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

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.