Discontinuous Galerkin methods for coupled flow and reactive transport problems

Primal discontinuous Galerkin methods with interior penalty are proposed to solve the coupled system of flow and reactive transport in porous media, which arises from many applications including miscible displacement and acid-stimulated flow. A cut-off operator is introduced in the discontinuous Galerkin schemes to treat the coupling of flow and transport and the coupling of transport and reaction. The uniform positive definitiveness and the uniform Lipschitz continuity are established for the commonly used dispersion-diffusion tensor. Interestingly, the polynomial degrees of approximation for the flow and the transport equations need to be in the same order in order to maintain the convergence of DG applied to the coupled system. Optimal or nearly optimal convergences for both flow and transport are obtained when the same polynomial degrees of approximation are chosen for flow and transport. That is, error estimate in L2(H1) for concentration is optimal in h and nearly optimal in p with a loss of 12; error estimates in semi-L∞(H1) for pressure and in L∞(L2) for velocity establish optimality in h and sub-optimality in p by 12; error estimates for concentration jump and pressure jump are optimal in both h and p.