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, Volume 52, Issue 2, pp 139–159 | Cite as

On numerical cubatures of nearly singular surface integrals arising in BEM collocation

  • W. Hackbusch
  • S. A. Sauter
Article

Abstract

In this paper we present efficient methods to approximate nearly singular surface integrals arising massively when discretizing boundary integral equations via the collocation method. The idea is to introduce local polar coordinates centred at a corner of the triangle. Thus it is possible to perform the inner integration analytically, where either the corresponding formulae can be evaluated numerically stable or can be replaced by simple (rational) approximation quite efficiently. We show that the outer integration can be performed by simple Gauß-Legendre quadrature and how to adapt the order of the Gauß formulae to a required order of consistency. Numerical tests will emphasize the efficiency of our method.

AMS Subject Classification

65N38 45B05 45E05 45Z10 65Y20 

Key words

Boundary element method numerical quadrature collocation panel method cubature 

Numerische Quadratur fast singulärer Oberflächenintegrale aus der Randelementkollokation

Zusammenfassung

Die Arbeit präsentiert effiziente Verfahren zur Bestimmung fast singulärer Integrale, wie sie in großer Anzahl bei der Diskretisierung von Integralgleichungen durch Kollokation auftreten. Die Methode basiert auf der Einführung von lokalen Polarkoordinaten um eine Dreiecksecke. Die innere Integration läßt sich analytisch durchführen, wobei man entweder entsprechende numerisch stabile Formeln oder Funktionsapproximationen verwenden kann. Die äußere Integration kann mit gewünschter Genauigkeit mittels Gauß-Legendre-Quadratur ermittelt werden. Numerische Tests unterstreichen die Effizienz unserer Methode.

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Copyright information

© Springer-Verlag 1994

Authors and Affiliations

  • W. Hackbusch
    • 1
  • S. A. Sauter
    • 1
  1. 1.Institut für Informatik und Praktische MathematikKielFederal Republic of Germany

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