A priori and a posteriori error estimates of the weak Galerkin finite element method for parabolic problems

We derive the priori and a posteriori error estimates of the weak Galerkin finite element method with the Crank-Nicolson time discretization for the parabolic equation in this paper. The priori error estimates are deduced based on existing priori error results of the corresponding elliptic projection problem. For the a posteriori error estimates, the elliptic reconstruction technique is introduced to decompose the true error into elliptic error and parabolic error. Then the elliptic part is bounded by the a posteriori error estimates of the auxiliary elliptic reconstruction problem. The a posteriori error estimator is further used to develop the temporal and spatial adaptive algorithm. Numerical results in the uniform and adaptive meshes are provided to validate the proposed estimators.