A Posteriori Error Estimates of Edge Residual-Type of Weak Galerkin Mixed FEM Solving Second-Order Elliptic Problems on Polytopal Mesh

Abstract

In this paper, we present a posteriori error estimator of edge residual-type Weak Galerkin mixed finite element method (WG-MFEM) solving second-order elliptic problems, where two different ways of a posteriori error estimator are presented, both of which hold on polygonal mesh. The two analyses are based on the standard FEM method and continuous local mass conservation property, respectively. In order to prove the reliability holding on polygonal meshes, the a posteriori error estimator is disposed by the post-processed numerical scalar solution. Compared with some other methods, the proposed a posteriori error estimator does not contain the post-processed solution and the efficiency can be obtained automatically. Numerical experiments are conducted to confirm the efficiency of this estimator.

A posteriori error estimate of the weak Galerkin finite element method solving the Stokes problems on polytopal meshes

“…which has been proved in [39]. Combining the estimates (3.8)-(3.10) and then taking a summation over T, Lemma 2.1 yields…”