Regularity of the Diffusion-Dispersion Tensor and Error Analysis of Galerkin FEMs for a Porous Medium 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 $D({\bf u}) = \Phi d_m I + |{\bf u}| ( \alpha_T I + (\alpha_L - \alpha_T) \frac{{\bf u} \otimes {\bf u}}{|{\bf u}|^2} )$. 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 $\nabla_x\partial_tD({\bf u}(x,t)) \in L^p(\Omega_T)$. However, the Bear--Scheidegger diffusion-dispersion tensor may not satisfy the regularity condition even for a smooth velocity field ${\bf u}$. A new approach is presented in this paper, in terms of a parabolic projection, which only requires the Lipschitz continuity of $D({\bf u})$. With the new approach, we establish optimal $L^p$ error estimates and an almost optimal $L^\infty$ error estimate.