Numerical study of the RBF-FD method for the Stokes equations

Abstract

We study the numerical behavior of the meshless Radial Basis Function Finite Difference method applied to the stationary incompressible Stokes equations in two spatial dimensions, using polyharmonic splines as radial basis functions with a polynomial extension on two separate node sets to discretize the velocity and the pressure. On the one hand,we show that the convergence rates of the method correspond to the known convergence rates of numerical differentiation by the polyharmonic splines. On the other hand, we show that the main condition for the stability of the numerical solution is that the distributions of the pressure nodes has to be coarser than that of the velocity everywhere in the domain. There seems to be no need for any complex assumptions similar to the Ladyzhenskaya-Babuška-Brezzi condition in the finite element method. Numerical results for the benchmark driven cavity problem are in a good agreement with those in the literature.