A LEAST SQUARES RADIAL BASIS FUNCTION FINITE DIFFERENCE METHOD WITH IMPROVED STABILITY PROPERTIES

Localized collocation methods based on radial basis functions (RBFs) for elliptic problems appear to be nonrobust in the presence of Neumann boundary conditions. In this paper, we overcome this issue by formulating the RBF-generated finite difference method in a discrete least squares setting instead. This allows us to prove high-order convergence under node refinement and to numerically verify that the least squares formulation is more accurate and robust than the collocation formulation. The implementation effort for the modified algorithm is comparable to that for the collocation method.

Automated summary

This paper introduces a formulation of the RBF-generated finite difference method in a discrete least squares setting to overcome the nonrobustness issue in the presence of Neumann boundary conditions. The high-order convergence under node refinement is proven, and numerical verification shows that the least squares formulation is more accurate and robust than the collocation formulation. The implementation effort for the modified algorithm is comparable to that for the collocation method.