Convergence, divergence, and inherent oscillations in MFS solutions of two-dimensional Laplace-Neumann problems

TL;DR

This work analyzes the MAS/MFS approach for 2D Laplace-Neumann problems, showing that intermediate MAS sources can diverge and oscillate while the final potential converges to the correct solution. It develops two MAS schemes for exterior circular problems (bounded and traditional fundamental solutions), derives exact finite-$N$ equations and large-$N$ asymptotics, and demonstrates that the final vector potential remains accurate despite unphysical intermediate behavior. The study extends to noncircular (elliptic) geometries and interior cavity problems, revealing both analogous convergence and distinct divergence phenomena, particularly for interior problems where the potential may diverge under certain parameter choices. These insights clarify how to interpret MAS oscillations, inform numerical practice, and motivate future work on insensitivity with oscillations and broader geometries.

Abstract

The method of fundamental solutions (MFS), also known as the method of auxiliary sources (MAS), is a well-known computational method for the solution of boundary-value problems. The final solution ("MAS solution") is obtained once we have found the amplitudes of $N$ auxiliary "MAS sources." Past studies have demonstrated that it is possible for the MAS solution to converge to the true solution even when the $N$ auxiliary sources diverge and oscillate. The present paper extends the past studies by demonstrating this possibility within the context of Laplace's equation with Neumann boundary conditions. One can thus obtain the correct solution from sources that, when $N$ is large, must be considered unphysical. We carefully explain the underlying reasons for the unphysical results, distinguish from other difficulties that might concurrently arise, and point to significant differences with time-dependent problems that were studied in the past.

Figures (5)

• Figure 1: Geometry of the exterior circular problem. Indicative positions of MAS currents and collocation points are depicted.
• Figure 1: Exterior circular problem with $\rho_{\rm cyl}/d_{\rm ref}=8$, $\rho_{\rm fil}/d_{\rm ref}=10$, $\rho_{\rm aux}/d_{\rm ref}=5.5$, and $N=81$. (a) Divergence of MAS currents: Normalized MAS currents $I_\ell/I$ vs the number $\ell$ of the auxiliary current. For symmetry reasons, the value of MAS current $I_0$ is repeated for $\ell=81$. Solid line: solution of the MAS system \ref{['sys1']}; dots: asymptotic formula \ref{['asyILodd']}. (b) Convergence of MAS potential: Normalized total vector potential $A^{\rm tot}_{z,N}/(\mu_0 I)$ vs normalized observation distance $\rho/\rho_{\rm cyl}$, for $\phi=45^{\circ}$. Solid line: MAS solution; dots: exact solution.
• Figure 1: Exterior elliptic problem with $a/d_{\rm ref}=6$, $b/d_{\rm ref}=3$, $\rho_{\rm fil}/d_{\rm ref}=7.5$, $a_{\rm aux}/d_{\rm ref}=5.2222$, $b_{\rm aux}/d_{\rm ref}=0.5205$, and $N=80$. (a) Divergence of MAS currents: Normalized MAS currents $I'_\ell/I$ vs the number $\ell$ of the auxiliary current. For symmetry reasons, the value of MAS current $I'_0$ is repeated for $\ell=80$. (b) Convergence of MAS potential: Normalized total vector potential $A'^{\rm tot}_{z,N}/(\mu_0 I)$ vs normalized observation distance $\rho/b$, for $\phi=90^{\circ}$. Solid line: MAS solution; dots: exact solution.
• Figure 2: Geometry of the interior circular problem. Indicative positions of MAS currents and collocation points are depicted.
• Figure 3: Interior circular problem with $\rho_{\rm cyl}/d_{\rm ref}=5$, $\rho_{\rm fil}/d_{\rm ref}=4$, and $\rho_{\rm aux}/d_{\rm ref}=6.5$. Divergence of MAS potential: Normalized scattered vector potential $A^{\rm sc}_{z,N}/(\mu_0 I)$ (as calculated from \ref{['AscNin']}) vs normalized observation distance $\rho/\rho_{\rm cyl}$, for $\phi=60^{\circ}$. Blue solid line: $N=59$; red dashed line: $N=60$; green dash-dotted line: $N=61$.