Efficient fast multipole method for low-frequency scattering

The solution of the Helmholtz and Maxwell equations using integral formulations requires to solve large complex linear systems. A direct solution of those problems using a Gauss elimination is practical only for very small systems with few unknowns. The use of an iterative method such as GMRES can reduce the computational expense. Most of the expense is then computing large complex matrix vector products. The cost can be further reduced by using the fast multipole method which accelerates the matrix vector product. 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 NlogN. There exist two versions of the fast multipole method: one which is based on a multipole expansion of the interaction kernel expιkr/r and which was first proposed by V. Rokhlin and another based on a plane wave expansion of the kernel, first proposed by W.C. Chew. In this paper, we propose a third approach, the stable plane wave expansion (SPW-FMM), which has a lower computational expense than the multipole expansion and does not have the accuracy and stability problems of the plane wave expansion. The computational complexity is NlogN as with the other methods.