A Class of Weak Galerkin Finite Element Methods for the Incompressible Fluid Model

Abstract

In this paper, we propose a weak Galerkin finite element method (WG) for solving the stationary incompressible Stokes equation in two or three dimensional space. The weak Galerkin finite element method is highly flexible by allowing the use of discontinuous functions on arbitrary polygons or polyhedra with certain shape regularity. However, since additional variables are introduced, the computational cost is much higher. Our new method can significantly reduce the computational cost  while maintaining the accuracy. Optimal error orders are established for the weak Galerkin finite element approximations in various norms. Some numerical results are presented to demonstrate the efficiency of the method.