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.

arXiv comment18 pages, 5 figures, accepted to Journal of Computational Physics

References in corpus (3)

• Stable computations with flat radial basis functions using vector-valued rational approximations
• The Overlapped Radial Basis Function-Finite Difference (RBF-FD) Method: A Generalization of RBF-FD
• Mesh-free Semi-Lagrangian Methods for Transport on a Sphere Using Radial Basis Functions

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• A General-Purpose, Inelastic, Rotation-Free Kirchhoff-Love Shell Formulation for Peridynamics
• RBF-FD discretization of the Navier-Stokes equations on scattered but staggered nodes
• An Efficient High-Order Meshless Method for Advection-Diffusion Equations on Time-Varying Irregular Domains
• First-Passage Time Statistics on Surfaces of General Shape: Surface PDE Solvers using Generalized Moving Least Squares (GMLS)
• A Meshfree RBF-FD Constant along Normal Method for Solving PDEs on Surfaces