Fourier Spectral Approximation to Global Attractor for 2D Convective Cahn–Hilliard Equation

Abstract

In this paper, we consider the long-time behavior of the convective Cahn–Hilliard equation in 2D case. A linear fully discrete Galerkin–Fourier spectral approximation scheme is constructed and the existence of the global attractors of the discrete system is obtained. All results in this paper are obtained without any restriction on the time step size.

1 Introduction

With the development of studies in the field of infinite-dimensional dynamical systems, the long-time behavior of solutions for nonlinear dissipative evolution equations has attracted more and more attention of scientists. As is known to all, the long-time dynamics of a dissipative systems is completely determined by the attractor of the equation. Hence, the existence and structure of the attractor are the most important characteristics for evolution equations. There are many papers related to the global attractor for dissipative nonlinear evolution equations, such as Navier–Stokes equation, Kuramoto–Sivashinsky equation, Cahn–Hilliard equation (see [1–6]).

However, the study of long-time behavior for nonlinear evolution equations depended on the results of numerical experimentation to a great extent. For this reason, it is worth studying whether the numerical results are reliable and the calculation schemes are suitable. During the past years, many authors have paid much attention to this problem. For example, in [7], based on Galerkin approximations, Hale, Lin, and Raugle studied the approximate system of a given evolutionary equation as a compact attractor which converges to the original one as the approximation is refined. Furthermore, using finite element approximation, Marion and Temam [8], Elliott and Larsson [9] studied the numerical approximation to attractor for some nonlinear evolution equations. Eden, Michaux, and Rakotoson [10] considered a time discretization of a doubly nonlinear parabolic type equations by the Euler forward scheme. They proved the existence of a compact attractor and estimated its Hausdorff dimension using CFT theory. Lü and Lu [11, 12] studied the dynamical properties of the discrete systems of Ginzburg–Landau type equation and generalized KdV–Burgers equation. They constructed the fully discrete scheme, proved the existence, convergence of global attractors of the discrete systems. There is much literature concerned with the approximation to global attractor for evolution equations, for more recent results we refer the reader to [13–20] and the references therein.

The convective Cahn–Hilliard equation, which arises naturally as a continuous model for the formation of facets and corners in crystal growth (see [21, 22]), is a kind of important nonlinear equations. For this reason, the research of it is of theoretical and practice significance. In this article, based on Fourier spectral approximation, we study the periodic initial value problem of the 2D convective Cahn–Hilliard equation

$$\begin{aligned}&\frac{\partial u}{\partial t}+\gamma \Delta ^2u-\Delta f(u)-\nabla \cdot g(u)=0,\quad x=(x_1,x_2)\in {\mathbb {R}}^2,\quad \,t>0, \end{aligned}$$

(1.1)

$$\begin{aligned}&u(x_1+2\pi ,x_2,t)=u(x_1,x_2+2\pi ,t)=u(x_1,x_2,t),\quad x\in {\mathbb {R}}^2,\,\quad t\ge 0, \end{aligned}$$

(1.2)

$$\begin{aligned}&u(x,0)=u_0(x),\quad \,x\in {\mathbb {R}}^2. \end{aligned}$$

(1.3)

In the following, we first construct a fully discrete Fourier spectral approximation scheme, which is a linear scheme. Then the existence and the convergence of approximate attractors, as well as the stability of discrete scheme, are proved. Throughout this paper, we use the following notation: $\Omega =[0,2\pi ]\times [0,2\pi ]$; $(\cdot ,\cdot )$ denotes the inner product of $L^2(\Omega )$, $\Vert \cdot \Vert _m$ the norm of $L^m(\Omega )$, and $\Vert \cdot \Vert =\Vert \cdot \Vert _2$, $\Vert \cdot \Vert _{\infty }=\Vert \cdot \Vert _{L^{\infty }(\Omega )}$. On the other hand, we point out one basic fact about problem (1.1)–(1.3): the spatial average of any solution u is preserved. Indeed,

$$\begin{aligned} \frac{\partial }{\partial t}\int _{\Omega }u(x,t)dx=0,\quad \int _{\Omega }u(x,t)dx=\int _{\Omega }u_0dx,\quad \forall t>0. \end{aligned}$$

We assume that the initial function satisfies $\int _{\Omega }u_0dx=m_0$.

