathematikTechnische Universität MünchenGarching b. MünchenGermany

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