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Constrained Optimization and Optimal Control for Partial Differential Equations pp 147–170Cite as

Birkhäuser

A Space-Time Multigrid Method for Optimal Flow Control

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Part of the book series: International Series of Numerical Mathematics ((ISNM,volume 160))

Abstract

We present a hierarchical solution concept for optimization problems governed by the time-dependent Navier–Stokes system. Discretisation is carried out with finite elements in space and a one-step-θ-scheme in time. By combining a Newton solver for the treatment of the nonlinearity with a space-time multigrid solver for linear subproblems, we obtain a robust solver whose convergence behaviour is independent of the refinement level of the discrete problem. A set of numerical examples analyses the solver behaviour for various problem settings with respect to efficiency and robustness of this approach.

Mathematics Subject Classification (2000). 35Q30, 49K20, 49M05, 49M15, 49M29, 65M55, 65M60, 76D05, 76D55.

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Author information

Authors and Affiliations

  1. Department of Mathematics, University of Hamburg, Bundesstrasse 55, D-20146, Hamburg, Germany

    Michael Hinze

  2. Institut für Angewandte Mathematik, Technische Universität Dortmund, Vogelpothsweg 87, D-44227, Dortmund, Germany

    Michael Köster & Stefan Turek

Authors
  1. Michael Hinze
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  2. Michael Köster
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  3. Stefan Turek
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Corresponding author

Correspondence to Michael Köster .

Editor information

Editors and Affiliations

  1. , Department Mathematik, Universität Erlangen-Nürnberg, Martensstr. 3, Erlangen, 91058, Germany

    Günter Leugering

  2. , Fakultät Bio- und Chemieingenieurwesen, Technische Universität Dortmund, Geschossbau 2, Dortmund, 44221, Germany

    Sebastian Engell

  3. , Institut für Mathematik, Humboldt-Universität zu Berlin, Rudower Chaussee 25, Berlin, 12489, Germany

    Andreas Griewank

  4. , Fachbereich Mathematik, Universität Hamburg, Bundesstr. 55, Hamburg, 20146, Germany

    Michael Hinze

  5. , Institut für Angewandte Mathematik, Ruprecht-Karls-Universität Heidelberg, Im Neuenheimer Feld 293, Heidelberg, 69120, Germany

    Rolf Rannacher

  6. , FB 4 - Department Mathematik, Universität Trier, Trier, 54286, Germany

    Volker Schulz

  7. Fakultät für Mathematik, M1, Lehrstuhl für Mathematische Optimierung, Technische Universität München, Boltzmannstr. 3, Garching bei München, 85748, Germany

    Michael Ulbrich

  8. , Fachbereich Mathematik, Technische Universität Darmstadt, Dolivostrasse 15, Darmstadt, 64293, Germany

    Stefan Ulbrich

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Hinze, M., Köster, M., Turek, S. (2012). A Space-Time Multigrid Method for Optimal Flow Control. In: Leugering, G., et al. Constrained Optimization and Optimal Control for Partial Differential Equations. International Series of Numerical Mathematics, vol 160. Birkhäuser, Basel. https://doi.org/10.1007/978-3-0348-0133-1_8

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