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

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.

Some discrete Gronwall inequalities

We will need the following discrete Gronwall inequality cited in [22, 29]:

Lemma 4.1

Fix $T>0$. Let M be a positive integer, and define $\tau \le \frac{T}{M}$. Suppose $\left\{ a_m\right\} _{m=0}^M$, $\left\{ b_m\right\} _{m=0}^M$ and $\left\{ c_m\right\} _{m=0}^{M-1}$ are non-negative sequences such that $\tau \sum _{m=0}^{M-1} c_m \le C_1$, where $C_1$ is independent of $\tau$ and M. Further suppose that,

$$\begin{aligned} a_\ell + \tau \sum _{m=0}^{\ell } b_m \le C_2 +\tau \sum _{m=0}^{\ell -1} a_m c_m, \quad \forall \, 1\le \ell \le M, \end{aligned}$$

(4.1)

where $C_2>0$ is a constant independent of $\tau$ and M. Then, for all $\tau >0$,

$$\begin{aligned} a_\ell +\tau \sum _{m=0}^{\ell } b_m \le C_2 \exp \left( \tau \sum _{m=0}^{\ell -1}c_m \right) \le C_2\exp (C_1), \quad \forall \, 1\le \ell \le M. \end{aligned}$$

(4.2)

Note that the sum on the right-hand-side of (4.1) must be explicit.

In addition, the following more general discrete Gronwall inequality is needed in the stability analysis.

Lemma 4.2

Fix $T>0$. Let M be a positive integer, and define $\tau \le \frac{T}{M}$. Suppose $\left\{ a_m\right\} _{m=0}^M$, $\left\{ b_m\right\} _{m=0}^M$ and $\left\{ c_m\right\} _{m=0}^{M-1}$ are non-negative sequences such that $\tau \sum _{m=0}^{M-1} c_m \le C_1$, where $C_1$ is independent of $\tau$ and M. Suppose that, for all $\tau >0$ and for some constant $0< \alpha < 1$,

$$\begin{aligned} a_\ell + \tau \sum _{m=0}^{\ell } b_m \le C_2 +\tau \sum _{m=0}^{\ell -1} c_m \sum _{j=0}^m \alpha ^{m-j} a_j, \quad \forall \, 1\le \ell \le M, \end{aligned}$$

(4.3)

where $C_2>0$ is a constant independent of $\tau$ and M. Then, for all $\tau >0$,

$$\begin{aligned} a_\ell +\tau \sum _{m=0}^{\ell } b_m \le ( C_2 + a_0 C_1) \exp \left( \frac{C_1}{1-\alpha }\right) , \quad \forall \, 1\le \ell \le M. \end{aligned}$$

(4.4)

Proof

We set $A_\alpha := \frac{1}{1-\alpha } > 1$. A careful application of induction, using (4.3), yields the following inequality:

$$\begin{aligned} a_\ell +\tau \sum _{m=0}^{\ell } b_m \le \prod _{m=1}^\ell d_{\ell ,m}, \quad \forall \, 1\le \ell \le M, \end{aligned}$$

(4.5)

where

$$\begin{aligned} d_{\ell ,m} = \left\{ \begin{array}{l@{\quad }l@{\quad }l} \prod _{k=0}^{m-1}\left( 1 + \tau \alpha ^k c_m \right) &{} \text{ if } &{} 1\le m\le \ell -1 \\ C_2 + a_0 \tau \sum _{k=0}^{\ell -1} c_k \alpha ^k &{} \text{ if } &{} m =\ell \end{array} \right. . \end{aligned}$$

(4.6)

Meanwhile, the following bound is available:

$$\begin{aligned} d_{\ell ,m} =&\ ( 1 + \tau c_m ) ( 1 + \alpha \tau c_m ) \cdots ( 1 + \alpha ^{m-1} \tau c_m ) \nonumber \\ \le&\ \exp ( \tau c_m ) \exp ( \alpha \tau c_k ) \cdots \exp ( \alpha ^{m-1} \tau c_m ) \nonumber \\ =&\ \exp \left( \tau ( 1 + \alpha + \cdots + \alpha ^{m-1} ) c_m \right) \le \exp \left( A_\alpha c_m \tau \right) , \quad \forall \, 1\le m < \ell -1, \end{aligned}$$

(4.7)

which in turn leads to

$$\begin{aligned} d_{\ell ,1} d_{\ell ,2} \cdots d_{\ell ,\ell -1} \le&\ \mathrm{e}^{A_\alpha c_1 \tau } \mathrm{e}^{A_\alpha c_2 \tau } \cdots \mathrm{e}^{A_\alpha c^{\ell -1} \tau } \nonumber \\ \le&\ \exp \left( A_\alpha \tau ( c_1 + c_2 + \cdots + c_{\ell -1} ) \right) \le \exp ( A_\alpha C_1 ). \end{aligned}$$

(4.8)

On the other hand, we also have

$$\begin{aligned} d_{\ell ,\ell }&= C_2 + a_0 \tau \left( c_0 + c_1 \alpha + \cdots + c_{\ell -1} \alpha ^{\ell -1} \right) \nonumber \\&\le \ C_2 + a_0 \tau \left( c_0 + c_1 + \cdots + c^{\ell -1} \right) \le C_2 + a_0 C_1. \end{aligned}$$

(4.9)

In turn, a substitution of (4.8) and (4.9) into (4.6) results in (4.4), the desired estimate. The proof of Lemma 4.2 is complete. $\square$