Accurate error estimates in the fast multipole method for electromagnetics

The fast multipole method (FMM) is useful for solving electromagnetic scattering problems through integral equations. Some asymptotic laws are given for the truncation of the series involved in the FMM. Those laws were rigorously proved in Carayol (2002). For the first time, the accuracy of the widely used empirical formulas was studied mathematically. Our results also give some elements to understand the difference of FMM error due to the configuration of points in FMM boxes (connected to the term u/spl I.cap//spl middot/ v/spl I.cap/). We also derived in Carayol some consequences on the number of quadrature points needed for the numerical integrations in the FMM.