A Locally Conservative Post-Processing Technique for the Crouzeix-Raviart Finite Element Method

Considering the Crouzeix-Raviart finite element method for solving elliptic equations, we propose a post-processing technique to obtain a locally conservative solution. The method only needs to solve a small linear system on each element and achieves the optimal convergence rate. The novelty of this paper is to generalize the post-processing method to the Crouzeix-Raviart element and to demonstrate that the boundary averaging term in most existing methods is not necessary here. We rigorously prove the local conservation property, the existence, the uniqueness, and the accuracy of the post-processed solution. Finally, our numerical experiments verify the theoretical results.