For any given positive integer N, $j=(j_1,j_2)$, $j\cdot x=j_1x_1+j_2x_2$, let $S_N=\hbox {span}\{e^{ij\cdot x}:|j|\le N\}$, where $|j|=\max \{|j_1|,|j_2|\}$. Denote by $P_N: L^2_p(\Omega )\rightarrow S_N$ the orthogonal projection operator(see [27]). Let $\tau$ be the mesh size in the variable t, $t_k=k\tau$, $u^k=u(x,t_k)$, $\bar{\partial }_tu^k=\frac{1}{\tau }(u^k-u^{k-1})$. The Fourier spectral scheme for solving (1.1)–(1.3) is to find $u_N^k\in S_N$ such that

$$\begin{aligned}&\displaystyle \left( \bar{\partial }_tu_N^k+\gamma \Delta ^2 u_{N}^k,\varphi \right) +\left( f^{\prime }\big (u_N^{k-1}\big )\nabla u_{N}^k,\nabla \varphi \right) -B\left( u_N^{k-1},u_N^k,\varphi \right) =0,\,\quad \forall \varphi \in S_N, \nonumber \\\end{aligned}$$

(1.4)

$$\begin{aligned}&\displaystyle u_N^0=P_Nu_0. \end{aligned}$$

(1.5)

where

$$\begin{aligned} B\left( u_N^{k-1},u_N^k,\varphi \right) =\left( G\big (u_N^{k-1}\big )\varphi ,\nabla \cdot u_{N}^k\right) -\left( G\big (u_N^{k-1}\big )u_N^k,\nabla \cdot \varphi \right) , \nonumber \end{aligned}$$

and

$$\begin{aligned} G(u)=\left\{ \begin{array}{ll}\frac{1}{u^2}\int _0^usg'(s)ds,&{}\quad \hbox {for}~u\ne 0,\\ \frac{1}{2}g'(0),&{}\quad \hbox {for}~u=0.\end{array}\right. \nonumber \end{aligned}$$

It is a linear iteration scheme, and then it needs only to solve a class of linear algebraic equations for every iteration.

Remark 1.1

The main tool for studying the numerical approximation to long-time behavior for evolution equations is nonlinear Galerkin methods. The nonlinear Galerkin methods sometimes can be linear schemes, but the dimension of which is many times that of the nonlinear Galerkin ones: consequently, the computation amount is very heavy. Here, we construct a linear full discrete Galerkin spectral scheme. In comparison with the nonlinear Galerkin methods with linear schemes or classical Galerkin methods with nonlinear schemes, the computation amount of this scheme can be greatly reduced. In addition, without any restriction on the time step size, the results about the uniform stability and convergence of this discrete scheme are obtained.

Remark 1.2

To be more specific in this paper, we restrict ourselves to study the Fourier spectral approximation to global attractor for the convective Cahn–Hilliard equation, which is a kind of important nonlinear equations. More generally, similar schemes and analysis are applicable to other higher-order nonlinear evolution equations, for example, molecular beam epitaxy model [23, 24], viscous Cahn–Hilliard equation [25, 26], and so on.

The following lemmas are useful in our analysis.

Lemma 1.3

(see [27]) If $u\in H_p^m(\Omega )$, then there exists a constant c independent of u, N such that

$$\begin{aligned} \Vert u-P_Nu\Vert _{\mu }\le cN^{\mu -m}\Vert D^mu\Vert ,\quad \forall 0\le \mu \le m. \end{aligned}$$

Lemma 1.4

(Poincaré inequality(see [28])) Let $\Omega \subset {\mathbb {R}}^n$ be a bounded domain and $\Vert \cdot \Vert$ be the norm of $L^2(\Omega )$, then $\forall v\in H^1(\Omega )$, we have

$$\begin{aligned} \Vert v\Vert ^2\le & {} \frac{|\Omega |^2}{2}\Vert Dv\Vert ^2+\frac{1}{|\Omega |}\left( \int _{\Omega } vdx\right) ^2,\quad n=1,\\ \Vert v\Vert ^2\le & {} C(\Omega )\left[ \Vert Dv\Vert ^2+\left( \int _{\Omega }v(x)dx\right) ^2\right] ,\quad n\ge 2. \end{aligned}$$

