Bound on global error of the fast multipole method for Helmholtz equation in 2-D

This paper analyze the global error of the fast multipole method(FMM) for two-dimensional Helmholtz equation. We first propose the global error of the FMM for the discretized boundary integral operator. The error is caused by truncating Graf's addition theorem, according to the limiting forms of Bessel and Neumann functions, we provide sharper and more precise estimates for the truncations of Graf's addition theorem. Finally, using the estimates we derive the explicit bound and convergence rate for the global error of the FMM for Helmholtz equation, numerical experiments show that the results are valid. The method in this paper can also be applied to the FMM for other problems such as potential problems, elastostatic problems, Stokes flow problems and so on