The Smoothing Property for Regular Splittings

  • Arnold Reusken
Part of the Notes on Numerical Fluid Mechanics (NNFM) book series (NNFM, volume 29)


In this paper we discuss convergence of multigrid methods with respect to the maximum norm for 2D elliptic boundary value problems. Our analysis uses Hackbusch’s framework based on the Smoothing Property and Approximation Property (cf. [4]). We present a rather general framework for establishing the Smoothing Property in the maximum norm. The analysis fits in nicely with the classical theory of diagonally dominant matrices and of M-matrices.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. [1]
    O. Axelsson, S. Brinkkemper, V.P. Il’in, On some versions of incomplete block-matrix factorization iterative methods ,Linear Algebra Appl., 58 (1984), pp. 3–15.CrossRefzbMATHMathSciNetGoogle Scholar
  2. [2]
    J. Descloux, On finite element matrices ,SIAM J. Numer. Anal., 9 (1972), pp. 260–265.CrossRefzbMATHMathSciNetGoogle Scholar
  3. [3]
    J. Frehse, R. Rannacher, Eine L1-Fehlerabschätzung für diskrete Grundlösungen in der Methode der finiten Elemente ,Tagungsband “Finite Elemente” Bonn. Math. Schr. 1976.Google Scholar
  4. [4]
    W. Hackbusch, Multi-grid Methods and Applications ,Springer, Berlin, 1985.Google Scholar
  5. [5]
    J.A. Meijerink, H.A. van der Vorst, An iterative solution method for linear systems of which the coefficient matrix is a symmetric M-matrix ,Math. Comp., 31 (1977), pp. 148–162.zbMATHMathSciNetGoogle Scholar
  6. [6]
    R. Rannacher, Zur L°°-Konvergenz linearer finiter Elemente beim Dirichlet-problem ,Math. Z., 149 (1976), pp. 69–77.CrossRefzbMATHMathSciNetGoogle Scholar
  7. [7]
    A. Reusken, A new lemma in multigrid convergence theory ,RANA Report 91–07, Department of Mathematics and Computing Science, Eindhoven University of Technology, 1991.Google Scholar
  8. [8]
    A. Reusken, On maximum norm convergence of multigrid methods for two-point boundary value problems ,to appear in SIAM J. Numer. Anal.Google Scholar
  9. [9]
    A. Reusken, On maximum norm convergence of multigrid methods for elliptic boundary value problems ,submitted.Google Scholar
  10. [10]
    R.S. Varga, Matrix Iterative Analysis ,Prentice-Hall, Englewood Cliffs, 1962.Google Scholar
  11. [11]
    G. Wittum, On the robustness of ILU-smoothing ,SIAM J. Sci. Stat. Comput., 10 (1989), pp. 699–717.CrossRefzbMATHMathSciNetGoogle Scholar
  12. [12]
    G. Wittum, Linear iterations as smoothers in multigrid methods: Theory with applications to incomplete decompositions ,Impact of Computing in Science and Engineering, 1 (1989), pp. 180–215.CrossRefzbMATHGoogle Scholar
  13. [13]
    D.M. Young, Iterative solution of large linear systems ,Academic Press, New York, 1971.zbMATHGoogle Scholar

Copyright information

© Friedr. Vieweg & Sohn Verlagsgesellschaft mbH, Braunschweig/Wiesbaden 1993

Authors and Affiliations

  • Arnold Reusken
    • 1
  1. 1.Department of Mathematics and Computing ScienceEindhoven University of TechnologyEindhovenThe Netherlands

Personalised recommendations