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

TL;DR

This work addresses ill-conditioning in the dipole-type method of fundamental solutions (DSM) for Laplace boundary-value problems by introducing DSM-QR, a QR-based basis reparameterization. It establishes unique solvability and $O(1)$ conditioning on disk domains and proves convergence with explicit error behavior tied to the decay of boundary data Fourier coefficients. The method is extended to general Jordan regions through conformal mappings, supported by numerical experiments that show exponential convergence and well-conditioned systems even for complex geometries. The results offer a robust, mesh-free solver for Dirichlet problems in irregular domains and lay a foundation for extensions to higher dimensions and other PDEs.

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.

Figures (5)

• Figure 1: Plots of the $L^\infty$-norms of errors and the condition numbers. (Left) $L^\infty$-error; (Right) condition number when $\Omega$ is a disk region.
• Figure 2: Arrangements of the collocation points, the singular points, and the dipole moments in Example \ref{['ex:Jordan_1']}.
• Figure 3: Plots of the $L^\infty$-norms of errors and the condition numbers in Example \ref{['ex:Jordan_1']}. (Left) $L^\infty$-error; (Right) condition number.
• Figure 4: Arrangements of the collocation points, the singular points, and the dipole moments in Example \ref{['ex:Jordan_2']}.
• Figure 5: Plots of the $L^\infty$-norms of errors and the condition numbers in Example \ref{['ex:Jordan_2']}. (Left) $L^\infty$-error; (Right) condition number.

Theorems & Definitions (14)

• Theorem 1.1
• Theorem 1.2
• Theorem 1.3
• Lemma 4.1
• Lemma 4.2
• proof
• proof : Proof of Theorem \ref{['thm:condition-number']}
• Lemma 4.3
• proof
• Lemma 4.4