A Mixed-Hybrid-Discontinuous Galerkin Finite Element Method for Convection-Diffusion Problems

We propose and analyse a new finite element method for convection diffusion problems based on the combination of a mixed method for the elliptic and a discontinuous Galerkin method for the hyperbolic part of the problem. The two methods are made compatible via hybridization and the combination of both is appropriate for the solution of intermediate convection-diffusion problems. By construction, the discrete solutions obtained for the limiting subproblems coincide with the ones obtained by the mixed method for the elliptic and the discontinuous Galerkin method for the limiting hyperbolic problem, respectively. We present a new type of analysis that explicitly takes into account the Lagrange-multipliers introduced by hybridization. The use of adequate energy norms allows to treat the purely diffusive, the convection dominated, and the hyperbolic regime in a unified manner. In numerical tests, we illustrated the efficiency of our approach and compare to results obtained with other methods for convection diffusion problems.