Lemma 1.5

(Sobolev’s interpolation inequality [29]) Suppose that $u\in L^q(\Omega )$, $D^mu\in L^r(\Omega )$, $\Omega \subset {\mathbb {R}}^n$, $1\le r\le \infty$, $0\le j\le m$. Then there exists a constant $c=c(j,m,\Omega , p,q,r)$ independent of u such that

$$\begin{aligned} \Vert D^ju\Vert _{L^p}\le c\Vert D^mu\Vert ^a_{W^{m,r}}\Vert u\Vert ^{1-a}_{L^q}, \end{aligned}$$

where

$$\begin{aligned} \frac{1}{p}=\frac{j}{n}+a\left( \frac{1}{r}-\frac{m}{n}\right) +(1-a)\frac{1}{q},\quad \frac{j}{m}<a<1. \end{aligned}$$

Lemma 1.6

(Discrete Gronwall’s inequality [10, 12, 13]) Let $y^k$, $g^k$, and $h^k$ be three series satisfying

$$\begin{aligned} \frac{y^{k+1}-y^k}{\tau }\le g^ky^k+h^k,\quad k=0,1,2,\ldots \end{aligned}$$

Hence, we have

$$\begin{aligned} y^n\le y^0\exp \left( \tau \sum _{k=0}^{n-1}g^k\right) + \tau \sum _{k=0}^{n-1}h^k\exp \left( \tau \sum _{i=k}^{n-1}g^i\right) , \quad \forall n\ge 1. \end{aligned}$$

Lemma 1.7

(Discrete uniform Gronwall’s inequality [12, 13]) Let $y^k$, $g^k$, and $h^k$ be three series satisfying

$$\begin{aligned} \frac{y^{k+1}-y^k}{\tau }\le g^ky^k+h^k,\quad \forall k\ge k_0, \end{aligned}$$

and

$$\begin{aligned} \tau \sum _{k=k_1}^{n_0+k_1}g^k\le \alpha _1,\quad \tau \sum _{k=k_1}^{n_0+k_1}h^k \le \alpha _2,\quad \tau \sum _{k=k_1}^{n_0+k_1}y^k\le \alpha _3,\quad \forall k_1\ge k_0, \end{aligned}$$

with $\tau n_0=r$. Hence, we have

$$\begin{aligned} y^k\le \left( \frac{\alpha _3}{r}+\alpha _2\right) e^{\alpha _1},\quad \forall k\ge n_0+k_1. \end{aligned}$$

The rest of the article is organized as follows. In the next section, the existence of discrete attractors ${\mathcal {A}}_N^{\tau }$ is obtained by the t-independent prior estimates of discrete solutions; In Sect. 3, the convergence of ${\mathcal {A}}_N^{\tau }$ is proved by the error estimates in $[0,+\infty )$ of the discrete solutions.

2 Existence of Global Attractor

In this section, we prove the existence of the global attractors ${\mathcal {A}}_N^{\tau }$ of problem (1.4)–(1.5).

Lemma 2.1

If $f\in C^1$, $f^{\prime }(s)>0$, $u_0\in L_p^2(\Omega )$. Then for the solution $u_N^n$ of problem (1.4)–(1.5), we have

$$\begin{aligned}&\displaystyle \Vert u_N^n\Vert ^2\le E_0^2,\quad \forall n\ge 1,\\&\displaystyle \quad \overline{\lim _{n\rightarrow \infty }}\Vert u_N^k\Vert ^2\le (\rho _0^{\prime })^2,\\&\displaystyle \quad \tau \sum _{k=1}^n\Vert \bar{\partial }_tu_N^k\Vert ^2\le c_1(1+t_n),\quad \forall n\ge 1, \end{aligned}$$

where the constant $c_1=c_1(\Vert u_0\Vert )$ is independent of N, n, and $\tau$.

Proof

Let $\varphi =u_N^k$ in (1.4), we derive that

$$\begin{aligned}&\displaystyle \frac{1}{2}\bar{\partial }_t\Vert u_N^k\Vert ^2+\frac{\tau }{2}\Vert \bar{\partial }_tu_N^k\Vert ^2+\gamma \Vert \Delta u_{N}^k\Vert ^2+\Big (f^{\prime }\big (u_N^{k-1}\big )\nabla u_{N}^k,\nabla u_N^k\Big )\\&\displaystyle -B\big (u_N^{k-1},u_N^k,u_N^k\big )=0. \end{aligned}$$

Note that $f^{\prime }(s)>0$, then

$$\begin{aligned} \left( f^{\prime }\big (u_N^{k-1}\big )\nabla u_{N}^k,\nabla u_N^k\right) >0. \end{aligned}$$

