Regularity of the diffusion-dispersion tensor and error analysis of Galerkin FEMS for a porous medium flow

We study Galerkin finite element methods for an incompressible miscible flow in porous media with the commonly used Bear-Scheidegger diffusion-dispersion tensor {equation presented} The traditional approach to optimal L^∞((0, T);L²) error estimates is based on an elliptic Ritz projection, which usually requires the regularity of ?x?tD(u(x, t)) ∈Lp(Ω_T ). However, the Bear-Scheidegger diffusion-dispersion tensor may not satisfy the regularity condition even for a smooth velocity field u. A new approach is presented in this paper, in terms of a parabolic projection, which only requires the Lipschitz continuity of D(u). With the new approach, we establish optimal L^p error estimates and an almost optimal L^∞ error estimate.