A penalty mechanism for discontinuous Galerkin processing of Darcy velocity

The Darcy velocity plays an important role in the flow in porous media, particularly when a miscible displacement is concerned. One major requirement when approximating this velocity is the continuity of its normal component. The discontinuous Galerkin methods, by nature, are not well designed for this challenge, since approximations are performed in space of totally discontinuous polynomials.We propose in such context a penalty approach, in order to enhance the continuity of the normal component of the Darcy velocity. The resulting formulation is shown to be stable whatever the origin of the pressure but requires the inversion of a global matrix. We then propose two modifications leading to the inversion of only local matrices. Error estimates are furnished and the analysis of the penalty parameter vis-a-vis the computed pressure is addressed. We show that the proposed reconstructions have better performance compared to the simple local differentiation of the computed pressure. Numerical tests are provided to illustrate the theoretical results.