RBF-Generated Finite Difference Method Coupled with Quadratic Programming for Solving PDEs on Surfaces with Derivative Boundary Conditions

TL;DR

This paper introduces a mesh-free, stable, high-accuracy method combining RBF-FD and quadratic programming for solving PDEs on surfaces with derivative boundary conditions, overcoming stability issues.

Contribution

It develops a novel coupling of RBF-FD with quadratic programming to improve stability and accuracy for surface PDEs with boundary conditions, avoiding ghost points.

Findings

Method achieves high-order accuracy on various surface PDEs.

Numerical experiments confirm stable performance on complex surfaces.

Approach handles boundary conditions without ghost points or quasi-uniform nodes.

Abstract

Derivative boundary conditions introduce challenges for mesh-free discretizations of PDEs on surfaces, especially when the domain is represented by randomly sampled point clouds. The recently developed two-step tangent-space RBF-generated finite difference (RBF-FD) method provides high accuracy on closed surfaces. However, it may lose stability when applied directly to surface PDEs with derivative boundary conditions. To enhance numerical stability, we develop a mesh-free method that couples the two-step tangent-space RBF-FD discretization with a quadratic programming (QP) procedure to stabilize the operator approximation for interior points near boundaries. For boundary points, we construct restricted nearest-neighbor stencils biased in the co-normal direction and employ a constrained quadratic program to approximate outward co-normal derivatives. The resulting method avoids using ghost points and does not require quasi-uniform node distributions. We validate the approach on elliptic problems, eigenvalue problems, time-dependent diffusion equations, and elliptic interface problems on surfaces with boundary. Numerical experiments demonstrate stable performance and high-order accuracy across a variety of surfaces.