Convergence analysis and error estimates for a second order accurate finite element method for the Cahn–Hilliard–Navier–Stokes system

TLDR: In this paper, a second order in time mixed finite element scheme for the Cahn-Hilliard-Navier-Stokes equations with matched densities was presented, which combines a standard second-order Crank-Nicolson method for the Navier-stokes equations and a modification to the Crank Nicolson algorithm for the cahn-hilliard equation.

Abstract: In this paper, we present a novel second order in time mixed finite element scheme for the Cahn–Hilliard–Navier–Stokes equations with matched densities. The scheme combines a standard second order Crank–Nicolson method for the Navier–Stokes equations and a modification to the Crank–Nicolson method for the Cahn–Hilliard equation. In particular, a second order Adams-Bashforth extrapolation and a trapezoidal rule are included to help preserve the energy stability natural to the Cahn–Hilliard equation. We show that our scheme is unconditionally energy stable with respect to a modification of the continuous free energy of the PDE system. Specifically, the discrete phase variable is shown to be bounded in $$\ell ^\infty \left( 0,T;L^\infty \right) $$ and the discrete chemical potential bounded in $$\ell ^\infty \left( 0,T;L^2\right) $$ , 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 along with the fluid velocity converge with optimal rates in the appropriate energy norms in both two and three dimensions.