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Petrov-Galerkin Crank-Nicolson Scheme for Parabolic Optimal Control Problems on Nonsmooth Domains

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Trends in PDE Constrained Optimization

Part of the book series: International Series of Numerical Mathematics ((ISNM,volume 165))

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Abstract

In this paper we transfer the a priori error analysis for the discretization of parabolic optimal control problems on domains allowing for H 2 regularity (i.e. either with smooth boundary or polygonal and convex) to a large class of nonsmooth domains. We show that a combination of two ingredients for the optimal convergence rates with respect to the spatial and the temporal discretization is required. First we need a time discretization scheme which has the desired convergence rate in the smooth case. Secondly we need a method to treat the singularities due to non-smoothness of the domain for the corresponding elliptic state equation. In particular we demonstrate this philosophy with a Crank-Nicolson time discretization and finite elements on suitably graded meshes for the spatial discretization. A numerical example illustrates the predicted convergence rates.

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Acknowledgements

The function u s for the numerical example in Sect. 6 was taken from a presentation by T. Apel and J. Pfefferer. The numerical experiments in Sect. 6 are carried out using both a MATLAB-FEM implementation based on [1, 12, 15] and the software packages RODOBO [30] and GASCOIGNE [16].

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Authors and Affiliations

  1. Institut für Mathematik und Bauinformatik, Universität der Bundeswehr München, Werner-Heisenberg-Weg 39, 85577, Neubiberg, Germany

    Thomas G. Flaig

  2. Fakultät für Mathematik, Lehrstuhl für Optimale Steuerung, Technische Universität München, Boltzmannstraße 3, 85748, Garching b. München, Germany

    Dominik Meidner & Boris Vexler

Authors
  1. Thomas G. Flaig
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  2. Dominik Meidner
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  3. Boris Vexler
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Corresponding author

Correspondence to Dominik Meidner .

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Editors and Affiliations

  1. Department Mathematik, Universität Erlangen-Nürnberg, Erlangen, Germany

    Günter Leugering

  2. Institut für Dynamik komplexer technisch, Max-Planck-Institut, Magdeburg, Germany

    Peter Benner

  3. Fakultät Bio- und Chemieingenieurwesen, Technische Universität Dortmund, Dortmund, Germany

    Sebastian Engell

  4. Institut für Mathematik, Humboldt-Universität zu Berlin, Berlin, Germany

    Andreas Griewank

  5. Mathematisches Institut, Universität Basel, Basel, Switzerland

    Helmut Harbrecht

  6. Fachbereich Mathematik Optimierung und Approximation, Universität Hamburg, Hamburg, Germany

    Michael Hinze

  7. Institut für Angewandte Mathematik, Ruprecht-Karls-Universität Heidelberg, Heidelberg, Germany

    Rolf Rannacher

  8. Fachbereich Mathematik, Technische Universität Darmstadt, Darmstadt, Germany

    Stefan Ulbrich

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Flaig, T.G., Meidner, D., Vexler, B. (2014). Petrov-Galerkin Crank-Nicolson Scheme for Parabolic Optimal Control Problems on Nonsmooth Domains. In: Leugering, G., et al. Trends in PDE Constrained Optimization. International Series of Numerical Mathematics, vol 165. Birkhäuser, Cham. https://doi.org/10.1007/978-3-319-05083-6_26

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