A well-conditioned method of fundamental solutions for Laplace equation

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 are typically ill-conditioned and this may prevent the method from achieving high accuracy. In this work, we propose a new algorithm to remove the ill-conditioning of the classical MFS in the context of the 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 when possible to be applied, this approach is much superior to previous approaches, such as the classical MFS or the MFS-QR.

Automated summary

The method of fundamental solutions (MFS) is a numerical method for solving linear partial differential equations. However, the ill-conditioning of the matrices in the original method limits its accuracy. This study proposes a new algorithm to overcome the ill-conditioning of classical MFS by expanding the basis functions in terms of harmonic polynomials.