Skip to main content
SpringerLink
Log in

Wavelet-based preconditioners for boundary integral equations

  • Published:
Advances in Computational Mathematics Aims and scope Submit manuscript

Abstract

We study simple preconditioners for the conjugate gradient method when used to solve matrix systems arising from some hypersingular and weakly singular integral equations. The preconditioners, which are of the type of hierarchical basis preconditioners, are based on the decomposition of the piecewise-linear (respectively piecewise-constant) functions as the sum of prewavelets (respectively derivatives of prewavelets). We prove that with these preconditioners the preconditioned systems have condition numbers uniformly bounded with respect to the degrees of freedom. Numerical experiments support our analysis.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. G. Beylkin, R. Coifman and V. Rokhlin, Fast wavelet transforms and numerical algorithms I, Comm. Pure Appl. Math. 44 (1991) 141-183.

    MATH  MathSciNet  Google Scholar 

  2. J.H. Bramble, Z. Leyk and J.E. Pasciak, The analysis of multigrid algorithms for pseudodifferential operators of order minus one, Math. Comp. 63 (1994) 461-478.

    Article  MATH  MathSciNet  Google Scholar 

  3. J.H. Bramble and J.E. Pasciak, New estimates for multilevel algorithms including the V-cycle, Math. Comp. 60 (1993) 447-471.

    Article  MATH  MathSciNet  Google Scholar 

  4. J.H. Bramble, J.E. Pasciak and A.H. Schatz, The construction of preconditioners for elliptic problems by substructuring, I, Math. Comp. 47 (1986) 103-134.

    Article  MATH  MathSciNet  Google Scholar 

  5. J.H. Bramble, J.E. Pasciak and J. Xu, Parallel multilevel preconditioners, Math. Comp. 55 (1990) 1-22.

    Article  MATH  MathSciNet  Google Scholar 

  6. C.K. Chui, An Introduction to Wavelets(Academic Press, New York, 1992).

    MATH  Google Scholar 

  7. M. Costabel, Boundary integral operators on Lipschitz domains: Elementary results, SIAM J. Math. Anal. 19 (1988) 613-626.

    Article  MATH  MathSciNet  Google Scholar 

  8. W. Dahmen, Wavelet and multiscale methods for operator equations, Acta Numerica 6 (1997) 55-228.

    Article  MATH  MathSciNet  Google Scholar 

  9. W. Dahmen, A. Kunoth and R. Schneider, Operator equations, multiscale concepts and complexity, in: The Mathematics of Numerical Analysis, Proc. 1995 AMS-SIAM Summer Seminar in Applied Mathematics, Park City, UT, eds. J. Renegar, M. Shub and S. Smale, Lectures in Applied Mathematics, Vol. 32 (Amer. Math. Soc., Providence, RI, 1996) pp. 225-261.

    Google Scholar 

  10. W. Dahmen, S. Prössdorf and R. Schneider, Wavelet approximation methods for pseudodifferential equations II: Matrix compression and fast solution, Adv. Comput. Math. 1 (1993) 259-335.

    Article  MATH  MathSciNet  Google Scholar 

  11. W. Dahmen, S. Prössdorf and R. Schneider, Multiscale methods for pseudodifferential equations, in: Recent Advances in Wavelet Analysis, eds. L.L. Schumaker and G. Webb, Wavelet Analysis and Its Applications, Vol. 3 (Academic Press, San Diego, 1994) pp. 191-235.

    Google Scholar 

  12. W. Dahmen, S. Prössdorf and R. Schneider, Wavelet approximation methods for pseudodifferential equations I: Stability and convergence, Math. Z. 215 (1994) 583-620.

    MATH  MathSciNet  Google Scholar 

  13. M. Dryja and O.B. Widlund, Multilevel additive methods for elliptic finite element problems, in: Parallel Algorithms for Partial Differential Equations, Proc. of the 6th GAMM-Seminar, Kiel, Germany, January 1990, ed. W. Hackbusch (Vieweg, Braunschweig, 1991) pp. 58-69.

