Fast multipole method and microlocal discretization for the 3-D Helmholtz equation

A numerical solution of the boundary integral equation for the exterior Helmholtz problem in three dimensions, leads to the solution of a dense linear system. In order to have a well conditioned system, we consider the Despres integral equations (see J. Electromag. Waves and Appl., vol.13, p.1553-68, 1999). An iterative method is given by Despres to solve this system. If Niter is the number of the iterations, the complexity of this resolution is of order Niter k/sup 4/, where k is the wave number. In order to speed up the iterative solution of the system, we have considered the coupling of two methods, the microlocal discretization method and the fast multipole method (FMM). The microlocal discretization method of T. Abboud, J.-C. Nedelec and B. Zhou (1995), enables one to consider a new system whose size is of order k/sup 2/3//spl times/k/sup 2/3/ instead of k/sup 2//spl times/k/sup 2/ for convex geometries. However, due to the geometrical approximation of the surface, the fine mesh of the standard case is still considered. Another method, the fast multipole method, is one of the most efficient and robust methods used to speed up the calculation of matrix-vector products, with a cost of order k/sup 3/ instead of k/sup 4/ for the one level FMM. In this paper, a coupling of these two methods is presented. It enables one to reduce the CPU time very efficiently for large wave number, with a complexity of order k/sup 3/+Niter k/sup 4/3/.