Optimal order convergence of a modified BDM1 mixed finite element scheme for reactive transport in porous media

We present a mass conservative finite element approach of second order accuracy for the numerical approximation of reactive solute transport in porous media modeled by a coupled system of advection-diffusion-reaction equations. The lowest order Brezzi–Douglas–Marini (BDM1) mixed finite element method is used. A modification based on the hybrid form of the approach is suggested for the discretization of the advective term. It is demonstrated numerically that this leads to optimal second order convergence of the flux variable. The modification improves the convergence behavior of the classical BDM1 scheme, which is known to be suboptimal of first order accuracy only for advection–diffusion problems; cf. [8]. Moreover, the new scheme shows more robustness for high Péclet numbers than the classical approach. A comparison with the Raviart–Thomas element (RT1) of second order accuracy for the approximation of the flux variable is also presented. For the case of strongly advection-dominated problems we propose a full upwind scheme. Various numerical studies, including also a nonlinear test problem, are presented to illustrate the numerical performance properties of the considered numerical methods.