The D-RBF-PU method for solving surface PDEs

Abstract

In this paper a new direct RBF partition of unity (D-RBF-PU) method is developed for numerical solution of partial differential equations defined on smooth orientable surfaces which are discretized with sets of scattered nodes and with approximations to normal vectors at each of the nodes. The accuracy, stability and efficiency of the new method are studied through some theoretical and experimental results. This method is a localized RBF based technique, results in a perfectly sparse final linear system, uses only scattered nodes on the surface rather than a connected mesh, and is applicable for a large class of PDEs on manifolds. Applications to some biological and chemical reaction-diffusion models are also given. Results show that the new method outperforms other comparable techniques for surface PDEs.