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

TL;DR

This paper introduces a second order accurate, energy-stable finite element scheme for the coupled Cahn-Hilliard-Navier-Stokes system, providing convergence analysis and error estimates in multiple dimensions.

Contribution

It develops a novel second order mixed finite element method with proven unconditional energy stability and optimal convergence rates for the Cahn-Hilliard-Navier-Stokes equations.

Findings

The scheme is unconditionally energy stable in 2D and 3D.

Discrete variables are bounded in relevant norms for all step sizes.

Optimal convergence rates are achieved for velocity and phase variables.

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-Nicholson method for the Navier-Stokes equations and a modification to the Crank-Nicholson 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.