Robust a-posteriori error estimates for weak Galerkin method for the convection-diffusion problem

Abstract

We present a robust a posteriori error estimator for a weak Galerkin finite element method applied to stationary convection-diffusion equations in the convection-dominated regime. The estimator provides global upper and lower bounds of the error and is robust in the sense that upper and lower bounds are uniformly bounded with respect to the diffusion coefficient. The weak Galerkin method we use was developed by Lin, Ye, Zhang, and Zhu (2018) for the convection-diffusion problem without assuming any additional conditions on the convection coefficient and, has a simple formulation. The motivation for our work comes from the fact that while this method performs very well in the strongly convection-dominated regime, it continues to exhibit poor behavior in the intermediate regime. In this proposed work we show that by relying on adaptively refined meshes based on a posteriori residual-type estimator, we can retrieve the optimal order of convergence for all the regimes not just for the strongly convection-dominated regime. Results of the numerical experiments are presented to illustrate the performance of the error estimator.