High-order discretization of integral equations with singular kernels

High-order discretization of integral equations with a singular kernel (such as the electric- or magnetic-field integral equation) requires a high-order quadrature method for singular integrands. "High-order" means, in this case, that one can obtain extra digits of precision at relatively modest cost. High-order basis functions alone are not sufficient. Use of low-order integration techniques may be responsible for reports of the limited utility of high-order basis functions. Often, in the computation of near matrix elements, Gaussian quadrature is employed in a way that destroys its high-order behavior, a practice often called "singularity subtraction". The author shows, by simple numerical example, how the high-order behavior of Gaussian quadrature is destroyed by crude "singularity subtraction", even with functions that appear, visually to be extraordinarily smooth. He then exhibits quadratures for one dimensional integrals (appropriate to two dimensional scattering problems) that are high-order for integrands with logarithmic singularities. Finally, he outlines a method of high-order integration for three dimensional surface scattering.