Error estimates in the numerical evaluation of some BEM singular integrals

In some applications of Galerkin boundary element methods one has to compute integrals which, after proper normalization, are of the form ∫ b a ∫ 1 −1 f(x, y) x− y dxdy, where (a, b) ≡ (−1, 1), or (a, b) ≡ (a,−1), or (a, b) ≡ (1, b), and f(x, y) is a smooth function. In this paper we derive error estimates for a numerical approach recently proposed to evaluate the above integral when a p−, or h − p, formulation of a Galerkin method is used. This approach suggests approximating the inner integral by a quadrature formula of interpolatory type that exactly integrates the Cauchy kernel, and the outer integral by a rule which takes into account the log endpoint singularities of its integrand. Some numerical examples are also given.