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The numerical evaluation of a $ 2$-D Cauchy principal value integral arising in boundary integral equation methods


Author: Giovanni Monegato
Journal: Math. Comp. 62 (1994), 765-777
MSC: Primary 65N38; Secondary 65D32
DOI: https://doi.org/10.1090/S0025-5718-1994-1212268-8
MathSciNet review: 1212268
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Abstract: In this paper we consider the problem of computing 2-D Cauchy principal value integrals of the form

$\displaystyle {\fint_S}F({P_0};P)\,dP,\qquad {P_0} \in S,$

where S is either a rectangle or a triangle, and $ F({P_0};P)$ is integrable over S, except at the point $ {P_0}$ where it has a second-order pole. Using polar coordinates, the integral is first reduced to the form

$\displaystyle \int_{{\theta _1}}^{{\theta _2}} {[unk]_0^{R(\theta )}\frac{{f(r,\theta )}}{r}dr\, d \theta ,} $

where $ [unk]$ denotes the finite part of the (divergent) integral. Then ad hoc products of one-dimensional quadrature rules of Gaussian type are constructed, and corresponding convergence results derived. Some numerical tests are also presented.

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DOI: https://doi.org/10.1090/S0025-5718-1994-1212268-8
Article copyright: © Copyright 1994 American Mathematical Society

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