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In this paper, we present a novel second order in time mixed finite elementscheme for the Cahn-Hilliard-Navier-Stokes equations with matched densities.The scheme combines a standard second order Crank-Nicholson method for theNavier-Stokes equations and a modification to the Crank-Nicholson method forthe Cahn-Hilliard equation. In particular, a second order Adams-Bashforthextrapolation and a trapezoidal rule are included to help preserve the energystability natural to the Cahn-Hilliard equation. We show that our scheme isunconditionally energy stable with respect to a modification of the continuousfree energy of the PDE system. Specifically, the discrete phase variable isshown to be bounded in $\ell^\infty \left(0,T;L^\infty\right)$ and the discretechemical potential bounded in $\ell^\infty \left(0,T;L^2\right)$, for any timeand space step sizes, in two and three dimensions, and for any finite finaltime $T$. We subsequently prove that these variables along with the fluidvelocity converge with optimal rates in the appropriate energy norms in bothtwo and three dimensions.

Convergence Analysis and Error Estimates for a Second Order Accurate Finite Element Method for the Cahn-Hilliard-Navier-Stokes System