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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 L 2 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. [ABSTRACT FROM AUTHOR]

We introduce a new weak Galerkin finite element method whose weak functions on interior edges are double-valued and approximation spaces are based on (P k (T), P k (e), RT k (T)) elements. It is natural to develop a semi-discrete stable scheme for parabolic problems, and then fully discrete approaches are formulated with implicit θ -schemes in time for 1/2 ≤ θ ≤ 1, which include first-order backward Euler (θ = 1) and second-order Crank-Nicolson schemes (θ = 1 / 2). Furthermore, optimal convergence rates in the L 2 and energy norms are derived. Numerical results are given to verify the theory. [ABSTRACT FROM AUTHOR]