Degenerate Two - Phase Incompressibleflow Ii : Optimal Error

This is the second paper of a series in which we analyze mathematical properties and develop numerical methods for a degenerate elliptic-parabolic partial diierential system which describes the ow of two incom-pressible, immiscible uids in porous media. In this paper we consider a nite element approximation for this system. The elliptic equation for the pressure and velocity is approximated by a mixed nite element method, while the degenerate parabolic equation for the saturation is approximated by a Galerkin nite element method. Both semidiscrete (continuous in time) and fully discrete approximations are analyzed. Optimal error estimates in energy norms are obtained for this approximation. The error analysis does not use any regularization of the saturation equation; the error estimates are derived directly from the fully degenerate equation. Also, the analysis does not impose any restriction on the nature of degeneracy. Finally, it respects the minimal regularity on the solution of the diierential system.