An efficient implementation of boundary element methods for computationally expensive Green's functions

We describe an implementation technique for boundary element methods that greatly reduces the required number of evaluations of Green's function. Assuming that evaluating Green's function is a costly operation, our approach has a computational cost that is almost independent of the chosen discretization scheme. Nyström methods, collocation methods and Galerkin methods lead to similar computation times, although the latter requires the computation of a large number of double integrals. The result is obtained mainly by constructing a specific family of quadrature and cubature rules that maximizes the possibility of sharing function evaluations.We apply the technique to the problem of scattering of electromagnetic waves by diffraction gratings in two dimensions. In this case Green's function is given by a slowly converging oscillatory series. We describe a novel approach for evaluating such series and we illustrate our implementation with numerical results.