A local target specific quadrature by expansion method for evaluation of layer potentials in 3D

TL;DR

This paper introduces a local, target-specific QBX method for evaluating layer potentials in 3D, improving accuracy and efficiency for boundary integral equations, especially in multiply connected domains.

Contribution

The paper develops a novel target-specific QBX approach that reduces expansion complexity by focusing on local neighborhoods, enhancing evaluation of layer potentials in 3D boundary problems.

Findings

The method achieves high accuracy in layer potential evaluation.

It effectively handles multiply connected domains.

The approach is validated through numerical experiments.

Abstract

Accurate evaluation of layer potentials is crucial when boundary integral equation methods are used to solve partial differential equations. Quadrature by expansion (QBX) is a recently introduced method that can offer high accuracy for singular and nearly singular integrals, using truncated expansions to locally represent the potential. The QBX method is typically based on a spherical harmonics expansion which when truncated at order $p$ has $O(p^2)$ terms. This expansion can equivalently be written with $p$ terms, however paying the price that the expansion coefficients will depend on the evaluation/target point. Based on this observation, we develop a target specific QBX method, and apply it to Laplace's equation on multiply connected domains. The method is local in that the QBX expansions only involve information from a neighborhood of the target point. An analysis of the truncation error in the QBX expansions is presented, practical parameter choices are discussed and the method is validated and tested on various problems.