Analysis of fully discrete FEM for miscible displacement in porous media with Bear--Scheidegger diffusion tensor
For researchers in numerical analysis of porous media flow, this work removes a restrictive regularity assumption, making error analysis applicable to a more realistic diffusion model.
The authors derived optimal and almost optimal error estimates for fully discrete finite element methods applied to miscible displacement in porous media with the Bear–Scheidegger diffusion tensor, relaxing previous regularity assumptions. They achieved these estimates under only Lipschitz continuity of the diffusion tensor with respect to velocity.
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: $$ D({\bf u}) = γd_m I + |{\bf u}|\bigg( α_T I + (α_L - α_T) \frac{{\bf u} \otimes {\bf u}}{|{\bf u}|^2}\bigg) \, . $$ Previous works on optimal-order $L^\infty(0,T;L^2)$-norm error estimate required the regularity assumption $\nabla_x\partial_tD({\bf u}(x,t)) \in L^\infty(0,T;L^\infty(Ω))$, 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 ${\bf u}$.