A numerical investigation of some RBF-FD error estimates

In a recent paper by Tominec, Larsson and Heryudono a convergence proof for an oversampled version of the RBF-FD method, using polyharmonic spline basis functions augmented with polynomials, was derived. In this paper, we take a closer look at the individual estimates involved in this proof. We investigate how large the bounds are and how they depend on the node layout, the stencil size, and the polynomial degree. We find that a moderate amount of oversampling is sufficient for the method to be stable when Halton nodes are used for the stencil approximations, while a random node layout may require a very high oversampling factor. From a practical perspective, this indicates the importance of having a locally quasi uniform node layout for the method to be stable and give reliable results. We see an overall growth of the error constant with the polynomial degree and with the stencil size.