Solving non-strongly elliptic pseudodifferential equations on a sphere using radial basis functions
Non-strongly elliptic pseudodifferential equations on the unit sphere arise from geodesy. An example of equations of this type is the boundary integral reformulation of a boundary value problem with the Laplace equation in the interior domain of the unit sphere, and a Robin boundary condition. Approximate solutions with spherical radial basis functions are found by the Galerkin and collocation methods. The paper presents a unified theory for error analysis of both approximation methods. The theoretical results are corroborated by numerical experiments. It is noted that the stiffness matrix arising from the Galerkin method for this problem resembles that arising from a least squares method.
elliptic pseudodifferential equation sphere radial basis function collocation method galerkin method