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.

翻译：本文关注二维散射问题中快速多极子方法(FMM)的误差估计。FMM的误差源于算法每一步中对Graf加法定理的截断，包括两次展开和三次平移。我们首先通过Bessel函数与Neumann函数的极限形式，给出了Graf加法定理截断误差的一个新界限，进而估计了FMM的误差。显式误差界及其收敛阶被推导得出。本文提出的方法也可应用于其他问题的FMM，例如势问题、弹性静力学问题、Stokes流问题等。