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: $$D(\mathbf{u})=\gamma d_{m}I+\mathrm{\mid }\mathbf{u}\mathrm{\mid }({\alpha }_{T}I+({\alpha }_L-{\alpha }_T)\frac{\mathbf{u}\otimes \mathbf{u}}{\mathrm{\mid }\mathbf{u}{\mathrm{\mid }}^2})\mathrm{.}$$ Previous works on optimal-order $L^{\mathrm{\infty }}(0,T;L^2)$-norm error estimate required the regularity assumption ${\mathrm{\nabla }}_x{\mathrm{\partial }}_{t}D(\mathbf{u}(x,t))\in L^{\mathrm{\infty }}(0,T;L^{\mathrm{\infty }}(\mathrm{\Omega }))$, 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 $L^{\mathrm{\infty }}(0,T;L^q)$-norm are established under the assumption of $D(\mathbf{u})$ being Lipschitz continuous with respect to $\mathbf{u}$.