Solving the Dirichlet-to-Neumann map on an oblate spheroid by a mesh-free method

In this paper, we study a mesh-free method using the Galerkin method with radial basis functions (RBFs) for the exterior Neumann problem of the Laplacian with boundary condition on an oblate spheroid. This problem is reformulated as a pseudo-differential equation on the spheroid by using the Dirichlet-to-Neumann map. We show convergence of the Galerkin scheme. Our approach is particularly suitable for handling scattered data. We also propose a fast solution technique based on a domain decomposition method (obtained by the additive Schwarz operator) to precondition the illconditioned matrices arising from the Galerkin scheme. We estimate the condition number of the preconditioned system. Numerical results supporting the theoretical results are presented.