Weak Galerkin method with (r,r−1,r−1)(r,r−1,r−1)-order finite elements for second order parabolic equations

In this paper, we propose a weak Galerkin method with a stabilization term for a model problem of second order parabolic differential equations. We establish both the continuous time and the discrete time weak Galerkin finite element schemes, which allow the use of totally discontinuous piecewise polynomial basis and the finite element partitions on shape regular polygons. In addition, we adapt the combination of polynomial spaces {Pr(T0),Pr−1(e¯),[Pr−1(T)]2} that reduces the number of unknowns in the numerical scheme without compromising the accuracy of the numerical approximation. We show as well that the continuous time weak Galerkin finite element method preserves the energy conservation law. The optimal convergence order estimates in both discrete H1 and L2 norms are obtained. We also present numerical experiments to illustrate the theoretical results.