Solution to the Neumann problem exterior to a prolate spheroid by radial basis functions

We consider the exterior Neumann problem of the Laplacian with boundary condition on a prolate spheroid. We propose to use spherical radial basis functions in the solution of the boundary integral equation arising from the Dirichlet-to-Neumann map. Our approach is particularly suitable for handling of scattered data, e.g. satellite data. We also propose a preconditioning technique based on domain decomposition methods to deal with ill-conditioness of the matrix arising from the Galerkin scheme.