Convergence analysis of a family of ELLAM schemes for a fully coupled model of miscible displacement in porous media

We analyse the convergence of numerical schemes in the GDM–ELLAM (Gradient Discretisation Method–Eulerian Lagrangian Localised Adjoint Method) framework for a strongly coupled elliptic-parabolic PDE which models miscible displacement in porous media. These schemes include, but are not limited to, Mixed Finite Element–ELLAM and Hybrid Mimetic Mixed–ELLAM schemes. A complete convergence analysis is presented on the coupled model, using only weak regularity assumptions on the solution (which are satisfied in practical applications), and not relying on L^∞ bounds (which are impossible to ensure at the discrete level given the anisotropic diffusion tensors and the general grids used in applications).