A Meshfree RBF-FD Constant along Normal Method for Solving PDEs on Surfaces

TL;DR

This paper presents a meshfree RBF-FD method for solving PDEs on surfaces that avoids grid use, achieves high-order convergence, and is effective for evolving surfaces and complex geometries.

Contribution

It introduces a novel meshfree RBF-FD approach using a constant normal extension, eliminating the need for a closest point mapping and enabling high-order solutions on surfaces.

Findings

High-order convergence on surface PDEs

Effective for evolving surfaces with particle tracking

Stable and flexible method with complex geometries

Abstract

This paper introduces a novel meshfree methodology based on Radial Basis Function-Finite Difference (RBF-FD) approximations for the numerical solution of partial differential equations (PDEs) on surfaces of codimension 1 embedded in $\mathbb{R}^3$. The method is built upon the principles of the closest point method, without the use of a grid or a closest point mapping. We show that the combination of local embedded stencils with these principles can be employed to approximate surface derivatives using polyharmonic spline kernels and polynomials (PHS+Poly) RBF-FD. Specifically, we show that it is enough to consider a constant extension along the normal direction only at a single node to overcome the rank deficiency of the polynomial basis. An extensive parameter analysis is presented to test the dependence of the approach. We demonstrate high-order convergence rates on problems involving surface advection and surface diffusion, and solve Turing pattern formations on surfaces defined either implicitly or by point clouds. Moreover, a simple coupling approach with a particle tracking method demonstrates the potential of the proposed method in solving PDEs on evolving surfaces in the normal direction. Our numerical results confirm the stability, flexibility, and high-order algebraic convergence of the approach.