Adaptive RBF-FD Method for Elliptic Problems with Point Singularities in 2D
For computational scientists solving elliptic PDEs with singularities, this work offers a meshless alternative to FEM with comparable accuracy, though it is an incremental improvement over prior RBF-FD methods.
The paper presents an adaptive meshless RBF-FD method that achieves competitive accuracy with finite element methods on 2D elliptic problems with point singularities, demonstrating up to 2x error reduction on benchmark problems with reentrant corners and sharp peaks.
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.