Computing

, Volume 37, Issue 2, pp 125–136 | Cite as

A method for preconditioning matrices arising from linear integral equations for elliptic boundary value problems

  • Lothar Reichel
Contributed Papers

Abstract

The discretization of linear integral equations for elliptic boundary value problems by the boundary element method yields linear systems of simultaneous equations with filled matrices. The structure of these matrices allows Fourier methods to be used to determine preconditioning matrices such that fast iterative solution of the linear system of algebraic equations is possible. The preconditioning method is applicable to Fredholm integral equations of the first kind with non-smooth convolutional principal part as well as to Fredholm integral equations of the second kind. Numerical examples are presented.

AMS (MOS) Subject Classification

65R20 65F10 

Key words

Integral equation iterative solution preconditioning Fourier methods 

Eine Methode zur Vorkonditionierung von Matrizen, die von linearen Integralgleichungen für elliptische Randwertaufgaben stammen

Zusammenfassung

Die Diskretisierung von linearen Integralgleichungen für elliptische Randwertaufgaben durch die Randelementmethode gibt lineare Gleichungssysteme mit gefüllten Matrizen. Die Struktur dieser Matrizen läßt Vorkonditionierung durch Fourier-Methoden zu, was schnelles iteratives Lösen des Gleichungssystems von algebraischen Gleichungen ermöglicht. Die Vorkonditionierungsmethode ist verwendbar für Fredholmsche Integralgleichungen erster Art mit unglattem Kern vom Faltungstyp, sowie für Fredholmsche Integralgleichungen zweiter Art. Numerische Beispiele werden präsentiert.

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References

  1. [1]
    Arnold, N. A., Wendland, W. L.: On the asymptotic convergence of collocation methods. Math. Comp.41, 349–381 (1984).Google Scholar
  2. [2]
    Atkinson, K. E.: A survey of numerical methods for the solution of Fredholm integral equations of the 2nd kind. Philadelphia: SIAM 1976.Google Scholar
  3. [3]
    de Boor, C.: A practical guide to splines. New York: Springer 1978.Google Scholar
  4. [4]
    Henrici, P.: Fast Fourier methods in computational complex analysis. SIAM Rev.21, 481–527 (1979).Google Scholar
  5. [5]
    Hemker, P. W., Schippers, H.: Multiple grid methods for the solution of Fredholm integral equations of the second kind. Math. Comp.36, 215–232 (1981).Google Scholar
  6. [6]
    Hoidn, H.-P.: Die Kollokationsmethode angewandt auf die Symmsche Integralgleichung. Thesis, Swiss Institute of Technology, Zürich (1983).Google Scholar
  7. [7]
    Hsiao, G. C., Kopp, P., Wendland, W. L.: A Galerkin collocation method for some integral equations of the first kind. Computing25, 89–130 (1980).Google Scholar
  8. [8]
    Hsiao, G. C., Kopp, P., Wendland, W. L.: Some applications of a Galerkin-collocation method for boundary integral equations of the first kind. Fachbereich Mathematik, Technische Hochschule Darmstadt, Preprint 768 (1983).Google Scholar
  9. [9]
    Lanczos, C.: Discourse on Fourier series. London: Oliver & Boyd 1966.Google Scholar
  10. [10]
    Reichel, L.: On the determination of collocation points for solving some problems for the Laplace opeator. J. Comput. Appl. Math.11, 175–196 (1984).Google Scholar
  11. [11]
    Reichel, L.: A fast method for solving certain integral equations of the first kind with application to conformal mapping. J. Comput. Appl. Math., to appear.Google Scholar
  12. [12]
    Wendland, W. L.: On Galerkin-collocation methods for integral equations of elliptic boundary value problems. In: Numerical treatment of integral equations (Albrecht, J., Collatz, L., eds.) pp. 244–275. Basel: Birkhäuser 1980.Google Scholar
  13. [13]
    Wendland, W. L.: On the asymptotic convergence of some boundary element methods. In: MAFELAP 1981 (Whiteman, J., ed.) pp. 281–312. London: Academic Press 1982.Google Scholar
  14. [14]
    Wendland, W. L.: Elliptic systems in the plane. London: Pitman 1979.Google Scholar

Copyright information

© Springer-Verlag 1986

Authors and Affiliations

  • Lothar Reichel
    • 1
  1. 1.Department of MathematicsUniversity of KentuckyLexingtonUSA

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