A Meshfree Generalized Finite Difference Method for Surface PDEs

TL;DR

This paper introduces a meshfree generalized finite difference method for solving PDEs on surfaces, avoiding the need for meshes or explicit manifold reconstruction, and enabling direct adaptation of volume-based methods to surface problems.

Contribution

It presents a novel meshfree approach that discretizes surface PDEs directly on tangent spaces, simplifying implementation and extending volume-based methods to manifolds.

Findings

Method effectively discretizes surface gradient, Laplacian, and diffusion operators.

Handles anisotropic and discontinuous surface properties.

Demonstrates practical applications on complex surface PDEs.

Abstract

In this paper, we propose a novel meshfree Generalized Finite Difference Method (GFDM) approach to discretize PDEs defined on manifolds. Derivative approximations for the same are done directly on the tangent space, in a manner that mimics the procedure followed in volume-based meshfree GFDMs. As a result, the proposed method not only does not require a mesh, it also does not require an explicit reconstruction of the manifold. In contrast to existing methods, it avoids the complexities of dealing with a manifold metric, while also avoiding the need to solve a PDE in the embedding space. A major advantage of this method is that all developments in usual volume-based numerical methods can be directly ported over to surfaces using this framework. We propose discretizations of the surface gradient operator, the surface Laplacian and surface Diffusion operators. Possibilities to deal with anisotropic and discontinous surface properties (with large jumps) are also introduced, and a few practical applications are presented.