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

TL;DR

This paper provides a comprehensive convergence analysis of GDM-ELLAM schemes for a coupled elliptic-parabolic PDE modeling miscible displacement in porous media, under weak regularity assumptions and without requiring $L^ fty$ bounds.

Contribution

It offers the first convergence proof for GDM-ELLAM schemes applied to strongly coupled models with minimal regularity assumptions and general grids.

Findings

Convergence established for GDM-ELLAM schemes in practical settings.

Analysis applies to mixed finite element and hybrid mimetic schemes.

No need for $L^ infty$ bounds due to anisotropic diffusion and complex grids.

Abstract

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).