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Energy corrected FEM for optimal Dirichlet boundary control problems

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Abstract

In the presence of re-entrant corners, the solution of PDE-constrained optimization problems, in general, has singular components, even when the given data are smooth. Consequently, the use of standard finite element discretization gives rise to the so-called ’pollution effect’. This means that only a reduced convergence rate, compared to the best approximation error, is obtained. We discuss how optimal convergence rates in weighted norms can be achieved using the idea of energy corrected finite element methods, applied to optimal Dirichlet boundary control problem in the energy space. We present optimal error estimates in weighted norms for the state variable and for the control. Several numerical examples illustrate the obtained theoretical results.

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Acknowledgements

The financial support by the German Research Foundation (DFG) trough grant WO 671/11-1 and through the International Research Training Group IGDK 1754 “Optimization and Numerical Analysis for Partial Differential Equations with Nonsmooth Structures” is gratefully acknowledged.

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Authors and Affiliations

  1. Institute for Numerical Mathematics, Technische Universität München, 85748, Garching bei München, Germany

    Lorenz John, Piotr Swierczynski & Barbara Wohlmuth

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  1. Lorenz John
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  2. Piotr Swierczynski
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  3. Barbara Wohlmuth
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Correspondence to Piotr Swierczynski.

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John, L., Swierczynski, P. & Wohlmuth, B. Energy corrected FEM for optimal Dirichlet boundary control problems. Numer. Math. 139, 913–938 (2018). https://doi.org/10.1007/s00211-018-0952-8

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