On the other hand, we have

$$\begin{aligned} B\left( u_N^{k-1},u_N^k,u_N^k\right) =0. \end{aligned}$$

Therefore

$$\begin{aligned} \bar{\partial }_t\Vert u_N^k\Vert ^2+\tau \Vert \bar{\partial }_tu_N^k\Vert ^2+2\gamma \Vert \Delta u_{N}^k\Vert ^2\le 0. \end{aligned}$$

(2.1)

By Poincaré’s inequality, we obtain

$$\begin{aligned} \Vert u_N^k\Vert ^2\le C(\Omega )\left( \Vert \nabla u_N^k\Vert ^2+m_0^2\right) \le C(\Omega )\left( \frac{1}{2C(\Omega )}\Vert u_N^k\Vert ^2+\frac{C(\Omega )}{2}\Vert \Delta u_N^k\Vert ^2+m_0^2\right) , \end{aligned}$$

that is

$$\begin{aligned} \Vert u_N^k\Vert ^2\le [C(\Omega )]^2\Vert \Delta u_N^k\Vert ^2+2C(\Omega )m_0^2. \end{aligned}$$

(2.2)

Then

$$\begin{aligned} \bar{\partial }_t\Vert u_N^k\Vert ^2+\tau \Vert \bar{\partial }_tu_N^k\Vert ^2+\gamma \left( \Vert \Delta u_{N}^k\Vert ^2+\frac{1}{[C(\Omega )]^2}\Vert u_N^k\Vert ^2\right) \le \frac{2\gamma m_0^2}{C(\Omega )}. \end{aligned}$$

(2.3)

Multiplying (2.3) by $[1+\frac{\gamma }{4\pi ^4}\tau ]^{k-1}$ and summing them for k from 1 to n, we have

$$\begin{aligned} \begin{aligned} \Vert u_N^n\Vert ^2\le&\left( 1+\frac{\gamma }{[C(\Omega )]^2}\tau \right) ^{-n}\left( \Vert u_N^0\Vert ^2+2 m_0^2C(\Omega )\right) +2 m_0^2C(\Omega )\\ \le&\Vert u_N^0\Vert ^2+2 m_0^2C(\Omega )\triangleq E_0^2, \end{aligned}\nonumber \end{aligned}$$

that is

$$\begin{aligned} \overline{\lim _{n\rightarrow \infty }}\Vert u_N^k\Vert ^2\le 2 m_0^2C(\Omega )\triangleq (\rho _0^{\prime })^2. \end{aligned}$$

Taking the sum of (2.3) for k from $k_0+1$ to n, we recover the proof of the lemma. $\square$

Corollary 2.2

For any given $\rho _0>\rho _0^{\prime }$ and $R_0>0$, if $\Vert u_0\Vert \le R_0$, then

$$\begin{aligned} \Vert u_N^n\Vert ^2\le \rho _0^2,\quad \forall n\ge n_0=\left( \ln \frac{R_0^2}{\rho _0^2-(\rho _0^{\prime })^2}\right) /\ln \left( 1+\frac{\gamma }{[C(\Omega )]^2}\tau \right) . \end{aligned}$$

Lemma 2.3

In addition to the conditions of Lemma 2.1, we suppose that $f,g\in C^1$, $|f^{\prime }(s)|\le A|s|^{\frac{3}{2}}$, $|g^{\prime }(s)|\le B|s|^2$, $u_0\in H^1_p(\Omega )$, $\Vert u_0\Vert _{1}\le R_0$. Then for the solution $u_N^n$ of problem (1.4)–(1.5), we have

$$\begin{aligned}&\Vert \nabla u_{N}^k\Vert ^2\le \rho _1^2,\quad \forall n\ge n_0+N_0\triangleq n_1,\\&\quad \Vert \nabla u_{N}^k\Vert ^2\le E_1^2,\quad \forall n\ge 1,\\&\quad \tau ^2\sum _{k=1}^n\Vert \bar{\partial }_t\nabla u_{N}^k\Vert ^2\le c_2(1+t_n),\quad \forall n\ge 1, \end{aligned}$$

where $n_0$ is given by Corollary 2.2, $N_0$ an arbitrary positive integer, r an arbitrary positive number such that $N_0\tau =r$, the constant $\rho _1$ independent of N, n, $\tau$ and $\Vert u_0\Vert _{1}$, $c_2=c_2(\Vert u_0\Vert _{1})$, and $E_1=E_1(\Vert u_0\Vert _{1})$ independent of N, n, and $\tau$.

