Convergence of the method of fundamental solutions for Poisson’s equation on the unit sphere

Solutions of boundary value problems of the Laplace equation on the unit sphere are constructed by using the fundamental solution \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Phi (\bf{x},\bf{y})=\frac{1}{4\pi \|\bf{x}-\bf{y}\|},\qquad \bf{x}, \bf{y}\in R^3.$$\end{document}With the use of radial basis approximation for finding particular solutions of Poisson's equation, the rate of convergence of the method of fundamental solutions is derived for solving the boundary value problems of Poisson’s equation.