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Mathematics of Computation

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Error estimates in the numerical evaluation of some BEM singular integrals

Authors: G. Mastroianni and G. Monegato
Journal: Math. Comp. 70 (2001), 251-267
MSC (2000): Primary 41A55; Secondary 65D32, 65N38
Published electronically: June 12, 2000
MathSciNet review: 1803127
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In some applications of Galerkin boundary element methods one has to compute integrals which, after proper normalization, are of the form \begin{equation*}\int _{a}^{b}\int _{-1}^{1}\frac{f(x,y)}{{x-y}}dxdy,\end{equation*}where $(a,b)\equiv (-1,1)$, or $(a,b)\equiv (a,-1)$, or $(a,b)\equiv (1,b)$, and $f(x,y)$ is a smooth function.

In this paper we derive error estimates for a numerical approach recently proposed to evaluate the above integral when a $p-$, or $h-p$, formulation of a Galerkin method is used. This approach suggests approximating the inner integral by a quadrature formula of interpolatory type that exactly integrates the Cauchy kernel, and the outer integral by a rule which takes into account the $\log $endpoint singularities of its integrand. Some numerical examples are also given.

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Additional Information

G. Mastroianni
Affiliation: Dipartimento di Matematica, Università della Basilicata, I-85100 Potenza, Italy

G. Monegato
Affiliation: Dipartimento di Matematica, Politecnico di Torino, I-10129 Torino, Italy

Keywords: Singular integrals, error estimates, boundary element methods
Received by editor(s): February 17, 1999
Published electronically: June 12, 2000
Additional Notes: Work supported by the Consiglio Nazionale delle Ricerche - Comitato Nazionale per le Ricerche Tecnologiche e l’Innovazione, under contract n.96.01875.CT11.
Article copyright: © Copyright 2000 American Mathematical Society