Numerische Mathematik

, Volume 37, Issue 3, pp 387–404

The contraction number of a multigrid method for solving the Poisson equation

  • Dietrich Braess
Article
  • 123 Downloads

Summary

The treatment of a multigrid method in the framework of numerical analysis elucidates that regularity of the solution is not necessary for the convergence of the multigrid algorithm but only for fast convergence. For the linear equations which arise from the discretization of the Poisson equation, a convergence factor 0,5 is established independent of the shape of the domain and of the regularity of the solution.

Subject Classifications

AMS(MOS): 65N20 CR: 5.17 

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References

  1. 1.
    Babuška, I., Rheinboldt, W.: Mathematical problems of computational decisions in the finite element method. Technical report TR-426, University of Maryland, 1975Google Scholar
  2. 2.
    Bank, R.E., Dupont, T.: An optimal order process for solving finite element equations. Math. Comput.36, 35–51 (1981) and: Analysis of a two-level scheme for solving finite element equations. Numer. Math. (1981, in press)Google Scholar
  3. 3.
    Collatz, L.: Numerische Behandlung von Differentialgleichungen. Berlin-Göttingen-Heidelberg: Springer 1951Google Scholar
  4. 4.
    Gunn, J.E.: The solution of difference equations by semi-explicit iterative techniques. SIAM J. Numer. Anal.2, 24–45 (1965)Google Scholar
  5. 5.
    Hackbusch, W.: On the multigrid method applied to difference equations. Computing20, 291–306 (1978)Google Scholar
  6. 6.
    Meis, T., Marcowitz, U.: Numerische Behandlung partieller Differentialgleichungen. Berlin-Heidelberg-New York: Springer 1978Google Scholar
  7. 7.
    Nicolaides, R.A.: On thel 2-convergence of an algorithm for solving finite element equations. Math. Comput.31, 892–906 (1977)Google Scholar
  8. 8.
    Nicolaides, R.A.: On some theoretical and practical aspects of multigrid methods. Math. Comput.33, 933–952 (1979)Google Scholar
  9. 9.
    Varga, R.S.: Matrix Iterative Analysis. Englewood Cliffs: Prentice-Hall 1962Google Scholar
  10. 10.
    Wesseling, P.: A convergence proof for a multiple grid method. In: Numerical analysis. Proceedings Dundee 1979 (G.A. Watson, ed.), pp. 164–183. Lecture Notes in Mathematics, Vol. 733. Berlin-Heidelberg-New York: Springer 1980Google Scholar

Copyright information

© Springer-Verlag 1981

Authors and Affiliations

  • Dietrich Braess
    • 1
  1. 1.Institut für MathematikRuhr-Universität BochumBochum 1Germany (Fed. Rep.)

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