Explicit error bound of the fast multipole method for scattering problems in 2-D

TL;DR

This paper provides explicit error bounds and convergence order estimates for the fast multipole method applied to 2-D scattering problems, based on truncation error analysis of Graf's addition theorem.

Contribution

It introduces a novel error bound for Graf's addition theorem truncation and applies it to derive explicit FMM error estimates for scattering problems.

Findings

Explicit error bounds for FMM in 2-D scattering problems.

Convergence order of the FMM error established.

Method applicable to other potential and flow problems.

Abstract

This paper is concerned with the error estimation of the fast multipole method (FMM) for scattering problems in 2-D. The FMM error is caused by truncating Graf's addition theorem in each step of the algorithm, including two expansions and three translations. We first give a novel bound on the truncation error of Graf's addition theorem by the limiting forms of Bessel and Neumann functions, and then estimate the error of the FMM. Explicit error bound and its convergence order are derived. The method proposed in this paper can also be used to the FMM for other problems, such as potential problems, elastostatic problems, Stokes flow problems and so on.