Extrinsic Meshless Collocation Methods for PDEs on Manifolds

Abstract

We proposed ways to implement meshless collocation methods extrinsically for solving elliptic PDEs on smooth, closed, connected, and complete Riemannian manifolds with arbitrary codimensions. Our methods are based on strong-form collocations with oversampling and least-squares minimizations, which can be implemented either analytically or approximately. By restricting global kernels to the manifold, our methods resemble their easy-to-implement domain-type analogies, i.e., Kansa methods. Our main theoretical contribution is the robust convergence analysis under some standard smoothness assumptions for high-order convergence. Numerical demonstrations are provided to verify the proven convergence rates, and we simulate reaction-diffusion equations for generating Turing patterns on manifolds.