A Fast Algorithm with Error Bounds for Quadrature by Expansion

TL;DR

This paper introduces an improved algorithm for Quadrature by Expansion (QBX) that provides rigorous error bounds and maintains computational efficiency, enhancing the evaluation of layer potentials in two dimensions.

Contribution

The paper presents a modified QBX algorithm with comprehensive error and cost analysis, improving upon previous methods for rapid and accurate integral evaluation.

Findings

Empirical cost-per-accuracy is comparable to prior approaches.

The new algorithm demonstrates scalability in numerical experiments.

Provides rigorous error bounds for QBX in 2D Laplace problems.

Abstract

Quadrature by Expansion (QBX) is a quadrature method for approximating the value of the singular integrals encountered in the evaluation of layer potentials. It exploits the smoothness of the layer potential by forming locally-valid expansion which are then evaluated to compute the near or on-surface value of the integral. Recent work towards coupling of a Fast Multipole Method (FMM) to QBX yielded a first step towards the rapid evaluation of such integrals (and the solution of related integral equations), albeit with only empirically understood error behavior. In this paper, we improve upon this approach with a modified algorithm for which we give a comprehensive analysis of error and cost in the case of the Laplace equation in two dimensions. For the same levels of (user-specified) accuracy, the new algorithm empirically has cost-per-accuracy comparable to prior approaches. We provide experimental results to demonstrate scalability and numerical accuracy.