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Numerische Mathematik

, Volume 139, Issue 4, pp 913–938 | Cite as

Energy corrected FEM for optimal Dirichlet boundary control problems

  • Lorenz John
  • Piotr Swierczynski
  • Barbara Wohlmuth
Article

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.

Mathematics Subject Classification

49J20 65N12 65N15 65N30 

Notes

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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Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  • Lorenz John
    • 1
  • Piotr Swierczynski
    • 1
  • Barbara Wohlmuth
    • 1
  1. 1.Institute for Numerical MathematicsTechnische Universität MünchenGarching bei MünchenGermany

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