The permanent 2D Stokes flow is considered. The applied solution technique is the classical pressure correction method, which converts the original problem to a sequence of Poisson equations. These Poisson equations are discretized and solved in a meshless way, using local interpolation based on radial basis functions. Further improvement can be achieved by using a direct multi-elliptic approach instead of local interpolation, which results in re-globalized, quadtree-based schemes. The number of unknowns can be reduced by applying the method of fundamental solutions. A special regularization technique is introduced which uses higher order fundamental solutions without singularities. This regularization is combined with the direct multi-elliptic interpolation, which significantly reduces the computational cost and makes it possible to avoid the use of dense and ill-conditioned matrices.