An overlapping additive Schwarz preconditioner for the Laplace-Beltrami equation using spherical splines (Q695632)

The paper deals with an overlapping domain decomposition technique based on a two-level mesh for solving the Laplace-Beltrami equation on the sphere. Spherical splines are used to solve the equation by Galerkin's method. The authors overcome the ill-conditionedness of the resulting linear system by preconditioning with additive Schwarz methods. The authors prove that the condition number of the additive Schwarz operator is bounded by \(O(H^{2}/h^{2})\) where \(H\) and \(h\) are the sizes of the coarse and fine meshes, respectively. A better bound is proved if the degree of the splines is even. Numerical experiments on large point sets taken from MAGSAT satellite data illustrate the method.