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The convergence rate of a multigrid method with Gauss-Seidel relaxation for the poisson equation

Part II: Specific Contributions

Part of the Lecture Notes in Mathematics book series (LNM,volume 960)

Abstract

The numerical solution of the Poisson equation is treated by a multigrid method for a uniform grid. The convergence rate can be estimated even for the iteration with a V-cycle independently of the shape of the domain as long as it is convex and polygonal. The smoothing effect of the Gauß-Seidel relaxation is described by a discrete seminorm which is weaker than the energy norm.

Keywords

  • Convergence Rate
  • Energy Norm
  • Multigrid Method
  • Convergence Factor
  • Finite Element Equation

These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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References

  1. R.E. Bank and T. Dupont, An optimal order process for solving finite element equations. Math.Comp. 36 (1981), 35–51.

    CrossRef  MathSciNet  MATH  Google Scholar 

  2. R.E.Bank and T.Dupont, Analysis of a two-level scheme for solving finite element equations. Report CNA-159, Austin 1980.

    Google Scholar 

  3. D. Braess, The contraction number of a multigrid method for solving the Poisson equation. Numer. Math. 37 (1981), 387–404.

    CrossRef  MathSciNet  MATH  Google Scholar 

  4. D.Braess, The convergence rate of a multigrid method with Gauß-Seidel relaxation for the Poisson equation. (submitted)

    Google Scholar 

  5. A. Brandt, Multi-level adaptive solutions to boundary-value problems. Math. Comp. 31 (1977) 333–390.

    CrossRef  MathSciNet  MATH  Google Scholar 

  6. W. Hackbusch, On the convergence of multi-grid iterations. Beiträge zur Numer. Math. 9 (1981), 213–239.

    MATH  Google Scholar 

  7. J.F.Maitre and F.Musy, The contraction number of a class of two-level methods; an exact evaluation for some finite element subspaces and model problems. (These proceedings).

    Google Scholar 

  8. Th.Meis and H.-W.Branca, Schnelle Lösung von Randwertaufgaben. ZAMM 62.

    Google Scholar 

  9. R.A. Nicolaides, On the l2-convergence of an algorithm for solving finite element equations. Math. Comp. 31 (1977), 892–906.

    MathSciNet  MATH  Google Scholar 

  10. R.A. Nicolaides, On some theoretical and practical aspects of multigrid methods. Math. Comp. 33 (1979), 933–952.

    CrossRef  MathSciNet  MATH  Google Scholar 

  11. M.Ries, U.Trottenberg and G.Winter, A note on MGR methods, Preprint, Bonn 1981.

    Google Scholar 

  12. U.Trottenberg, private communication.

    Google Scholar 

  13. R.Verfürth, The contraction number of a multigrid method with mesh ratio 2 for solving Poisson’s equation. (in preparation).

    Google Scholar 

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© 1982 Springer-Verlag

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Braess, D. (1982). The convergence rate of a multigrid method with Gauss-Seidel relaxation for the poisson equation. In: Hackbusch, W., Trottenberg, U. (eds) Multigrid Methods. Lecture Notes in Mathematics, vol 960. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0069934

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  • DOI: https://doi.org/10.1007/BFb0069934

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  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-11955-5

  • Online ISBN: 978-3-540-39544-7

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