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

TL;DR

This paper introduces a mesh-free numerical method leveraging local kernel theory to approximate solutions of linear elliptic PDEs on smooth manifolds, providing theoretical convergence guarantees and demonstrating high accuracy in diverse examples.

Contribution

It presents a novel mesh-free approach using local kernels to solve elliptic PDEs on manifolds, with proven convergence and practical effectiveness.

Findings

Accurate PDE solutions on manifolds with error proportional to kernel bandwidth.

Method works on flat and curved manifolds with known or unknown embeddings.

Theoretical convergence guarantees under standard PDE solution conditions.

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.