Abstract
Localized collocation methods based on radial basis functions (RBFs) for
elliptic problems appear to be non-robust 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.