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

Abstract

In this paper, we propose an a posteriori error estimate of the weak Galerkin finite element method (WG-FEM) solving the Stokes problems with variable coefficients. Its error estimator, based on the property of Stokes' law conservation, Helmholtz decomposition and bubble functions, yields global upper bound and local lower bound for the approximation error of the WG-FEM. Error analysis is proved to be valid under the mesh assumptions of the WG-FEM and the way can be extended to other FEMs with the property of Stokes' law conservation, for example, discontinuous Galerkin (DG) FEMs. Finally, we verify the performance of error estimator by performing a few numerical examples.