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Error estimates for the finite element approximation of bilinear boundary control problems

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  • Max Winkler

    (Technische Universität Chemnitz)

Abstract

In this article a special class of nonlinear optimal control problems involving a bilinear term in the boundary condition is studied. These kind of problems arise for instance in the identification of an unknown space-dependent Robin coefficient from a given measurement of the state, or when the Robin coefficient can be controlled in order to reach a desired state. Necessary and sufficient optimality conditions are derived and several discretization approaches for the numerical solution of the optimal control problem are investigated. Considered are both a full discretization and the postprocessing approach meaning that we compute an improved control by a pointwise evaluation of the first-order optimality condition. For both approaches finite element error estimates are shown and the validity of these results is confirmed by numerical experiments.

Suggested Citation

  • Max Winkler, 2020. "Error estimates for the finite element approximation of bilinear boundary control problems," Computational Optimization and Applications, Springer, vol. 76(1), pages 155-199, May.
    • Handle: RePEc:spr:coopap:v:76:y:2020:i:1:d:10.1007_s10589-020-00171-5
    DOI: 10.1007/s10589-020-00171-5
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    References listed on IDEAS

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    1. Egger, H. & Fellner, K. & Pietschmann, J.-F. & Tang, B.Q., 2018. "Analysis and numerical solution of coupled volume-surface reaction-diffusion systems with application to cell biology," Applied Mathematics and Computation, Elsevier, vol. 336(C), pages 351-367.
    2. K. Krumbiegel & J. Pfefferer, 2015. "Superconvergence for Neumann boundary control problems governed by semilinear elliptic equations," Computational Optimization and Applications, Springer, vol. 61(2), pages 373-408, June.
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