Stabilization and a posteriori error analysis of a mixed FEM for convection–diffusion problems with mixed boundary conditions

Abstract

We introduce a new augmented dual-mixed finite element method for the linear convection-diffusion equation with mixed boundary conditions. The approach is based on adding suitable residual type terms to a dual-mixed formulation of the problem. We prove that for appropriate values of the stabilization parameters, that depend on the diffusivity and the magnitude of the convective velocity, the new variational formulation and the corresponding Galerkin scheme are well-posed and a Céa estimate can be derived. We establish the rate of convergence when the flux and the concentration are approximated, respectively, by Raviart–Thomas/Brezzi–Douglas–Marini and continuous piecewise polynomials. In addition, we develop an a posteriori error analysis of residual type. We derive a simple a posteriori error indicator and prove that it is reliable and locally efficient. Finally, we provide some numerical experiments that illustrate the performance of the method.