Fast multipole method solution of three dimensional integral equation

Abstract

The fast multipole method (FMM) speeds up the matrix-vector multiplication in the conjugate gradient (CG) method when it is used to solve the matrix equation iteratively. The FMM is applied to solve the problem of electromagnetic scattering from three dimensional arbitrary shape conducting bodies. The electric field integral equation (EFIE), magnetic field integral equation (MFIE), and the combined field integral equation (CFIE) are considered. The FMM formula for the CFIE has been derived, which reduces the complexity of the matrix-vector multiplication from O(N/sup 2/) to O(N/sup 1/.5), where N is the number of unknowns. With a nonnested method, using the ray-propagation fast multipole algorithm (RPFMA), the cost of the FMM matrix-vector multiplication is reduced to O(N/sup 4/3/). We have implemented a multilevel fast multipole algorithm (MLFMA), whose complexity is further reduced to O(NlogN). The FMM also requires less memory, and hence, can solve a larger problem on a small computer.