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Numerische Mathematik

, Volume 66, Issue 1, pp 199–214 | Cite as

On the numerical solution of a logarithmic integral equation of the first kind for the Helmholtz equation

  • Rainer Kress
  • Ian H. Sloan
Article

Summary

We describe a quadrature method for the numerical solution of the logarithmic integral equation of the first kind arising from the single-layer approach to the Dirichlet problem for the two-dimensional Helmholtz equation in smooth domains. We develop an error analysis in a Sobolev space setting and prove fast convergence rates for smooth boundary data.

Mathematics Subject Classification (1991)

35J05 65N38 65R20 65T05 

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References

  1. 1.
    Atkinson, K.E. (1988): A discrete Galerkin method for first kind integral equations with a logarithmic kernel. J. Integral Equations Appl.1, 343–363Google Scholar
  2. 2.
    Atkinson, K.E., Sloan, I.H. (1991): The numerical solution of first-kind logarithmic-kernel integral equations on smooth open arcs. Math. Comput.56, 119–139Google Scholar
  3. 3.
    Chandler, G.A., Sloan, I.H. (1990): Spline qualocation methods for boundary integral equations. Numer. Math.58, 537–567Google Scholar
  4. 4.
    Colton, D., Kress, R. (1983): Integral Equation Methods in Scattering Theory. Wiley, New YorkGoogle Scholar
  5. 5.
    Kress, R. (1989): Linear Integral Equations. Springer Berlin Heidelberg New YorkGoogle Scholar
  6. 6.
    Kress, R. (1990): A Nyström method for boundary integral equations in domains with corners. Numer. Math.58, 145–161Google Scholar
  7. 7.
    Kress, R. (1991): Boundary integral equations in time-harmonic acoustic scattering. Math. Comput. Modelling15, 229–243Google Scholar
  8. 8.
    Saranen, J. (1991): The modified quadrature method for logarithmic-kernel integral equations on closed curves. J. Integral Equations Appl.3, 575–600Google Scholar
  9. 9.
    Saranen, J., Sloan, I.H. (1991): Quadrature methods for logarithmic-kernel integral equations on closed curves. IMA J. Numer. Anal.12, 167–187Google Scholar
  10. 10.
    Sloan, I.H. (1988): A quadrature-based approach to improving the collocation method. Numer. Math.54, 41–56.Google Scholar
  11. 11.
    Sloan, I.H. (1992): Error analysis of boundary integral methods. Acta Numerical1, 287–339Google Scholar
  12. 12.
    Sloan, I.H., Burn, B.J. (1992): An unconventional quadrature method for logarithmic-kernel integral equations of closed curves. J. Integral Equations Appl.4, 117–151Google Scholar
  13. 13.
    Sloan, I.H., Wendland, W.L. (1989): A quadrature-based approach to improving the collocation method for splines of even degree. Zeit. Anal. Anwend.8, 362–376Google Scholar

Copyright information

© Springer-Verlag 1993

Authors and Affiliations

  • Rainer Kress
    • 1
  • Ian H. Sloan
    • 2
  1. 1.Institut für Numerische und Angewandte MathematikUniversität GöttingenGöttingenGermany
  2. 2.School of MathematicsUniversity of New South WalesSydneyAustralia

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