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)


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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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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