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 D( u) = Φ dₘ I + | u| ( αT I + (αL - αT) u ⊗ u| u|² ). 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(ΩT). 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 Lᵖ error estimates and an almost optimal L^∞ error estimate.