The solution of Helmholtz and Maxwell equations by integral formulations (kernel in exp(i kr)/r) leads to large dense linear systems. Using direct solvers requires large computational costs in O(N3). Using iterative solvers, the computational cost is reduced to large matrix–vector products. The fast multipole method provides a fast numerical way to compute convolution integrals. Its application to Maxwell and Helmholtz equations was initiated by Rokhlin, based on a multipole expansion of the interaction kernel. A second version, proposed by Chew, is based on a plane–wave expansion of the kernel.
We propose a third approach, the stable–plane–wave expansion, 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 Nlog N as with the other methods.