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, Volume 50, Issue 3, pp 199–211 | Cite as

Multigrid with matrix-dependent transfer operators for a singular perturbation problem

  • A. Reusken
Article

Abstract

We consider multigrid applied to a class of singularly perturbed two-point boundary value problems. In the multigrid method we use a matrix-dependent prolongation and restriction. For a class of two-grid method we prove uniform convergence for allh (mesh size parameter) and ε (perturbation parameter).

AMS subject classification

65N20 

Key words

Convection-diffusion problem multigrid matrix-dependent prolongation and restriction 

Mehrgitterverfahren mit matrixabhängigen Transferoperatoren für ein singulär gestöres Problem

Zusammenfassung

In dieser Arbeit wird ein Mehrgittierverfahren für eine Klasse singulär gestörter Randwertprobleme untersucht. In dem mehrgitterverfahren verwenden wir eine matrixabhängige prolongation und Restriktion. Für eine Klasse von Zweigittermethoden beweisen wir gleichmäßige Konvergenz für alleh (Schrittweitenparameter) und ε (Parameter der singuläre Störung).

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References

  1. [1]
    van Asselt, E. J.: The multi-grid method and artificial viscosity. In: Hackbusch, W., Trottenberg, U. (eds.) Multigrid methods, 313–326. Berlin, Heidelberg, New York: Springer 1982 (Lecture Notes in Mathematicsd, 960).Google Scholar
  2. [2]
    Bank, R. E., Benbourenane, M.: The hierarchical basis multigrid method for convection-diffusion equations. Numer. Math.61, 7–37 (1992).CrossRefGoogle Scholar
  3. [3]
    Dendy, J. E., Jr.: Black box multigrid for nonsymmetric problems. Appl. Math. Comp.13, 261–283 (1983).CrossRefGoogle Scholar
  4. [4]
    Doolan, E. P., Miller, J. J. H., Schilkders, W. H. A.: Uniform numerical methods for problems with initial and boundary layers. Dublin: Boole Press 1980.Google Scholar
  5. [5]
    Eckhaus, W.: Matched aseymptotic expansions and singular perturbations. Amsterdam: North-Holland 1973.Google Scholar
  6. [6]
    Hackbusch, W.: Multi-grid methods and applications. Berlin, Heidelberg, New York, Tokyo: Springer 1985.Google Scholar
  7. [7]
    Hackbusch, W.: Multigrid convergence for a singular perturbation problem. Linear Algebra Appl.58, 125–145 (1984).CrossRefGoogle Scholar
  8. [8]
    Hemker, P. W.: A numerical study of stiff two-point boundary problems, Preprint MC Tracts80, Amsterdam: 1970.Google Scholar
  9. [9]
    Hemker, P. W., Kettler, R., Wesseling, P., de Zeeuw, P. M.: Multigrid methods: development of fast solvers. Appl. Math. Comp.13, 331–326 (1983).CrossRefGoogle Scholar
  10. [10]
    McCormick, S. F. (ed.): Multigrid Methods. Philadelphia: SIAM 1987.Google Scholar
  11. [11]
    O'Malley, R. E., Jr.: Introduction to singular perturbations. New York: Academic Press 1974.Google Scholar
  12. [12]
    Reusken, A.: On maximum nom convergence of multigrid methods for two-point boundary value problems. SIAM J. Numer. Anal.29, 1569–1578 (1992).CrossRefGoogle Scholar
  13. [13]
    Reusken, A.: The smoothing property for regular splittings. (to appear in Proceedings of the Eight GAMM-Seminar Kieel in Incomplete Decompositions, 1922).Google Scholar
  14. [14]
    Wesseling, P., In-introduction to multigrid methods. Chichester: Wiley 1992.Google Scholar
  15. [15]
    de Zeeuw, P. M.: Matrix-dependent prolongations and restrictions in a blackbox multigrid solver. J. Comput.Appl. Math.33, 1–27 (1990).CrossRefGoogle Scholar
  16. [16]
    de Zeeuw, P. M., van Asselt E. J.: The convergence rate of multi-level algorithms applied to the convection-diffusion equation. SIAM J. Sci. Stat. Comput.6, 492–503 (1985).CrossRefGoogle Scholar

Copyright information

© Springer-Verlag 1993

Authors and Affiliations

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

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