  In this paper we devise and analyze a mixed finite element method for a
modified Cahn-Hilliard equation coupled with a non-steady 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 phase variable is bounded in $L^\infty
\left(0,T,L^\infty\right)$ and the chemical potential is bounded in $L^\infty
\left(0,T,L^2\right)$ absolutely unconditionally in two and three dimensions,
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.
