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# NUMERICAL QUADRATURES FOR NEAR-SINGULAR AND NEAR-HYPERSINGULAR INTEGRALS IN BOUNDARY ELEMENT METHODS

Michael Carley
Mathematical Proceedings of the Royal Irish Academy
Vol. 109A, No. 1 (APRIL 2009), pp. 49-60
Stable URL: http://www.jstor.org/stable/40656988
Page Count: 12
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## Abstract

A method of deriving quadrature rules has been developed which gives nodes and weights for a Gaussian-type rule which integrates functions of the form: $f(x,y,t) = \frac{{a(x,y,t)}}{{(x - t)^2 + y^2 }} + \frac{{b(x,y,t)}}{{[(x - t)^2 + y^2 ]^{1/2} }} + c(x,y,t)\log [(x - t)^2 + y^2 ]^{1/2} + d(x,y,t)$without having to explicitly analyze the singularities of f(x,y,t) or separate it into its components. The method extends previous work on a similar technique for the evaluation of Cauchy principal value or Hadamard finite part integrals, in the case when y ≡ 0. The method is tested by evaluating standard reference integrals and its error is found to be comparable to machine precision in the best case.

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