Galerkin Methods for Miscible Displacement Problems in Porous Media

A priori error estimates for Galerkin methods for numerical approximation of the coupled quasilinear system for $c = c(x,t)$ and $p = p(x,t)$ given by \[{gathered} ∇ · [a(x,c)\{ ∇ p - γ (x,c)∇ z\} ] = q(x,t), \\ ∇ · [b(x,c,∇ p)∇ c] - u(x,c,∇ p) · ∇ c = φ (x){{∂ c}}{{∂ t}} + g(x,t,c) \\ {gathered} \] for x ∈ Ω ,t ∈ (0,T), and appropriate Neumann boundary and initial conditions are considered. Equations of this type arise in models for the miscible displacement of one incompressible fluid by another in a porous medium. Estimates for both continuous time and fully-discrete time Galerkin methods are presented.