Space-Time Newton-Multigrid Strategies for Nonstationary Distributed and Boundary Flow Control Problems

  • Michael Hinze
  • Michael KösterEmail author
  • Stefan Turek
Part of the International Series of Numerical Mathematics book series (ISNM, volume 165)


This paper considers a Newton-type solver strategy for optimal flow control problems using space-time multigrid solution techniques. Based on the standard Newton approach for optimal control, a space-time multigrid preconditioner is derived and numerically analysed for distributed and boundary control.


Distributed control Boundary control Finite elements Time-dependent Navier–Stokes Newton Space-time multigrid Optimal control 

Mathematics Subject Classification (2010)

35Q30 49K20 49M05 49M15 49M29 49M37 65F08 65F10 65K05 65M55 65M60 65R20 65R32 76D05 76D55 90C06 90C30 



This work was financed by the program SPP1253 from the DFG, projects HI689/5-2 and TU102/24-1+2.


  1. 1.
    R.E. Bank, T.F. Dupond, An optimal order process for solving finite element equations. Math. Comput. 36(153), 35–51 (1981)CrossRefzbMATHGoogle Scholar
  2. 2.
    A. Borzi, V. Schulz, Multigrid methods for PDE optimization. SIAM Rev. 51(2), 361–395 (2009)CrossRefzbMATHMathSciNetGoogle Scholar
  3. 3.
    A. Borzi, V. Schulz, Computational Optimization of Systems Governed by Partial Differential Equations (SIAM, Philadelphia, 2011)CrossRefGoogle Scholar
  4. 4.
    G. Büttner, Ein Mehrgitterverfahren zur optimalen Steuerung parabolischer Probleme. PhD thesis, Fakultät II – Mathematik und Naturwissenschaften der Technischen Universität Berlin (2004),
  5. 5.
    M.D. Gunzburger, Perspectives in Flow Control and Optimization (SIAM, Philadelphia, 2003). ISBN:089871527XzbMATHGoogle Scholar
  6. 6.
    W. Hackbusch, Fast solution of elliptic control problems. J. Opt. Theory Appl. 31(4), 565–581 (1980)CrossRefzbMATHMathSciNetGoogle Scholar
  7. 7.
    W. Hackbusch, Die schnelle Auflösung der Fredholmschen Integralgleichung zweiter Art. Beiträge zur numerischen Mathematik, 9, 47–62 (1981)zbMATHGoogle Scholar
  8. 8.
    W. Hackbusch, Numerical solution of linear and nonlinear parabolic optimal control problems, in Optimization and Optimal Control, eds. by A. Auslender, W. Oettli, J. Stoer. Lecture Notes in Control and Information Science, vol. 30 (Springer, Berlin/New York, 1981), pp. 179–185Google Scholar
  9. 9.
    W. Hackbusch, Multi-Grid Methods and Applications. Springer Series in Computational Mathematics (Springer, Berlin, 1985). ISBN 3-540-12761-5Google Scholar
  10. 10.
    M. Hinze, Optimal and instantaneous control of the instationary Navier–Stokes equations. Habilitation thesis, Institut für Numerische Mathematik, Technische Universität Dresden, 2000Google Scholar
  11. 11.
    M. Hinze, M. Köster, S. Turek, A hierarchical space-time solver for distributed control of the Stokes equation. Preprint SPP1253-16-01, SPP1253, 2008Google Scholar
  12. 12.
    M. Hinze, M. Köster, S. Turek. A space-time multigrid solver for distributed control of the time-dependent Navier–Stokes system. Preprint SPP1253-16-02, SPP1253, 2008Google Scholar
  13. 13.
    M. Hinze, M. Köster, S. Turek, A hierarchical space-time solver for optimal distributed control of fluid flow, 2009. Proceedings of the Conference on Modeling, Simulation and Optimization of Complex Processes, Heidelberg, Accepted 21–25 July 2008Google Scholar
  14. 14.
    V. John, G. Matthies, Higher order finite element discretizations in a benchmark problem for incompressible flows. Int. J. Numer. Methods Fluids 37, 885–903 (2001)CrossRefzbMATHGoogle Scholar
  15. 15.
    M. Köster, A hierarchical flow solver for optimisation with PDE constraints. PhD thesis, TU Dortmund, Lehrstuhl III für Angewandte Mathematik und Numerik, 2011. Slightly corrected version with an additional appendix concerning prolongation/restrictionGoogle Scholar
  16. 16.
    H. Yserentant, Old and new convergence proofs for multigrid methods. Acta Numer. 2, 285–326 (1993)CrossRefMathSciNetGoogle Scholar

Copyright information

© Springer International Publishing Switzerland 2014

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of HamburgHamburgGermany
  2. 2.Institute of Applied MathematicsTU DortmundDortmundGermany

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