Proof

Let $\varphi =\Delta u_{N}^k$ in (1.4), we derive that

$$\begin{aligned} \begin{aligned}&\frac{1}{2}\bar{\partial }_t\Vert \nabla u_{N}^k\Vert ^2+\frac{\tau }{2}\Vert \bar{\partial }_t\nabla u_{N}^k\Vert ^2+\gamma \Vert \nabla \Delta u_{N}^k\Vert ^2 \\ =&\left( f^{\prime }(u_{N}^{k-1})\nabla u^k_{N},\nabla \Delta u^k_{N}\right) -B\left( u_N^{k-1},u_N^k,-\Delta u_{N}^k\right) . \end{aligned} \end{aligned}$$

(2.4)

By Sobolev’s interpolation inequality, we deduce that

$$\begin{aligned} \Big (f^{\prime }\Big (u_{N}^{k-1}\Big )\nabla u^k_{N},\nabla \Delta u^k_{N}\Big )\le&\frac{\gamma }{8}\Vert \nabla \Delta u^k_{N}\Vert ^2+\frac{2}{\gamma }\Vert f^{\prime }\big (u_N^{k-1}\big )\Vert ^2_{\infty }\Vert \nabla u^k_{N}\Vert ^2\\ \le&\frac{\gamma }{8}\Vert \nabla \Delta u^k_{N}\Vert ^2+\frac{2A^2C}{\gamma }\Vert u_N^{k-1}\Vert ^{3}_{\infty }\Vert \nabla \Delta u^k_{N}\Vert ^{\frac{2}{3}}\Vert u_N^k\Vert ^{\frac{4}{3}}\\ \le&\frac{\gamma }{4}\Vert \nabla \Delta u^k_{N}\Vert ^2+C\Vert u_{N}^{k-1}\Vert ^{4}_{\infty }\Vert u_N^k\Vert ^2\\ \le&\frac{\gamma }{4}\Vert \nabla \Delta u^k_{N}\Vert ^2+C\Vert \Delta u_{N}^{k-1}\Vert ^2\Vert u_N^{k-1}\Vert ^2\Vert u_N^k\Vert ^2. \end{aligned}$$

We also have

$$\begin{aligned}\nonumber -B\Big (u_N^{k-1},u_N^k,-u_{Nxx}^k\Big )=2\Big (G\big (u_N^{k-1}\big )\nabla u_{N}^k,\Delta u_{N}^k\Big )+\Big (G^{\prime }\Big (u_N^{k-1}\Big )\nabla u_{N}^{k-1}u_N^k,\Delta u_{N}^k\Big ). \end{aligned}$$

Using Sobolev’s interpolation inequality again, we get

$$\begin{aligned}&2\Big (G\Big (u_N^{k-1}\Big )\nabla u_{N}^k,\Delta u_{N}^k\Big )=2\left( \frac{\nabla u_{N}^k}{\big (u_N^{k-1}\big )^2}\int _0^{u_N^{k-1}}sg'(s)ds,\Delta u_{N}^k\right) \\&\quad =\Big (g^{\prime }\Big (\theta u_N^{k-1}\Big )\nabla u_{N}^k,\Delta u_{N}^k\Big )\le \Vert g^{\prime }\Big (\theta u_N^{k-1}\Big )\Vert _{\infty }\Vert \nabla u_{N}^k\Vert \Vert \Delta u_{N}^k\Vert \\&\quad \le B\Vert u_N^{k-1}\Vert _{\infty }^2\Vert \nabla u_N^k\Vert \Vert \Delta u_N^k\Vert \le C\Vert \Delta u_N^{k-1}\Vert \Vert u_N^{k-1}\Vert \Vert \nabla \Delta u_N^k\Vert \Vert u_N^k\Vert \\&\quad \le \frac{\gamma }{8}\Vert \nabla \Delta u_{N}^k\Vert ^2+C\Vert \Delta u_{N}^{k-1}\Vert ^2\Vert u_N^{k-1}\Vert ^2\Vert u_N^k\Vert ^2, \end{aligned}$$

