Fast algorithms for Quadrature by Expansion I: Globally valid expansions

TL;DR

This paper introduces a unified fast algorithm combining Quadrature by Expansion with a customized Fast Multipole Method for efficient, accurate evaluation of layer potentials in boundary value problems for the Helmholtz equation in two dimensions.

Contribution

It presents a novel integrated numerical scheme that couples Quadrature by Expansion with a specialized FMM, enabling linear-time evaluation of layer potentials with uniform accuracy.

Findings

Achieves linear-time complexity for potential evaluation

Provides uniform high-accuracy results

Demonstrates speed and accuracy through numerical examples

Abstract

The use of integral equation methods for the efficient numerical solution of PDE boundary value problems requires two main tools: quadrature rules for the evaluation of layer potential integral operators with singular kernels, and fast algorithms for solving the resulting dense linear systems. Classically, these tools were developed separately. In this work, we present a unified numerical scheme based on coupling Quadrature by Expansion, a recent quadrature method, to a customized Fast Multipole Method (FMM) for the Helmholtz equation in two dimensions. The method allows the evaluation of layer potentials in linear-time complexity, anywhere in space, with a uniform, user-chosen level of accuracy as a black-box computational method. Providing this capability requires geometric and algorithmic considerations beyond the needs of standard FMMs as well as careful consideration of the accuracy of multipole translations. We illustrate the speed and accuracy of our method with various numerical examples. Keywords: Layer Potentials; Singular Integrals; Quadrature; High-order accuracy; Integral equations; Helmholtz equation; Fast multipole method.