A fast multipole boundary integral equation method for two-dimensional diffusion problems

The Fast Multipole Method (FMM) is a well-established and effective method for accelerating numerical solutions of the boundary integral equations (BIE). The BIE method, accelerated by the FMM, can solve large-scale electromagnetic wave propagation and diffusion problems with up to a million unknowns on a personal computer. The conventional BIE method requires O(N-2) operations to compute the system of equations and another O(N-2) operations to solve The system using iterative solvers, with N being the number of unknowns; in contrast, the BIE method accelerated by the two-level FMM can potentially reduce the operations and memory requirement to O(N-3/2). This paper introduces the procedure of the FMM accelerated BIE method, which is not only efficient in meshing complicated geometries, accurate for solving singular fields or fields in infinite domains, but also practical and often superior to other methods in solving large-scale problems. The two-dimensional Helmholtz equation with a complex wave number for non-trivial boundary geometries has been specifically studied as a test problem. Computational tests of the numerical solutions against the conventional BIE results and exact solutions are presented. It is shown that, in the thin skin limit, the far interactions can be discarded approximately due to the rapid decay of the kernel in the long distance.