Adaptive RBF-FD Method for Elliptic Problems with Point Singularities in 2D

TL;DR

This paper introduces an adaptive meshless RBF-FD method for solving 2D elliptic problems with point singularities, demonstrating competitive accuracy and efficiency against traditional finite element methods.

Contribution

The paper presents a novel adaptive RBF-FD approach with an improved error indicator, enhancing solution accuracy near singularities in elliptic problems.

Findings

Achieves high accuracy on benchmark problems with boundary reentrant corners.

Effectively handles sharp peaks and oscillations near point singularities.

Outperforms earlier algorithms in adaptive meshless methods.

Abstract

We describe and test numerically an adaptive meshless generalized finite difference method based on radial basis functions that competes well with the finite element method on standard benchmark problems with reentrant corners of the boundary, sharp peaks and rapid oscillations in the neighborhood of an isolated point. This is achieved thanks to significant improvements introduced into the earlier algorithms of [Oleg Davydov and Dang~Thi Oanh, Adaptive meshless centers and RBF stencils for Poisson equation, Journal of Computational Physics, 230:287--304, 2011], including a new error indicator of Zienkiewicz-Zhu type.