Integral evaluation in the BEM solution of (hyper)singular integral equations. 2D problems on polygonal domains

The formulation of certain classes of boundary value problems in terms of hypersingular integral equations is currently gaining increasing interest. In this paper we consider such type of equations on 2D polygonal domains, and assume we have to solve them by a collocation or a Galerkin BEM. In particular, given any (polynomial) local basis, we show how to compute efficiently, using a very low number of points, all integrals required by these methods. These integrals have kernels of the type log r, r−1 and r−2. adrature rules we propose to compute the above-mentioned integrals require the user to specify only the local polynomial degrees; therefore, they are quite suitable for the construction of a p or h−p version of the BEM.