Regularity of the diffusion-dispersion tensor and error analysis of Galerkin FEMs for a porous media flow

Abstract

We study Galerkin finite element methods for an incompressible miscible flow in porous media with the commonly-used Bear-Scheidegger diffusion-dispersion tensor $D(\mathbf{u})=\Phi d_{m}I+|\mathbf{u}|({\alpha }_{T}I+({\alpha }_L-{\alpha }_T)\frac{\mathbf{u}\otimes \mathbf{u}}{|\mathbf{u}{|}^2})$. The traditional approach to optimal $L^{\infty }((0,T);L^2)$ error estimates is based on an elliptic Ritz projection, which usually requires the regularity of ${\nabla }_x{\partial }_{t}D(\mathbf{u}(x,t))\in L^p({\Omega }_T)$. However, the Bear-Scheidegger diffusion-dispersion tensor may not satisfy the regularity condition even for a smooth velocity field $\mathbf{u}$. A new approach is presented in this paper, in terms of a parabolic projection, which only requires the Lipschitz continuity of $D(\mathbf{u})$. With the new approach, we establish optimal $L^p((0,T);L^q)$ error estimates and an almost optimal $L^{\infty }((0,T);L^{\infty })$ error estimate.