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Discrete qualocation methods for logarithmic-kernel integral equations on a piecewise smooth boundary

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Abstract

We consider a fully discrete qualocation method for Symm’s integral equation. The method is that of Sloan and Burn (1992), for which a complete analysis is available in the case of smooth curves. The convergence for smooth curves can be improved by a subtraction of singularity (Jeon and Kimn, 1996). In this paper we extend these results for smooth boundaries to polygonal boundaries. The analysis uses a mesh grading transformation method for Symm’s integral equation, as in Elschner and Graham (1995) and Elschner and Stephan (1996), to overcome the singular behavior of solutions at corners.

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References

  1. G.A. Chandler and I.H. Sloan, Spline qualocation methods for boundary integral equations, Numer. Math. 58 (1990) 537–567.

    Article  MATH  MathSciNet  Google Scholar 

  2. M. Costabel and E.P. Stephan, On the convergence of collocation methods for boundary integral equations on polygons, Math. Comp. 49 (1987) 461–478.

    Article  MATH  MathSciNet  Google Scholar 

  3. P.J. Davis and P. Rabinowitz, Numerical Integration (Blaisdell, Waltham, 1967).

    Google Scholar 

  4. J. Elschner and I.G. Graham, An optimal order collocation method for first kind boundary integral equations on polygons, Numer. Math. 70 (1995) 1–31.

    Article  MATH  MathSciNet  Google Scholar 

  5. J. Elschner and I.G. Graham, Quadrature methods for Symm's integral equation on polygons, IMA J. Numer. Anal., to appear.

  6. J. Elschner, S. Prössdorf and I.H. Sloan, The qualocation method for Symm's integral equation on a polygon, Math. Nachr. 177 (1996) 81–108.

    MATH  MathSciNet  Google Scholar 

  7. J. Elschner and E.P. Stephan, A discrete collocation method for Symm's integral equation on curves with corners, J. Comput. Appl. Math. 75 (1996) 131–146.

    Article  MATH  MathSciNet  Google Scholar 

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

    Google Scholar 

  9. Y. Jeon, A Nyström method for boundary integral equations on domains with a piecewise smooth boundary, J. Integral Equations Appl. 5 (1993) 221–242.

    MATH  MathSciNet  Google Scholar 

  10. Y. Jeon and H.-J. Kimn, A quadrature method for logarithmic-kernel integral equations on closed curves, J. Korean Math. Soc. 33 (1996) 929–954.

    MATH  MathSciNet  Google Scholar 

  11. R. Kress, A Nyström method for boundary integral equations in domains with corners, Numer. Math. 58 (1990) 145–161.

    Article  MATH  MathSciNet  Google Scholar 

  12. V.G. Maz'ya, Boundary integral equations, in: Analysis IV, eds. V.G. Maz'ya and S.M. Nikolskii, Encyclopaedia Math. Sci. 27 (Springer, Berlin, 1991) pp. 127–222.

    Google Scholar 

  13. S. Prössdorf and A. Rathsfeld, Quadrature methods for strongly elliptic Cauchy singular integral equations on an interval, in: Oper. Theory Adv. Appl. 41, eds. H. Dym et al. (Birkhäuser, Basel, 1989).

    Google Scholar 

  14. A. Rathsfeld, A quadrature method for a Cauchy singular integral equation, in: Seminar Analysis. Operator Equations and Numerical Analysis 1987/88 (Karl Weierstraß Inst. Math., Akad. Wiss. DDR, Berlin, 1988) pp. 107–117.

    Google Scholar 

  15. J. Saranen and I.H. Sloan, Quadrature methods for logarithmic-kernel equations on closed curves, IMA J. Numer. Anal. 12 (1992) 167–187.

    MATH  MathSciNet  Google Scholar 

  16. I.H. Sloan and B.J. Burn, An unconventional quadrature method for logarithmic-kernel equations on closed curves, J. Integral Equations Appl. 4 (1992) 117–151.

    MATH  MathSciNet  Google Scholar 

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Authors and Affiliations

  1. Department of Mathematics, Ajou University, Suwon, 441-749, Korea

    Y. Jeon

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

    I.H. Sloan

  3. Institut für Angewandte Mathematik, Universität Hannover, Postfach 6009, D-30060, Hannover, Germany

    E.P. Stephan

  4. Weierstraß-Institut für Angewandte Analysis und Stochastik, Mohrenstraße, 39, D-10117, Berlin, Germany

    J. Elschner

Authors
  1. Y. Jeon
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  2. I.H. Sloan
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  3. E.P. Stephan
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  4. J. Elschner
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Jeon, Y., Sloan, I., Stephan, E. et al. Discrete qualocation methods for logarithmic-kernel integral equations on a piecewise smooth boundary. Advances in Computational Mathematics 7, 547–571 (1997). https://doi.org/10.1023/A:1018967424040

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  • DOI: https://doi.org/10.1023/A:1018967424040

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