and

$$\begin{aligned}&\Big (G^{\prime }\Big (u_N^{k-1}\Big )\nabla u_{N}^{k-1}u_N^k,\Delta u_{N}^k\Big )\\&\quad =\left( \left[ \frac{g^{\prime }\Big (u_N^{k-1}\Big )}{u_N^{k-1}}-\frac{g^{\prime }\Big (\theta u_N^{k-1}\Big )}{u_N^{k-1}}\right] u_N^k\nabla u_{N}^{k-1},\Delta u_{N}^k\right) \\ \quad \le&B\Vert u_N^{k-1}\Vert _{\infty }\Vert u_N^k\Vert _{\infty }\Vert \nabla u_{N}^{k-1}\Vert \Vert \Delta u_{N}^k\Vert \\&\quad \le C\Vert \Delta u_{N}^{k-1}\Vert ^{\frac{1}{2}}\Vert u_N^{k-1}\Vert ^{\frac{1}{2}}\Vert \nabla \Delta u_{N}^k\Vert \Vert u_N^k\Vert \Vert \nabla u_N^{k-1}\Vert \\&\quad \le \frac{\gamma }{8}\Vert \nabla \Delta u_N^k\Vert ^2+C\Vert \Delta u_N^{k-1}\Vert \Vert u_N^{k-1}\Vert \Vert u_N^k\Vert ^2\Vert \nabla u_N^{k-1}\Vert ^2 \\&\quad \le \frac{\gamma }{8}\Vert \nabla \Delta u_N^k\Vert ^2+C\Vert \nabla u_N^{k-1}\Vert ^2\Vert u_N^k\Vert ^2\big (\Vert \Delta u_N^{k-1}\Vert ^2\Vert u_N^{k-1}\Vert ^2+1\big ). \end{aligned}$$

Hence, (2.4) can be rewritten as

$$\begin{aligned}&\bar{\partial }_t\Vert \nabla u_{N}^k\Vert ^2+\tau \Vert \bar{\partial }_t\nabla u_{N}^k\Vert ^2+\gamma \Vert \nabla \Delta u_{N}^k\Vert ^2\nonumber \\&\quad \le C\Vert \nabla u_N^{k-1}\Vert ^2\Vert u_N^k\Vert ^2\Big (\Vert \Delta u_N^{k-1}\Vert ^2\Vert u_N^{k-1}\Vert ^2+1\Big )+C\Vert \Delta u_{N}^{k-1}\Vert ^2\Vert u_N^{k-1}\Vert ^2\Vert u_N^k\Vert ^2. \end{aligned}$$

(2.5)

Using (2.3), Lemma 2.1, and Corollary 2.2, we obtain for all $k_0>n_0$

$$\begin{aligned}&C\tau \sum _{k=k_0+1}^{k_0+N_0}\left[ \Vert u_N^k\Vert ^2\Big (\Vert \Delta u_N^{k-1}\Vert ^2\Vert u_N^{k-1}\Vert ^2+1\Big )\right] \\&\quad \le C\left( \rho _0^4\tau \sum _{k=k_0+1}^{k_0+N_0}\Vert \Delta u_N^{k-1}\Vert ^2+\rho _0^2r\right) \le C\left[ \rho _0^4\left( \frac{2m_0^2r}{C(\Omega )}+\frac{r}{\gamma }\rho _0^2\right) +\rho _0^2r\right] \triangleq \alpha _1,\\&\quad C\tau \sum _{k=k_0+1}^{k_0+N_0}\left[ \Vert \Delta u_{N}^{k-1}\Vert ^2\Vert u_N^{k-1}\Vert ^2\Vert u_N^k\Vert ^2\right] \le C\rho _0^4\left( \frac{2m_0^2r}{C(\Omega )}+\frac{r}{\gamma }\rho _0^2\right) \triangleq \alpha _2,\\&\quad \tau \sum _{k=k_0+1}^{k_0+N_0}\Vert \nabla u_{N}^{k-1}\Vert ^2\le \frac{\tau }{2}\sum _{k=k_0+1}^{k_0+N_0}\left( \Vert u_N^{k-1}\Vert ^2+\Vert \Delta u_{N}^{k-1}\Vert ^2\right) \\&\quad \le \frac{1}{2}\left( \frac{2m_0^2r}{C(\Omega )}+\frac{r}{\gamma }\rho _0^2+\rho _0^2r\right) \triangleq \alpha _{3}. \end{aligned}$$

