Analysis of fully discrete FEM for miscible displacement in porous media with Bear--Scheidegger diffusion tensor

TL;DR

This paper develops and analyzes fully discrete finite element methods for simulating miscible displacement in porous media, achieving optimal error estimates without requiring overly restrictive regularity assumptions on the diffusion tensor.

Contribution

It establishes optimal error estimates for finite element solutions under Lipschitz continuity of the Bear--Scheidegger diffusion tensor, relaxing previous regularity requirements.

Findings

Optimal error estimates in L^p(0,T;L^q) norms.

Almost optimal error estimates in L^ 0,T;L^q) norms.

Method applicable with Lipschitz continuous diffusion tensor.

Abstract

Fully discrete Galerkin finite element methods are studied for the equations of miscible displacement in porous media with the commonly-used Bear--Scheidegger diffusion-dispersion tensor: $$ D({\bf u}) = \gamma d_m I + |{\bf u}|\bigg( \alpha_T I + (\alpha_L - \alpha_T) \frac{{\bf u} \otimes {\bf u}}{|{\bf u}|^2}\bigg) \, . $$ Previous works on optimal-order $L^\infty(0,T;L^2)$-norm error estimate required the regularity assumption $\nabla_x\partial_tD({\bf u}(x,t)) \in L^\infty(0,T;L^\infty(\Omega))$, while the Bear--Scheidegger diffusion-dispersion tensor is only Lipschitz continuous even for a smooth velocity field ${\bf u}$. In terms of the maximal $L^p$-regularity of fully discrete finite element solutions of parabolic equations, optimal error estimate in $L^p(0,T;L^q)$-norm and almost optimal error estimate in $L^\infty(0,T;L^q)$-norm are established under the assumption of $D({\bf u})$ being Lipschitz continuous with respect to ${\bf u}$.