Full Numerical Quadrature 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 nor the same, the kernel is regular and a Gauss-Legendre quadrature can be employed. But, when they have a common edge or a common vertex, then edge and vertex weak sin- gularities appear, while autointegrals have both facets coincidents and the whole integration domain is weakly singular. Taylor (IEEE Trans. on Antennas and Propagation, 51(7):1630–1637 (2003)) proposed a systematic evaluation, based on a reordering and partition of the integration domain, together 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 one in the fourth one. Since this scheme is a bit restrictive because it was designed for electromagnetic wave propagation kernels, a full numer- ical quadrature on the four coordinates is proposed here in order to handle other kernels, like those of creeping flows. A numerical test is also proposed based on slight modification of the Wang-Atalla one (Comm. in Num. Meth. Eng., 13(0):1–7 (1997)). [Submitted for publicaciont to Communications in Numerical Methods in Engineering]