An overlapping additive Schwarz preconditioner for the Laplace-Beltrami equation using spherical splines

We present an overlapping domain decomposition technique for solving the Laplace–Beltrami equation on the sphere with spherical splines. We prove that the condition number of the additive Schwarz operator is bounded by \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$O\left(H^2/h^2\right)$\end{document}, where H and h are the sizes of the coarse and fine meshes, respectively. In the case that the degree of the splines is even, a better bound \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$O\left(\max_{1\leq k \leq J}\left(1+H_k/\delta_k\right)\right)$\end{document} is proved. Here J is the number of subdomains, Hk is the size of the kth subdomain, and δk is the size of the overlap of the kth subdomain. The method is illustrated by numerical experiments on large point sets taken from magsat satellite data.