Fast multipole method for low-frequency electromagnetic scattering

Abstract

There exist two versions of the fast multipole method—one that is based on a multipole expansion of the interaction kernel exp(ikr)/r, and another based on a plane wave expansion of the kernel. The stable plane wave expansion has a lower computational expense than the multipole expansion and does not have the accuracy and stability problems of the plane wave expansion. For a linear system of size N, the use of an iterative method combined with the fast multipole method reduces the total expense of the computation to N log N. The solution of the Helmholtz and Maxwell equations, using integral formulations, requires solving large complex linear systems. A direct solution of such problems, using Gauss elimination, is practical only for very small systems with few unknowns. The use of an iterative method can reduce the computational expense.