Weak Galerkin Finite Element Method for Second Order Parabolic Equations

Abstract

We apply in this paper the weak Galerkin method to the second order parabolic differential equations based on a discrete weak gradient operator. We establish both the continuous time and the discrete time weak Galerkin finite element schemes, which allow using the totally discrete functions in approximation space and the finite element partitions of arbitrary polygons with certain shape regularity. 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 $H^1$ and $L^2$ norms are obtained. Numerical experiments are performed to confirm the theoretical results.