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

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 ${\bf u}$.