Divergence-free meshless local Petrov-Galerkin method for Stokes flow

The purpose of the present paper is development of an efficient meshless solution of steady incompressible Stokes flow problems with constant viscosity in two dimensions, with algebraic order of accuracy. This is achieved by employing a weak formulation with divergence-free matrix-valued quadratic Matern (QM) radial basis function (RBF) for the shape function and divergence-free matrix-valued compactly supported Gaussian (CSG) RBF for the weight function on the computational domain and its boundary. The continuity equation is inherently built-in in the formulation and the pressure is eliminated from the formulation with the aid of divergence theorem and the choice of divergence-free weight function. The developed method is thus iteration free, and results in a banded system of equations to be solved jointly for both velocity components. Gauss-Legendre cell integration is performed in the current investigation. The characteristics of the method are assessed by changing its free parameters, i.e., weight functions' sub-domain radius and shape functions' support domain radius and the shape parameter. A sensitivity test for several choices of shape functions with regular centers arrangement is done to identify the appropriate support size for the shape and weight functions and stagnation errors are reported accordingly. To the best of our knowledge, this article is initiative in introducing the application of divergence-free MLPG method to incompressible flows, aiming at elimination of pressure from the governing equations in primitive variables, with the aid of divergence-free RBFs through weak formulation. Only the momentum equation needs to be solved. Hence, the formulation of the problem is much simpler than the building of divergence-free elements in the related mesh-based methods.