A Weak Galerkin Mixed Finite Element Method for Second-Order Elliptic Equations with Robin Boundary Conditions

Abstract

In this paper, we present a weak Galerkin (WG) mixed finite element method for solving the second-order elliptic equations with Robin boundary conditions. Stability and a priori error estimates for the coupled method are derived. We present the optimal order error estimate for the WG-MFEM approximations in a norm that is related to the $L^2$ for the flux and $H^1$ for the scalar function. Also an optimal order error estimate in $L^2$ is derived for the scalar approximation by using a duality argument. A series of numerical experiments is presented that verify our theoretical results.