Error estimates in the numerical evaluation of some BEM singular integrals

Abstract:

In some applications of Galerkin boundary element methods one has to compute integrals which, after proper normalization, are of the form \begin{equation*}\int _{a}^{b}\int _{-1}^{1}\frac {f(x,y)}{{x-y}}dxdy,\end{equation*} where $(a,b)\equiv (-1,1)$, or $(a,b)\equiv (a,-1)$, or $(a,b)\equiv (1,b)$, 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.