Method of fundamental solutions for Neumann problems of the modified Helmholtz equation in disk domains

TL;DR

This paper rigorously demonstrates the existence and exponential convergence of the method of fundamental solutions for Neumann problems of the modified Helmholtz equation in disk domains, with error estimates and numerical validation.

Contribution

It provides a rigorous proof of the existence, convergence, and error bounds of the MFS for Neumann problems in disk domains, including exponential convergence rates.

Findings

Existence of approximate solutions is rigorously established.

Error between approximate and exact solutions decreases exponentially with more collocation points.

Numerical simulations confirm exponential convergence as the number of collocation points increases.

Abstract

The method of the fundamental solutions (MFS) is used to construct an approximate solution for a partial differential equation in a bounded domain. It is demonstrated by combining the fundamental solutions shifted to the points outside the domain and determining the coefficients of the linear sum to satisfy the boundary condition on the finite points of the boundary. In this paper, the existence of the approximate solution by the MFS for the Neumann problems of the modified Helmholtz equation in disk domains is rigorously demonstrated. We reveal the sufficient condition of the existence of the approximate solution. Applying Green's theorem to the Neumann problem of the modified Helmholtz equation, we bound the error between the approximate solution and exact solution into the difference of the function of the boundary condition and the normal derivative of the approximate solution by boundary integrations. Using this estimate of the error, we show the convergence of the approximate solution by the MFS to the exact solution with exponential order, that is, $N^2a^N$ order, where $a$ is a positive constant less than one and $N$ is the number of collocation points. Furthermore, it is demonstrated that the error tends to $0$ in exponential order in the numerical simulations with increasing number of collocation points $N$.