# Numerical quadratures for near-singular and near-hypersingular integrals in boundary element methods

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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 $y\equiv0$. The method is tested by evaluating standard reference integrals and its error is found to be comparable to machine precision in the best case.
Original language English 49-60 12 Mathematical Proceedings of the Royal Irish Academy 109 1 https://doi.org/10.3318/PRIA.2008.109.1.49 Published - 2009