On Quadrature Methods and Spline Approximation of Singular Integral Equations

  • S. Prössdorf
  • A. Rathsfeld
Part of the Boundary Elements IX book series (BOUNDARY, volume 9/1)

Abstract

In many practical computations with boundary integral equations for two-dimensional problems or with singular integral equations on an interval one uses spline approximations for the unknown solutions. The most popular numerical procedures are collocation and Galerkin methods as well as quadrature methods. For collocation and Galerkin procedures, the corresponding mathematical foundation and error analysis in Sobolev spaces has been developed only rather recently in the work by D.N. Arnold, M. Costabel, J. Elschner, G.C. Hsiao, J.C. Nedelec, J. Saranen, G. Schmidt, E. Stephan, K.S. Thomas, W.L. Wendland and the authors (see e.g. the surveys Elschner, Prössdorf, Rathsfeld and Schmidt1, Wendland2, Mikhlin and Prössdorf3, Chapter XVII). In the present paper, we put special emphasis on different quadrature methods, e.g. the “method of discrete whirls”, which are frequently used for approximate solving problems in aerodynamics, elasticity, electrodynamics and many other engineering problems (see e.g. Belotserkovski and Lifanov4). We present a uniform approach for obtaining necessary and sufficient stability conditions in LP as well as Sobolev norm error estimates for all approximation methods mentioned above. The central idea is a certain localization principle for numerical schemes defined by sequences of paired circulant matrices. The results are applied to the spline approximation of singular integral equations on a finite interval. The detailed proofs and the connections with other boundary element methods can be found in Prössdorf and Rathsfeld5, Rathsfeldl l.

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References

  1. 1.
    ELSCHNER, J., PROSSDORF, S., Rathsfeld, A. and Schmidt, G., Spline approximation of singular integral equations. Demonstratio Math., Vol. XVIII, 3: 661–671, 1985.MathSciNetGoogle Scholar
  2. 2.
    WENDLAND, W.L., On the spline approximation of singular integral equations and one-dimensional pseudo-differential equations on closed curves. Numerical solution of singular integral equations (Ed. Gerasoulis A. and Vichnevetsky R.) pp.113–119, Proc. of an IMACS International Symposium, Lehigh University, Bethlehem, PA, U.S.A. June 21–22, 1984.Google Scholar
  3. 3.
    MIKHLIN, S.G. and PROSSDORF, S., Singular integral operators, Springer-Verlag, Berlin and New York, 1986.CrossRefGoogle Scholar
  4. 4.
    BELOTSERKOVSKI, S.M. and LIFANOV, I.K., Numerical Methods for singular integral equations, in Russian, Nauka, Moscow, 1985.Google Scholar
  5. 5.
    PROSSDORF, S. and RATHSFELD, A., Stabilitätskriterien für NNherungsverfahren bei singulären Integralgleichungen in L. Z. Anal. Anw., Leipzig, 6, 1987.Google Scholar
  6. 6.
    SCHMIDT, G., On spline collocation methods for boundary integral equations in the plane. Math. Methods in Appl. Sci., 7: 74–89, 1985.MATHCrossRefGoogle Scholar
  7. 7.
    ARNOLD, D.N. and WENDLAND, W.L., The convergence of spline collocation for strongly elliptic equations on curves. Numer. Math., 46: 317–341, 1985.MathSciNetCrossRefGoogle Scholar
  8. 8.
    PROSSDORF, S., Ein Lokalisierungsprinzip in der Theorie der Spline-approximationen und einige Anwendungen. Math. Nachr. 119: 239–255, 1984.MathSciNetCrossRefGoogle Scholar
  9. 9.
    PROSSDORF, S. and RATHSFELD, A., On spline Galerkin methods for singular integral equations with piecewise continuous coefficients. Numer. Math., 48: 99–118, 1986.MathSciNetCrossRefGoogle Scholar
  10. 10.
    DE BOOR, C., The quasi-interpolant as a tool in elementary polynomial spline theory, in Approximation theory (Ed. Lorentz, G.G.), pp.269–276, Proc. of an International Symposium, Austin, 1973. Academic Press, New York and London, 1973.Google Scholar
  11. 11.
    RATHSFELD, A., Quadraturformelverfahren für eindimensionale singuläre Integralgleichungen. Seminar Analysis, Op.equ. and numer. anal. 85/86, Karl-Weierstrass-Institut für Mathematik, Berlin: 147–186, 1986.Google Scholar
  12. 12.
    GOHBERG, I. and KRUPNIK, N., Einführung in die Theorie der eindimensionalen singulären Integraloperatoren. Birkhäuser Verlag. Basel, Boston and Stuttgart, 1979.Google Scholar
  13. 13.
    SILBERMANN, B., Lokale Theorie des Reduktionsverfahrens für Toeplitzoperatoren. Math. Nachr., 104: 137–146, 1981.MathSciNetMATHCrossRefGoogle Scholar
  14. 14.
    SCHMIDT, G., On c-collocation for pseudodifferential equations on a closed curve. Math. Nachr., 126: 183–196, 1986.MathSciNetMATHCrossRefGoogle Scholar
  15. 15.
    PROSSDORF, S. and SCHMIDT, G., A finite element collocation method for singular integral equations. Math. Nachr., 100: 33–60, 1981.MathSciNetCrossRefGoogle Scholar
  16. 16.
    PROSSDORF, S. and RATHSFELD, A., A spline collocation method for singular integral equations with piecewise continuous coefficients. Integral Equations Oper. Theory, 7, 4: 536560, 1984.Google Scholar
  17. 17.
    PROSSDORF, S. and RATHSFELD, A., Strongly elliptic singular integral operators with piecewise continuous coefficients. Integral Equations Oper. Theory, 8: 825–841, 1985.MathSciNetGoogle Scholar
  18. 18.
    SCHMIDT, G., The convergence of Galerkin and collocation methods with splines for pseudodifferential equations on closed curves. Z. Anal. Anw., 3: 371–384, 1984.MATHGoogle Scholar
  19. 19.
    ELCHNER, J. and SCHMIDT, G., On spline interpolation in periodic Sobolev spaces. Preprint P-Math-01/83, Inst. f. Math. d.AdW d.DDR, Berlin, 1983.Google Scholar
  20. 20.
    SCHMIDT, G., Spline collocation for singular integro-differential equations over (0,1). Numer. Math., 50: 337352, 1987.Google Scholar
  21. 21.
    ELSCHNER, J., On spline approximation for singular integral equations on an interval. Preprint P-Math-04/87, KarlWeierstrass-Inst. f. Math. d.AdW d.DDR, Berlin, 1987.Google Scholar
  22. 22.
    ELSCHNER, J., Galerkin methods with splines for singular integral equations over (0,1). Numer. Math., 43: 265–281, 1984.MathSciNetMATHCrossRefGoogle Scholar
  23. 23.
    RATHSFELD, A., Quadraturformelverfahren für MellinOperatoren nullter Ordnung. Math. Nach., to appear.Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1987

Authors and Affiliations

  • S. Prössdorf
    • 1
  • A. Rathsfeld
    • 1
  1. 1.Karl-Weierstrass-Institut für Mathematik der Akademie der Wissenschaften der DDRBerlinGerman Democratic Republic

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