Skip to main content
SpringerLink
Log in

Fast Fourier-Galerkin methods for first-kind logarithmic-kernel integral equations on open arcs

  • Articles
  • Published:
Science in China Series A: Mathematics Aims and scope Submit manuscript

Abstract

We propose a fully discrete fast Fourier-Galerkin method for solving an integral equation of the first kind with a logarithmic kernel on a smooth open arc, which is a reformulation of the Dirichlet problem of the Laplace equation in the plane. The optimal convergence order and quasi-linear complexity order of the proposed method are established. A precondition is introduced. Combining this method with an efficient numerical integration algorithm for computing the single-layer potential defined on an open arc, we obtain the solution of the Dirichlet problem on a smooth open arc in the plane. Numerical examples are presented to confirm the theoretical estimates and to demonstrate the efficiency and accuracy of the proposed method.

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. Atkinson K. The Numerical Solution of Integral Equations of the Second Kind. Cambridge: Cambridge University Press, 1997

    MATH  Google Scholar 

  2. Atkinson K, Sloan I. The numerical solution of first-kind logarithmic-kernel integral equations on smooth open arcs. Math Comp, 1991, 56: 119–139

    Article  MATH  MathSciNet  Google Scholar 

  3. Cai H, Xu Y. A fast Fourier-Galerkin method for solving singular boundary integral equations. SIAM J Numer Anal, 2008, 46: 1965–1984

    Article  MATH  MathSciNet  Google Scholar 

  4. Chen Z, Micchelli C, Xu Y. A construction of interpolating wavelets on invariant sets. Math Comput, 1999, 68: 1569–1587

    Article  MATH  MathSciNet  Google Scholar 

  5. Chen Z, Wu B, Xu Y. Multilevel augmentation methods for solving operator equations. Numer Math J Chinese Univ, 2005, 14: 31–55

    MATH  MathSciNet  Google Scholar 

  6. Chen Z, Wu B, Xu Y. Multilevel augmentation methods for differential equations. Adv Comput Math, 2006, 24: 213–238

    Article  MATH  MathSciNet  Google Scholar 

  7. Cooley J, Tukey J. An algorithm for the machine calculation of complex Fourier series. Math Comp, 1965, 19: 297–301

    Article  MATH  MathSciNet  Google Scholar 

  8. Costabel M, Ervin V, Stephan E. On the convergence of collocation methods for Symm’s integral equation on open curves. Math Comp, 1988, 51: 167–179

    Article  MATH  MathSciNet  Google Scholar 

  9. Iserles A. On the numerical quadrature of highly-oscillating integrals I: Fourier transforms. IMA J Numer Anal, 2004, 24: 365–391

    Article  MATH  MathSciNet  Google Scholar 

  10. Iserles A. On the numerical quadrature of highly-oscillating integrals II: irregular oscillators. IMA J Numer Anal, 2005, 25: 25–44

    Article  MATH  MathSciNet  Google Scholar 

  11. Iserles A, Nörsett S. On quadrature methods for highly oscillatory integrals and their implementation. BIT, 2004, 44: 755–772

    Article  MATH  MathSciNet  Google Scholar 

  12. Jiang Y, Xu Y. Fast discrete algorithms for sparse Fourier expansions of high dimensional functions. J Complexity, 2010, 26: 51–81

    Article  MATH  Google Scholar 

  13. Jiang Y, Xu Y. Fast Fourier-Galerkin methods for solving singular boundary integral equations: numerical integration and precondition. J Comput Appl Math, to appear

  14. Joe S, Yan Y. A piecewise constant collocation method using cosine mesh grading for Symm’s equation. Numer Math, 1993, 65: 423–433

    Article  MATH  MathSciNet  Google Scholar 

  15. Kress R. Linear Integral Equations. New York: Springer-Verlag, 1989

    MATH  Google Scholar 

  16. Kaneko H, Xu Y. Gauss type quadratures for weakly singular integrals and their application to Fredholm integral equations of the second kind. Math Comp, 1994, 62: 739–753

    Article  MATH  MathSciNet  Google Scholar 

  17. Levin D. Procedures for computing one-and-two dimensional integrals of functions with rapid irregular oscillations. Math Comp, 1982, 38: 531–538

    Article  MATH  MathSciNet  Google Scholar 

  18. Prössdorf S, Saranen J, Sloan I. A discrete method for the logarithmic-kernel integral equation on an open arc. J Aust Math Soc, 1993, 34: 401–418

    Article  MATH  Google Scholar 

  19. Sloan I, Spence A. The Galerkin method for integral equations of the first kind with logrithmic kernel: theory. IMA J Numer Anal, 1988, 8: 105–122

    Article  MATH  MathSciNet  Google Scholar 

  20. Sloan I, Stephan E. Collocation with Chebyshev polynomials for Symm’s integral equation on an interval. J Austral Math Soc, 1992, 34: 199–211

    Article  MATH  MathSciNet  Google Scholar 

  21. Saranen J, Vainikko G. Fast collocation solvers for integral equations on open arcs. J Integral Equations Appl, 1999, 11: 57–102

    Article  MATH  MathSciNet  Google Scholar 

  22. Tolimieri R, An M, Lu C. Mathematics of Multidimensional Fourier Transform Algorithms. New York: Springer-Verlag, 1997

    Google Scholar 

  23. Xu Y, Chen H, Zou Q. Limit values of derivatives of the Cauchy integrals and computation of the logarithmic potentials. Computing, 2004, 73: 295–327

    Article  MATH  MathSciNet  Google Scholar 

  24. Xu Y, Zhao Y. Quadratures for boundary integral equations of the first kind with logarithmic kernels. J Integral Equations Appl, 1996, 8: 239–268

    Article  MATH  MathSciNet  Google Scholar 

  25. Yun B, Lee S. Double layer potential scheme for Dirichlet problems on smooth open arcs. Comput Math Appl, 1999, 37: 31–40

    Article  MATH  MathSciNet  Google Scholar 

  26. Yan Y, Sloan I. On integral equations of the first kind with logarithmic kernels. J Integral Equations Appl, 1988, 1: 549–579

    Article  MATH  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing, 100190, China

    Bo Wang

  2. School of Information Science and Engineering, Graduate University of the Chinese Academy of Sciences, Beijing, 100190, China

    Rui Wang

  3. Department of Mathematics, Syracuse University, Syracuse, NY, 13244, USA

    YueSheng Xu

  4. Department of Scientific Computing and Computer Applications, Sun Yat-sen University, Guangzhou, 510275, China

    YueSheng Xu

Authors
  1. Bo Wang
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Rui Wang
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. YueSheng Xu
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to YueSheng Xu.

Additional information

This work was partially supported by the President Fund of GUCAS and the US National Science Foundation (Grant No. CCR-0407476, DMS-0712827) and National Natural Science Foundation of China (Grant No. 10371122, 10631080)

Rights and permissions

Reprints and permissions

About this article

Cite this article

Wang, B., Wang, R. & Xu, Y. Fast Fourier-Galerkin methods for first-kind logarithmic-kernel integral equations on open arcs. Sci. China Ser. A-Math. 53, 1–22 (2010). https://doi.org/10.1007/s11425-010-0014-x

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11425-010-0014-x

Keywords

MSC(2000)

Search

Navigation