Highly localized RBF Lagrange functions for finite difference methods on spheres (Q6579311)

The authors study how rapidly decaying RBF Lagrange functions on the sphere can be used to create a numerically feasible, stable finite difference method based on radial basis functions (an RBF-FD-like method) and the application to certain PDE. The RBF-FD method and its variants are modifications of the classical finite difference method suitable for working with unstructured point sets. The authors consider a time-independent partial differential equation on a manifold without boundary (such as the sphere \(\mathbb{S}^2\)), \(\mathscr Lu = f\). The PDE is replaced by a linear system \(\mathbf Mu = y\) which can then be solved for a discrete solution. The aim of this paper is to give conditions on the operator \(\mathscr L\) which guarantee stable invertibility of \(\mathbf M\), and provide satisfying convergence rates.