Full numerical quadrature of weakly singular double surface integrals in Galerkin boundary element methods

Abstract

When a Galerkin discretization of a boundary integral equation with a weakly singular kernel is performed over triangles, a double surface integral must be evaluated for each pair of them. If these pairs are not contiguous or not coincident, the kernel is regular and a Gauss–Legendre quadrature can be employed. When the pairs have a common edge or a common vertex, then edge and vertex weak singularities appear. If the pairs have both facets coincident, the whole integration domain is weakly singular. Taylor (IEEE Trans. Antenn. Propag. 2003; 51(7):1630–1637) proposed a systematic evaluation based on a reordering and partitioning of the integration domain, together with a use of the Duffy transformations in order to remove the singularities, in such a way that a Gauss–Legendre quadrature was performed on three coordinates with an analytic integration in the fourth coordinate. Since this scheme is a bit restrictive because it was designed for electromagnetic kernels, a full numerical quadrature is proposed in order to handle kernels with a weak singularity with a general framework. Numerical tests based on modifications of the one proposed by Wang and Atalla (Commun. Numer. Meth. Engng 1997; 13(11):885–890) are included. Copyright © 2009 John Wiley & Sons, Ltd.