RBF-LOI: Augmenting Radial Basis Functions (RBFs) with Least Orthogonal Interpolation (LOI) for Solving PDEs on Surfaces

TL;DR

This paper introduces RBF-LOI, a stable and coordinate-free method combining RBFs and LOI for solving PDEs on surfaces, demonstrating high accuracy and robustness on spheres, tori, and biological reaction-diffusion models.

Contribution

The paper develops a novel RBF-LOI technique that augments RBFs with polynomial restrictions via LOI, enabling stable, high-order PDE solutions on manifolds without intrinsic coordinates.

Findings

Achieves high convergence orders on sphere and torus PDEs.

Demonstrates robustness to stagnation errors in RBF interpolation.

Successfully applies to biological reaction-diffusion PDEs.

Abstract

We present a new method for the solution of PDEs on manifolds $\mathbb{M} \subset \mathbb{R}^d$ of co-dimension one using stable scale-free radial basis function (RBF) interpolation. Our method involves augmenting polyharmonic spline (PHS) RBFs with polynomials to generate RBF-finite difference (RBF-FD) formulas. These polynomial basis elements are obtained using the recently-developed \emph{least orthogonal interpolation} technique (LOI) on each RBF-FD stencil to obtain \emph{local} restrictions of polynomials in $\mathbb{R}^3$ to stencils on $\mathbb{M}$. The resulting RBF-LOI method uses Cartesian coordinates, does not require any intrinsic coordinate systems or projections of points onto tangent planes, and our tests illustrate robustness to stagnation errors. We show that our method produces high orders of convergence for PDEs on the sphere and torus, and present some applications to reaction-diffusion PDEs motivated by biology.