Stability and Accuracy of a Meshless Finite Difference Method for the Stokes Equations

Abstract

We study the behavior of the meshless finite difference method based on radial basis functions applied to the stationary incompressible Stokes equations in two spatial dimensions, with the velocity and the pressure discretized on their own node sets. We demonstrate that the main condition for the stability of the numerical solution is that the distribution of the pressure nodes has to be coarser than that of the velocity both globally and locally in the domain, and there is no need for any more complex assumptions similar to the Ladyzhenskaya-Babuška-Brezzi condition in the finite element method. Optimal stability is achieved when the relative local density of the velocity to pressure nodes is about 4:1. The convergence rates of the method correspond to the convergence rates of numerical differentiation for both low and higher order discretizations. The method works well on both mesh-based and irregular nodes, such as those generated by random or quasi-random numbers and on nodes with varying density. There is no need for special staggered arrangements, which suggests that node generation algorithms may produce just one node set and obtain the other by either refinement or coarsening. Numerical results for the benchmark Driven Cavity Problem confirm the robustness and high accuracy of the method, in particular resolving a cascade of multiple Moffatt Eddies at the tip of the wedge by using nodes obtained from the quasi-random Halton sequence.