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On Preconditioned Uzawa-type Iterations for a Saddle Point Problem with Inequality Constraints

  • Carsten Gräser
  • Ralf Kornhuber
Part of the Lecture Notes in Computational Science and Engineering book series (LNCSE, volume 55)

Abstract

We consider preconditioned Uzawa iterations for a saddle point problem with inequality constraints as arising from an implicit time discretization of the Cahn-Hilliard equation with an obstacle potential. We present a new class of preconditioners based on linear Schur complements associated with successive approximations of the coincidence set. In numerical experiments, we found superlinear convergence and finite termination.

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References

  1. 1.
    R. E. Bank, B. D. Welfert, and H. Yserentant, A class of iterative methods for solving saddle point problems, Numer. Math., 56 (1989), pp. 645–666.CrossRefMathSciNetGoogle Scholar
  2. 2.
    J. W. Barrett, R. Nürnberg, and V. Styles, Finite element approximation of a phase field model for void electromigration, SIAM J. Numer. Anal., 42 (2004), pp. 738–772.zbMATHCrossRefMathSciNetGoogle Scholar
  3. 3.
    J. F. Blowey and C. M. Elliott, The Cahn-Hilliard gradient theory for phase separation with non-smooth free energy Part I: Mathematical analysis, European J. Appl. Math., 2 (1991), pp. 233–280.zbMATHMathSciNetGoogle Scholar
  4. 4.
    J. F. Blowey, The Cahn-Hilliard gradient theory for phase separation with non-smooth free energy Part II: Numerical analysis, European J. Appl. Math., 3 (1992), pp. 147–179.zbMATHCrossRefMathSciNetGoogle Scholar
  5. 5.
    D. Braess and R. Sarazin, An efficient smoother for the Stokes problem, Appl. Numer. Math., 23 (1997), pp. 3–19.zbMATHCrossRefMathSciNetGoogle Scholar
  6. 6.
    J. H. Bramble, J. E. Pasciak, and A. T. Vassilev, Analysis of the inexact Uzawa algorithm for saddle point problems, SIAM J. Numer. Anal., 34 (1997), pp. 1072–1092.zbMATHCrossRefMathSciNetGoogle Scholar
  7. 7.
    J. W. Cahn and J. E. Hilliard, Free energy of a nonuniform system I. interfacial energy, J. Chem. Phys., 28 (1958), pp. 258–267.CrossRefGoogle Scholar
  8. 8.
    X. Chen, Global and superlinear convergence of inexact Uzawa methods for saddle point problems with nondifferentiable mappings, SIAM J. Numer. Anal., 35 (1998), pp. 1130–1148.zbMATHCrossRefMathSciNetGoogle Scholar
  9. 9.
    X. Chen, On preconditioned Uzawa methods and SOR methods for saddle-point problems, J. Comput. Appl. Math., 100 (1998), pp. 207–224.zbMATHCrossRefMathSciNetGoogle Scholar
  10. 10.
    I. Ekeland and R. Temam, Convex analysis and variational problems, North-Holland, Amsterdam, 1976.zbMATHGoogle Scholar
  11. 11.
    C. M. Elliott, The Cahn-Hilliard model for the kinetics of phase separation, in Mathematical models for phase change problems, J. F. Rodrigues, ed., Basel, 1989, Birkhäuser, pp. 35–73.Google Scholar
  12. 12.
    H. C. Elman and G. H. Golub, Inexact and preconditioned Uzawa algorithms for saddle point problems, SIAM J. Numer. Anal., (1994), pp. 1645–1661.Google Scholar
  13. 13.
    D. J. Eyre, An unconditionally stable one-step scheme for gradient systems, tech. rep., University of Utah, Salt Lake City, UT, 1998.Google Scholar
  14. 14.
    R. Glowinski, J. L. Lions, and R. Trémolières, Numerical Analysis of Variational Inequalities, no. 8 in Studies in Mathematics and its Applications, North-Holland Publishing Company, Amsterdam, 1981.Google Scholar
  15. 15.
    C. Gräser and R. Kornhuber, Preconditioned Uzawa iterations for the Cahn-Hilliard equation with obstacle potential. To appear.Google Scholar
  16. 16.
    Q. Hu and J. Zou, Two new variants of nonlinear inexact Uzawa algorithms for saddle-point problems, Numer. Math., 93 (2002), pp. 333–359.zbMATHCrossRefMathSciNetGoogle Scholar
  17. 17.
    R. Kornhuber, Monotone multigrid methods for elliptic variational inequalities I, Numer. Math., 69 (1994), pp. 167–184.zbMATHMathSciNetGoogle Scholar
  18. 18.
    P. Lions and B. Mercier, Splitting algorithms for the sum of two nonlinear operators, SIAM J. Numer. Anal., 16 (1979), pp. 964–979.zbMATHCrossRefMathSciNetGoogle Scholar
  19. 19.
    A. Novick-Cohen, The Cahn-Hilliard equation: Mathematical and modeling perspectives, Adv. Math. Sci. Appl., 8 (1998), pp. 965–985.zbMATHMathSciNetGoogle Scholar
  20. 20.
    J. Schöberl and W. Zulehner, On Schwarz-type smoothers for saddle point problems, Numer. Math., 95 (2003), pp. 377–399.zbMATHCrossRefMathSciNetGoogle Scholar
  21. 21.
    S. P. Vanka, Block-implicit multigrid solution of Navier-Stokes equations in primitive variables, J. Comput. Phys., 65 (1986), pp. 138–158.zbMATHCrossRefMathSciNetGoogle Scholar
  22. 22.
    S. J. Wright, Primal-dual interior-point methods, SIAM, Philadelphia, PA, 1997.zbMATHGoogle Scholar
  23. 23.
    W. Zulehner, A class of smoothers for saddle point problems, Computing, 65 (2000), pp. 227–246.zbMATHCrossRefMathSciNetGoogle Scholar
  24. 24.
    W. Zulehner, Analysis of iterative methods for saddle point problems: A unified approach, Math. Comp., 71 (2002), pp. 479–505.zbMATHCrossRefMathSciNetGoogle Scholar

Copyright information

© Springer 2007

Authors and Affiliations

  • Carsten Gräser
    • 1
  • Ralf Kornhuber
    • 1
  1. 1.Fachbereich Mathematik und InformatikFreie Universität BerlinBerlinGermany

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