On Robust Multigrid Methods for Non-Smooth Variational Problems

  • Ralf Kornhuber
Part of the Lecture Notes in Computational Science and Engineering book series (LNCSE, volume 3)


We consider the fast solution of large piecewise smooth minimization problems as resulting from the approximation of elliptic free boundary problems. The most delicate question in constructing a multigrid method for a nonlinear non-smooth problem is how to represent the nonlinearity on the coarse grids. This process usually involves some kind of linearization. The basic idea of monotone multigrid methods to be presented here is first to select a neighborhood of the actual smoothed iterate in which a linearization is possible and then to constrain the coarse grid correction to this neighborhood. Such a local linearization allows to control the local corrections at each coarse grid node in such a way that the energy functional is monotonically decreasing. This approach leads to globally convergent schemes which are robust with respect to local singularities of the given problem. The numerical performance is illustrated by approximating the well-known Barenblatt solution of the porous medium equation.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    R.E. Bank and D.J. Rose. Analysis of a multilevel iterative method for nonlinear finite element equations. Math. Comp., 39:453–665, 1982.zbMATHCrossRefMathSciNetGoogle Scholar
  2. 2.
    R.E. Bank and J. Xu. An algorithm for coarsening unstructured meshes. Azurner. Math., 73:1–36, 1996.zbMATHMathSciNetGoogle Scholar
  3. 3.
    E. Bänsch. Numerical experiments with adaptivity for the porous medium equation. Acta Math. Univ. Comenianae, LXIV:157–172, 1995.Google Scholar
  4. 4.
    G.I. Barenblatt. On self-similar solutions of compressible fluids in porous media. (In russian), Prikl. Mat. Mech., 16:679–698, 1952.zbMATHMathSciNetGoogle Scholar
  5. 5.
    R. Beck, B. Erdmann, and R. Roitzsch. KASKADE Manual, Version 3.0. Technical Report TR95-4, Konrad-Zuse-Zentrum (ZIB), Berlin, 1995.Google Scholar
  6. 6.
    F.A. Bornemann, B. Erdmann, and R. Kornhuber. Adaptive multilevel methods in three space dimensions. Int. J. Numer. Meth. Engrg., 36:3187–3203, 1993.zbMATHCrossRefMathSciNetGoogle Scholar
  7. 7.
    F.A. Bornemann, B. Erdmann, and R. Kornhuber. A posteriori error estimates for elliptic problems in two and three space dimensions. SIAM J. Numer. Anal., 33, 1996.Google Scholar
  8. 8.
    P. Dcuflhard and M. Weiser. Local inexact Newton multilevel FEM for nonlinear elliptic problems. Preprint SC 96-29, Konrad-Zuse-Zentrum (ZIB), Berlin, 1996.Google Scholar
  9. 9.
    I. Ekeland and R. Temam. Convex Analysis and Variational Problems. North-Holland, Amsterdam, 1976.zbMATHGoogle Scholar
  10. 10.
    R. Glowinski. Numerical Methods for Nonlinear Variational Problems. Springer, New York, 1984.zbMATHGoogle Scholar
  11. 11.
    W. Hackbusch. Multi-Grid Methods and Applications. Springer, Berlin, 1985.zbMATHGoogle Scholar
  12. 12.
    W. Hackbusch and A. Reusken. Analysis of a damped nonlinear multilevel method. Numer. Math., 55:225–246, 1989.zbMATHCrossRefMathSciNetGoogle Scholar
  13. 13.
    W. Hackbusch and S.A. Sauter. Composite finite elements for the approximation of PDEs on domains with complicated micro-structures. Numer. Math., to appear.Google Scholar
  14. 14.
    W. Jäger and J. Kacur. Solution of porous medium type systems by linear approximation schemes. Numer. Math., 60:407–427, 1991.zbMATHCrossRefMathSciNetGoogle Scholar
  15. 15.
    W. Jäger and J. Kacur. Solution of doubly nonlinear and degenerate parabolic problems by relaxation schemes, M 2 AN, 29:605–627, 1995.zbMATHGoogle Scholar
  16. 16.
    R. Kornhuber. Monotone multigrid methods for elliptic variational inequalities I. Numer. Math., 69:167–184, 1994.zbMATHMathSciNetGoogle Scholar
  17. 17.
    R. Kornhuber. Monotone multigrid methods for elliptic variational inequalities II. Numer. Math., 72:481–499, 1996.zbMATHCrossRefMathSciNetGoogle Scholar
  18. 18.
    R. Kornhuber. Adaptive Monotone Multigrid Methods for Nonlinear Variational Problems. Teubner, 1997.Google Scholar
  19. 19.
    R. Kornhuber. Globally convergent multigrid methods for porous medium type problems. To appear.Google Scholar
  20. 20.
    R. Kornhuber and H. Yserentant. Multilevel methods for elliptic problems on domains not resolved by the coarse grid. In D.E. Keyes and J. Xu, editors, Proceedings of the 7th International Conference on Domain Decomposition Methods 1993, pages 49–60, Providence, 1994. AMS.Google Scholar
  21. 21.
    S.F. McCormick. Multilevel Projection Methods for Partial Differential Equations. SIAM, Philadalphia, 1992.zbMATHCrossRefGoogle Scholar
  22. 22.
    J.L. Vazquez. An introduction to the mathematical théorie of the porous medium equation. In M.C. Delfour, editor, Shape Optimization and Free Boundaries, volume 380 of Mathematical and Physical Sciences, pages 347–389, Dordrecht, 1992. Kluwer.Google Scholar
  23. 23.
    J. Xu. Iterative methods by space decomposition and subspace correction. SIAM Review, 34:581–613, 1992.zbMATHCrossRefMathSciNetGoogle Scholar
  24. 24.
    H. Yserentant. Old and new convergence proofs for multigrid methods. Acta Numerica, pages 285–326, 1993.Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1998

Authors and Affiliations

  • Ralf Kornhuber
    • 1
  1. 1.Universität StuttgartStuttgartGermany

Personalised recommendations