A well conditioned Method of Fundamental Solutions

TL;DR

This paper introduces a new algorithm that significantly improves the conditioning of the Method of Fundamental Solutions for Laplace equations in planar domains by expanding basis functions with harmonic polynomials and applying SVD and Arnoldi orthogonalization.

Contribution

The paper presents a novel approach to eliminate ill-conditioning in the classical MFS by using harmonic polynomial expansions and advanced matrix decompositions.

Findings

The new method achieves better numerical stability than classical MFS.

Numerical examples demonstrate superior accuracy and conditioning.

The approach outperforms previous methods like MFS-QR.

Abstract

The method of fundamental solutions (MFS) is a numerical method for solving boundary value problems involving linear partial differential equations. It is well known that it can be very effective assuming regularity of the domain and boundary conditions. The main drawback of the MFS is that the matrices involved typically are ill-conditioned and this may prevent to achieve high accuracy. In this work, we propose a new algorithm to remove the ill conditioning of the classical MFS in the context of Laplace equation defined in planar domains. The main idea is to expand the MFS basis functions in terms of harmonic polynomials. Then, using the singular value decomposition and Arnoldi orthogonalization we define well conditioned basis functions spanning the same functional space as the MFS's. Several numerical examples show that this approach is much superior to previous approaches, such as the classical MFS or the MFS-QR.