Convergence analysis of a family of ELLAM schemes for a fully coupled model of miscible displacement in porous media
This work offers rigorous convergence guarantees for numerical schemes used in porous media flow, addressing a known gap in analysis for strongly coupled models with anisotropic diffusion and general grids.
The paper provides a complete convergence analysis for a family of ELLAM schemes applied to a coupled elliptic-parabolic model of miscible displacement in porous media, under weak regularity assumptions and without relying on L∞ bounds.
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 $L^\infty$ bounds (which are impossible to ensure at the discrete level given the anisotropic diffusion tensors and the general grids used in applications).