A posteriori error estimates for weak Galerkin methods for second order elliptic problems on polygonal meshes

Abstract

In this paper, a posteriori error estimates for the Weak Galerkin finite element methods (WG-FEMs) for second order elliptic problems are derived in terms of an H1−equivalent energy norm. Corresponding estimators based on the helmholtz decomposition yield globally upper and locally lower bounds for the approximation errors of the WG-FEMs. Especially, the error analysis of our methods is proved to be valid for polygonal meshes (e.g., hybrid, polytopal non-convex meshes and those with hanging nodes) under general assumptions. In addition, the work can make adaptive WG-FEMs solving partial differential equations such as stokes equations and biharmonic equations on polygonal meshes possible. Finally, we verify the theoretical findings by a few numerical examples.