Solving partial differential equations on (evolving) surfaces with radial basis functions

摘要

Meshfree, kernel-based spatial discretisations are recent tools to discretise partial differential equations on surfaces. The goals of this paper are to analyse and compare three different meshfree kernel-based methods for the spatial discretisation of semilinear parabolic partial differential equations (PDEs) on surfaces, i.e. on smooth, compact, connected, orientable, and closed (d - 1)-dimensional submanifolds of Rd. The three different methods are collocation, the Galerkin, and the RBF-FD method, respectively. Their advantages and drawbacks are discussed, and previously known theoretical results are extended and numerically verified. Finally, a significant part of this paper is devoted to solving PDEs on evolving surfaces with RBF-FD, which has not been done previously.