Approximating solutions of linear elliptic PDE's on a smooth manifold using local kernels

Abstract

A mesh-free numerical method for solving linear elliptic PDE's using the local kernel theory that was developed for manifold learning is proposed. In particular, this novel approach exploits the local kernel theory which allows one to approximate the Kolmogorov operator associated with It\^o processes on compact Riemannian manifolds without boundary or with Neumann boundary conditions using an integral operator. Theoretical justification for the convergence of this numerical technique is provided under the standard conditions for the existence of the weak solutions of the PDEs. Numerical results on various instructive examples, ranging from PDE's defined on flat and non-flat manifolds with known and unknown embedding functions show accurate approximation with error on the order of the kernel bandwidth parameter.