The Overlapped Radial Basis Function-Finite Difference (RBF-FD) Method: A Generalization of RBF-FD

TL;DR

This paper introduces a generalized RBF-FD method with an overlap parameter, enabling flexible stencil computations, improved stability, and significant speedups in solving PDEs compared to traditional RBF-FD approaches.

Contribution

The paper proposes a new generalized RBF-FD method with an overlap parameter, providing stability and efficiency improvements over standard RBF-FD techniques.

Findings

Achieves up to 60x speedup in 3D PDE differentiation matrix formation.

Provides a stabilization procedure based on local Lebesgue functions.

Demonstrates effectiveness on parabolic PDEs with complex boundary conditions.

Abstract

We present a generalization of the RBF-FD method that computes RBF-FD weights in finite-sized neighborhoods around the centers of RBF-FD stencils by introducing an overlap parameter $\delta \in [0,1]$ such that $\delta=1$ recovers the standard RBF-FD method and $\delta=0$ results in a full decoupling of stencils. We provide experimental evidence to support this generalization, and develop an automatic stabilization procedure based on local Lebesgue functions for the stable selection of stencil weights over a wide range of $\delta$ values. We provide an a priori estimate for the speedup of our method over RBF-FD that serves as a good predictor for the true speedup. We apply our method to parabolic partial differential equations with time-dependent inhomogeneous boundary conditions-- Neumann in 2D, and Dirichlet in 3D. Our results show that our method can achieve as high as a 60x speedup in 3D over existing RBF-FD methods in the task of forming differentiation matrices.