On numerical cubatures of nearly singular surface integrals arising in BEM collocation

Abstract

In this paper we present efficient methods to approximate nearly singular surface integrals arising massively when discretizing boundary integral equations via the collocation method. The idea is to introduce local polar coordinates centred at a corner of the triangle. Thus it is possible to perform the inner integration analytically, either using a numerically stable evaluation of the corresponding formulae or replacing them quite efficiently by simple (rational) approximations. We show how the outer integration can be performed by simple Gauss-Legendre quadrature and how to adapt the order of the Gauss formulae to a required order of consistency. Numerical tests highlight the efficiency of our method.