  This article presents a simplified formulation for the weak Galerkin finite
element method for the Stokes equation without using the degrees of freedom
associated with the unknowns in the interior of each element as formulated in
the original weak Galerkin algorithm. The simplified formulation preserves the
important mass conservation property locally on each element and allows the use
of general polygonal partitions. A particular application of the simplified
weak Galerkin on rectangular partitions yields a new class of 5- and 7-point
finite difference schemes for the Stokes equation. An explicit formula is
presented for the computation of the element stiffness matrices on arbitrary
polygonal elements. Error estimates of optimal order are established for the
simplified weak Galerkin finite element method in the H^1 and L^2 norms.
Furthermore, a superconvergence of order O(h^{1.5}) is established on
rectangular partitions for the velocity approximation in the H^1 norm at cell
centers, and a similar superconvergence is derived for the pressure
approximation in the L^2 norm at cell centers. Some numerical results are
reported to confirm the convergence and superconvergence theory.