    Google Scholar 

  14. P. Grisvard, Elliptic Problems in Nonsmooth Domains(Pitman, Boston, 1985).

    MATH  Google Scholar 

  15. M. Hahne and E.P. Stephan, Schwarz iterations for the efficient solution of screen problems with boundary elements, Computing 56 (1996) 61-85.

    Article  MATH  MathSciNet  Google Scholar 

  16. J.L. Lions and E. Magenes, Non-Homogeneous Boundary Value Problems and Applications I(Springer, New York, 1972).

    MATH  Google Scholar 

  17. P.L. Lions, On the Schwarz alternating method, in: Proc. of the 1st International Symposium on Domain Decomposition Methods for Partial Differential Equations, eds. R. Glowinski, G.H. Golub, G.A. Meurant and J. Périaux (SIAM, Philadelphia, PA, 1988) pp. 1-42.

    Google Scholar 

  18. W. McLean and T. Tran, A preconditioning strategy for boundary element Galerkin methods, Numer. Methods Partial Differential Equations 13 (1997) 283-301.

    Article  MATH  MathSciNet  Google Scholar 

  19. E.P. Stephan and W.L. Wendland, An augmented Galerkin procedure for the boundary integral method applied to two-dimensional screen and crack problems, Appl. Anal. 18 (1984) 183-219.

    MathSciNet  Google Scholar 

  20. T. Tran and E.P. Stephan, Additive Schwarz methods for the hversion boundary element method, Appl. Anal. 60 (1996) 63-84.

    MATH  MathSciNet  Google Scholar 

  21. T. Tran, E.P. Stephan and P. Mund, Hierarchical basis preconditioners for first kind integral equations, Appl. Anal. 65 (1997) 353-372.

    MATH  MathSciNet  Google Scholar 

  22. T. von Petersdorff and C. Schwab, Wavelet approximations for first kind boundary integral equations on polygons, Numer. Math. 74 (1996) 479-516.

    Article  MATH  MathSciNet  Google Scholar 

  23. T. von Petersdorff and E.P. Stephan, On the convergence of the multigrid method for a hypersingular integral equation of the first kind, Numer. Math. 57 (1990) 379-391.

    Article  MATH  MathSciNet  Google Scholar 

  24. T. von Petersdorff and E.P. Stephan, Multigrid solvers and preconditioners for first kind integral equations, Numer. Methods Partial Differential Equations 8 (1992) 443-450.

    Article  MATH  MathSciNet  Google Scholar 

  25. W.L. Wendland and E.P. Stephan, A hypersingular boundary integral method for two-dimensional screen and crack problems, Arch. Rational Mech. Anal. 112 (1990) 363-390.

    Article  MATH  MathSciNet  Google Scholar 

  26. O.B. Widlund, Optimal iterative refinement methods, in: Proc. of the 2nd International Symposium on Domain Decomposition Methods for Partial Differential Equations, eds. T.F. Chan, R. Glowinski, J. Périaux and O.B. Widlund (SIAM, Philadelphia, PA, 1989) pp. 114-125.

    Google Scholar 

  27. S. Zaprianov, Wavelet approximations for weakly singular and hypersingular integral equations and wavelet-based preconditioners, Ph.D. thesis, University of Hannover, Hannover (1996).

  28. X. Zhang, Multilevel Schwarz methods, Numer. Math. 63 (1992) 521-539.

    Article  MATH  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. School of Mathematics, University of New South Wales, Sydney, 2052, Australia

    Thanh Tran

  2. Institut für Angewandte Mathematik, University of Hannover, Welfengarten 1, D-30167, Hannover, Germany

    Ernst P. Stephan & Stefan Zaprianov

Authors
  1. Thanh Tran
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Ernst P. Stephan
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Stefan Zaprianov
    View author publications

    You can also search for this author in PubMed Google Scholar

Rights and permissions

Reprints and permissions

About this article

Cite this article

Tran, T., Stephan, E.P. & Zaprianov, S. Wavelet-based preconditioners for boundary integral equations. Advances in Computational Mathematics 9, 233–249 (1998). https://doi.org/10.1023/A:1018993624466

Download citation

  • Issue Date:

  • DOI: https://doi.org/10.1023/A:1018993624466

Search

Navigation