Generalized weak Galerkin methods for Stokes equations

A finite element method, called generalized Weak Galerkin (gWG), is introduced for the Stokes equation by using a new weak gradient notion in the usual weak Galerkin approach. The gWG method allows polynomial elements of arbitrary order in any combination for the velocity variable, but with proper constraints on the pressure element due to the inf-sup condition. Error estimates are derived for the velocity approximation in a mesh-dependent energy norm, as well as in the L2 norm for both the velocity and the pressure approximations. Some numerical examples are presented to verify the accuracy, theoretical convergence order, and robustness of the proposed numerical scheme.