The Fast Multipole Method: Numerical Implementation

We study integral methods applied to the resolution of the Maxwell equations where the linear system is solved using an iterative method which requires only matrix–vector products. The fast multipole method (FMM) is one of the most efficient methods used to perform matrix–vector products and accelerate the resolution of the linear system. A problem involving N degrees of freedom may be solved in CNiterNlogN floating operations, where C is a constant depending on the implementation of the method. In this article several techniques allowing one to reduce the constant C are analyzed. This reduction implies a lower total CPU time and a larger range of application of the FMM. In particular, new interpolation and anterpolation schemes are proposed which greatly improve on previous algorithms. Several numerical tests are also described. These confirm the efficiency and the theoretical complexity of the FMM.