A quasi-RBF technique for numerical discretization of PDE's

TL;DR

This paper introduces a meshfree numerical method combining quasi-RBF and Fourier techniques to efficiently solve PDEs with complex geometries, avoiding artificial boundaries and achieving high accuracy.

Contribution

It proposes a novel quasi-RBF approach that approximates homogeneous solutions with nonsingular general solutions, enhancing accuracy and simplicity for complex PDE geometries.

Findings

High accuracy and spectral convergence demonstrated

Avoids artificial boundary issues of fundamental solutions

Efficient evaluation using NlogN algorithms

Abstract

Atkinson developed a strategy which splits solution of a PDE system into homogeneous and particular solutions, where the former have to satisfy the boundary and governing equation, while the latter only need to satisfy the governing equation without concerning geometry. Since the particular solution can be solved irrespective of boundary shape, we can use a readily available fast Fourier or orthogonal polynomial technique O(NlogN) to evaluate it in a regular box or sphere surrounding physical domain. The distinction of this study is that we approximate homogeneous solution with nonsingular general solution RBF as in the boundary knot method. The collocation method using general solution RBF has very high accuracy and spectral convergent speed and is a simple, truly meshfree approach for any complicated geometry. More importantly, the use of nonsingular general solution avoids the controversial artificial boundary in the method of fundamental solution due to the singularity of fundamental solution.