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

Abstract:

In this paper we consider the problem of computing 2-D Cauchy principal value integrals of the form \[ {\fint _S}F({P_0};P) dP,\qquad {P_0} \in S,\] 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 \[ \int _{{\theta _1}}^{{\theta _2}} {[unk]_0^{R(\theta )}\frac {{f(r,\theta )}}{r}dr d \theta ,} \] 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.