A Class Of Weak Galerkin Finite Element Methods For The Incompressible Fluid Model

The weak Galerkin finite element method (WG) is a new and efficient numerical method for partial differential equations. It is designed by using the generalized functions and their weak derivatives which are defmed as distributions. The variational form we considered is based on gradient operators and divergence operator which are different from the usual gradient-divergence operators. The WG method is highly flexible and efficient by allowing the use of discontinuous functions on arbitrary polygons or polyhedra with certain shape regularity. We use the Schur Complement skill to reduce the degrees of freedom, which makes the solving is more simpler and faster.The theory and numerical calculation of incompressible Stokes equations are very important issues in the field of physics, engineering and mathematics, and have extremely extensive applications. This paper discusses the stationary incompressible Stokes equation in two or three dimensional spaces with two kinds of variational forms by weak Galerkin finite element method. Optimal-order error orders are established for the approximated WG finite element solutions in various norms. Some numerical results are presented to demonstrate the efficiency of the presented method.