A Nyström-like Approach to Integral Equations with Singular Kernels

Traditional boundary element methods use panel-based discretization and exhibit low order convergence. In this paper, a new approach is proposed to discretize a singular integral equation. Global, numerically orthogonal bases are used to represent a solution, and mapping functions are used to represent the geometry. This method is capable of achieving spectral convergence, similar to the Nyström method for integral equations with non-singular kernels. In test case of a sphere, six digits of accuracy is achieved with 500 unknowns, which is about three orders of magnitude fewer than required by a panel method.