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: $$D(\mathbf{u})=\gamma d_{m}I+|\mathbf{u}|({\alpha }_{T}I+({\alpha }_L-{\alpha }_T)\frac{\mathbf{u}\otimes \mathbf{u}}{|\mathbf{u}{|}^2}).$$ Previous works on optimal-order $L^{\infty }(0,T;L^2)$-norm error estimate required the regularity assumption ${\nabla }_x{\partial }_{t}D(\mathbf{u}(x,t))\in L^{\infty }(0,T;L^{\infty }(\Omega ))$, 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 $\mathbf{u}$.