On the Convergence of the Multipole Expansion Method

TL;DR

This paper analyzes the convergence rate of the multipole expansion method (MEM), providing the first precise asymptotic characterization of how quickly the approximation error diminishes as the truncation number increases.

Contribution

The paper offers a rigorous analysis of the asymptotic convergence rate of MEM, applicable to various boundary conditions and formulations, filling a longstanding gap in understanding.

Findings

Established the asymptotic rate of convergence for MEM

Results apply to all boundary conditions and formulations

Provides theoretical foundation for MEM accuracy assessment

Abstract

The multipole expansion method (MEM) is a spatial discretization technique that is widely used in applications that feature scattering of waves from circular cylinders. Moreover, it also serves as a key component in several other numerical methods in which scattering computations involving arbitrarily shaped objects are accelerated by enclosing the objects in artificial cylinders. A fundamental question is that of how fast the approximation error of the MEM converges to zero as the truncation number goes to infinity. Despite the fact that the MEM was introduced in 1913, and has been in widespread usage as a numerical technique since as far back as 1955, to the best of the authors' knowledge, a precise characterization of the asymptotic rate of convergence of the MEM has not been obtained. In this work, we provide a resolution to this issue. While our focus in this paper is on the Dirichlet scattering problem, this is merely for convenience and our results actually establish convergence rates that hold for all MEM formulations irrespective of the specific boundary conditions or boundary integral equation solution representation chosen.