Analysis of fully discrete FEM for miscible displacement in porous media with Bear–Scheidegger 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:

$$\begin{aligned} D(\mathbf{u}) = \gamma d_m I + |\mathbf{u}|\bigg ( \alpha _T I + (\alpha _L - \alpha _T) \frac{\mathbf{u} \otimes \mathbf{u}}{|\mathbf{u}|^2}\bigg ) \, . \end{aligned}$$

Previous works on optimal-order $L^\infty (0,T;L^2)$-norm error estimate required the regularity assumption $\nabla _x\partial _tD(\mathbf{u}(x,t)) \in L^\infty (0,T;L^\infty (\Omega ))$, while the Bear–Scheidegger diffusion–dispersion tensor is only Lipschitz continuous even for a smooth velocity field $\mathbf{u}$. 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 $D(\mathbf{u})$ being Lipschitz continuous with respect to $\mathbf{u}$.