Strongly elliptic pseudodifferential equations on the sphere with radial basis functions

Abstract

Spherical radial basis functions are used to define approximate solutions to strongly elliptic pseudodifferential equations on the unit sphere. These equations arise from geodesy. The approximate solutions are found by the Galerkin and collocation methods. A salient feature of the paper is a unified theory for error analysis of both approximation methods.

Error Bounds for a Least Squares Meshless Finite Difference Method on Closed Manifolds

“…The proof of Theorem 1 will be given in Section 4. Note that convergence in H 2κ+r (M) as h A → 0 follows from (47) only when 0 ≤ r < s − 2κ − d, thus under a stricter assumption than (43), s > 2κ + d, and the discrete convergence follows from (48) under the assumption that s > max{2κ, d 2 + 1} + d.…”

“…Meshless numerical methods are particularly attractive for solving operator equations on manifolds, in particular on the sphere [22,23], because of the difficulties of creating and maintaining suitable meshes or grids in this setting. Error bounds for meshless methods for the approximation of functions and specifically solutions of operator equations on manifolds have been studied for example in [24,26,27,32,33,35,36,37,43]. However, these results do not apply to localized finite difference type methods considered below.…”

A Posteriori Error Estimates for Hypersingular Integral Equation on Spheres with Spherical Splines

“…Spherical radial basis functions appear to be more suitable for solving problems with scattered data, see e.g. [29,32,37,42] and references therein. However, the resulting matrix system from this approximation is very ill-conditioned.…”

Error bounds for a least squares meshless finite difference method on closed manifolds

“…Communicated by: Tobin Driscoll Meshless numerical methods are particularly attractive for solving operator equations on manifolds, in particular on the sphere [32,33], because of the difficulties of creating and maintaining suitable meshes or grids in this setting. Error bounds for meshless methods for the approximation of functions and specifically solutions of operator equations on manifolds have been studied for example in [12,14,34,36,37,42,43,[45][46][47]53]. However, these results do not apply to localized finite difference type methods considered below.…”