Well-conditioned dipole-type method of fundamental solutions: derivation and its mathematical analysis

TL;DR

This paper introduces a well-conditioned dipole-type fundamental solutions method, removing ill-conditionality and extending it to general regions, with mathematical analysis and numerical validation.

Contribution

It develops a new approach to eliminate ill-conditionality in dipole-type fundamental solutions and extends the method to general Jordan regions using conformal mapping.

Findings

The method effectively removes ill-conditionality.

Mathematical analysis confirms stability for disk regions.

Numerical experiments demonstrate practical efficacy.

Abstract

In this paper, we examine the dipole-type method of fundamental solutions, which can be conceptualized as a discretization of the "singularity-removed" double-layer potential. We present a method for removing the ill-conditionality, which was previously considered a significant challenge, and provide a mathematical analysis in the context of disk regions. Moreover, we extend the proposed method to the general Jordan region using conformal mapping, demonstrating the efficacy of the proposed method through numerical experiments.