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 $ abla_xpartial_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 ${bf u}$.