Error Analysis for a Galerkin Finite Element Method Applied to a Coupled Nonlinear Degenerate System of Advection-diffusion Equations

We consider a standard Galerkin Method applied to both the pressure equation and the saturation equation of a coupled nonlinear system of degenerate advection-diffusion equations modeling a two-phase immiscible flow through porous media. After regularizing the problem and establishing some regularity results, we derive error estimates for a semi-discretized Galerkin Method. A decoupled nonlinear scheme is then proposed for a fully discretized (backward in time) Galerkin Method, and error estimates are derived for that method. We also prove the existence and uniqueness for the nonlinear operator intervening in the backward time discretization.