ANALYSIS OF A MIXED FINITE ELEMENT METHOD FOR A CAHN-HILLIARD-DARCY-STOKES SYSTEM

In this paper we devise and analyze a mixed finite element method for a modified Cahn-Hilliard equation coupled with a nonsteady Darcy-Stokes flow that models phase separation and coupled fluid flow in immiscible binary fluids and diblock copolymer melts. The time discretization is based on a convex splitting of the energy of the equation. We prove that our scheme is unconditionally energy stable with respect to a spatially discrete analogue of the continuous free energy of the system and unconditionally uniquely solvable. We prove that the discrete phase variable is bounded in L-infinity (0, T; L-infinity) and the discrete chemical potential is bounded in L-infinity (0, T; L-2), for any time and space step sizes, in two and three dimensions, and for any finite final time T. We subsequently prove that these variables converge with optimal rates in the appropriate energy norms in both two and three dimensions.