A comparison of transformation methods for evaluating two-dimensional weakly singular integrals

Abstract

Accurate numerical evaluation of integrals arising in the boundary element method is fundamental to achieving useful results via this solution technique. In this paper, a number of techniques are considered to evaluate the weakly singular integrals which arise in the solution of Laplace's equation in three dimensions and Poisson's equation in two dimensions. Both are two-dimensional weakly singular integrals and are evaluated using (in a product fashion) methods which have recently been used for evaluating one-dimensional weakly singular integrals arising in the boundary element method. The methods used are based on various polynomial transformations of conventional Gaussian quadrature points where the transformation polynomial has zero Jacobian at the singular point. Methods which split the region of integration into sub-regions are considered as well as non-splitting methods. In particular, the newly introduced and highly accurate generalized composite subtraction of singularity and non-linear transformation approach (GSSNT) is applied to various two-dimensional weakly singular integrals. A study of the different methods reveals complex relationships between transformation orders, position of the singular point, integration kernel and basis function. It is concluded that the GSSNT method gives the best overall results for the two-dimensional weakly singular integrals studied.