A hybrid RBF-FD and WLS mesh-free strong-form approximation method

TL;DR

This paper introduces a hybrid mesh-free strong-form method combining RBF-FD and DAM techniques, demonstrating improved stability and efficiency for solving PDEs in complex domains.

Contribution

A novel hybrid method that integrates RBF-FD and DAM for enhanced stability and computational efficiency in mesh-free PDE solutions.

Findings

The hybrid method improves accuracy over traditional approaches.

It reduces computational complexity in 2D and 3D PDE problems.

Implementation overhead is justified by performance gains.

Abstract

Since the advent of mesh-free methods as a tool for the numerical analysis of systems of Partial Differential Equations (PDEs), many variants of differential operator approximation have been proposed. In this work, we propose a local mesh-free strong-form method that combines the stability of Radial Basis Function-Generated Finite Differences (RBF-FD) with the computational effectiveness of Diffuse Approximation Method (DAM), forming a so-called hybrid method. To demonstrate the advantages of a hybrid method, we evaluate its computational complexity and accuracy of the obtained numerical solution by solving a two-dimensional Poisson problem with an exponentially strong source in the computational domain. Finally, we employ the hybrid method to solve a three-dimensional Boussinesq's problem on an isotropic half-space and show that the implementation overhead can be justified.