An arbitrary-order scheme on generic meshes for miscible displacements in porous media

We design, analyze, and implement an arbitrary-order scheme applicable to generic meshes for a coupled elliptic-parabolic PDE system describing miscible displacement in porous media. The discretization is based on several adaptations of the hybrid-high-order (HHO) method due to Di Pietro, Ern, and Lemaire [Comput. Methods Appl. Math., 14 (2014), pp. 461–472]. The equation governing the pressure is discretized using an adaptation of the HHO method for variable diffusion, while the discrete concentration equation is based on the HHO method for advection-diffusion-reaction problems combined with numerically stable flux reconstructions for the advective velocity that we have derived using the results of Cockburn, Di Pietro, and Ern [ESAIM Math. Model. Numer. Anal., 50 (2016), pp. 635–650]. We perform some rigorous analysis of the method to demonstrate its L² stability under the irregular data often presented by reservoir engineering problems and present several numerical tests to demonstrate the quality of the results that are produced by the proposed scheme.