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.

在本文中，我们使用 Crank-Nicolson 时间离散化推导了弱伽辽金有限元方法的先验和后验误差估计，用于抛物线方程。先验误差估计是根据相应椭圆投影问题的现有先验误差结果推导出来的。对于后验误差估计，引入椭圆重构技术将真实误差分解为椭圆误差和抛物线误差。然后椭圆部分以辅助椭圆重建问题的后验误差估计为界。后验误差估计器进一步用于开发时间和空间自适应算法。提供了均匀和自适应网格中的数值结果以验证建议的估计器。