Applying discrete uniform Gronwall’s inequality to (2.5), we have

$$\begin{aligned} \Vert \nabla u_{N}^k\Vert ^2\le \Big (\frac{\alpha _3}{r}+\alpha _2\Big ) e^{\alpha _1}\triangleq \rho _1^2,\quad \forall n\ge n_1=n_0+N_0. \end{aligned}$$

For $n\le n_1$, using Discrete Gronwall’s inequality to (2.5), we deduce that

$$\begin{aligned} \Vert \nabla u_{N}^n\Vert ^2\le&\left( \Vert \nabla u_{0}\Vert ^2 +\frac{2m_0^2}{C(\Omega )}E_0^4t_{n}+\frac{E_0^2}{\gamma }\right) \exp \left[ E_0^{4}\left( \frac{2m_0^2}{C(\Omega )}t_{n_1} +\frac{1}{\gamma }\Vert u_0\Vert ^2\right) +\rho _0^2t_{n_1}\right] \\ \triangleq&(E_1^{\prime })^2. \end{aligned}$$

Set $E_1^2=\max \{\rho _1^2,(E_1^{\prime })^2\}$. Hence, the second relation is proved. Taking the sum of (2.5), we complete the proof of this lemma. $\square$

Corollary 2.4

In addition to the conditions of Lemma 2.3, we have the following estimates

$$\begin{aligned} \begin{aligned}&\Vert u_N^n\Vert _{L^p}\le C(\rho _0,\rho _1),\quad \forall n\ge n_1,~~1\le p<\infty , \\&\Vert u_N^n\Vert _{L^p}\le C(E_0,E_1),\quad \forall n\ge 1,~~1\le p<\infty . \end{aligned} \nonumber \end{aligned}$$

Lemma 2.5

In addition to the condition of Lemma 2.3, we suppose that $f\in C^2$, $|f{^{\prime \prime }}(s)|\le A^{\prime } |s|^{\frac{1}{2}}$, $u_0\in H_p^2(\Omega )$ satisfying $\Vert u_{0}\Vert _2^2\le R_0^2$. Then

$$\begin{aligned}&\displaystyle \Vert \Delta u_{N}^n\Vert ^2\le \rho _2^2,\quad \forall n\ge n_2=n_1+N_0, \\&\displaystyle \Vert \Delta u_{N}^n\Vert ^2\le E_2^2,\quad \forall n\ge 1, \\&\displaystyle \tau ^2\sum _{k=1}^n\Vert \bar{\partial }_t\Delta u_{N}^k\Vert ^2+\tau \sum _{k=1}^n\Vert \Delta ^2u_{N}^k\Vert ^2\le c_3(1+t_n),\quad \forall n\ge 1,\nonumber \end{aligned}$$

where $n_1$ is given by Lemma 2.3, $N_0$ an arbitrary positive integer, r an arbitrary positive number such that $N_0\tau =r$, the constant $\rho _2$ independent of N, n, $\tau$, and $\Vert u_0\Vert _{2}$, $E_2=E_2(\Vert u_0\Vert _{2})$, and $c_3=c_3(\Vert u_0\Vert _{2})$ independent of N, n and $\tau$.

Proof

Let $\varphi =\Delta ^2u_{N}^k$ in (1.4), we have

$$\begin{aligned} \begin{aligned}&\frac{1}{2}\bar{\partial }_t\Vert \Delta u_{N}^k\Vert ^2+\frac{\tau }{2}\Vert \bar{\partial }_t\Delta u_{N}\Vert ^2+\gamma \Vert \Delta ^2u_{N}^k\Vert ^2\\&\quad =\left( \nabla \cdot [f^{\prime }(u_{N}^{k-1})\nabla \cdot u_{N}^k],\Delta ^2u_{N}^k\right) +B\left( u_{N}^{k-1},u_N^k,\Delta ^2u_{N}^k\right) . \end{aligned} \end{aligned}$$

When $k\ge n_1$: Based on Lemma 2.1, Corollary 2.2, Lemma 2.3, Corollary 2.6, and Sobolev’s interpolation inequality, we deduce that

$$\begin{aligned} \begin{aligned}&\left( \nabla \cdot \left[ f^{\prime }(u_{N}^{k-1})\nabla \cdot u_{N}^k\right] ,\Delta ^2u_{N}^k\right) \\&\quad =\Big (f^{\prime }\Big (u_N^{k-1}\Big )\Delta u_{N}^k,\Delta ^2u_{N}^k\Big )+\Big (f{^{\prime \prime }}\Big (u_N^{k-1}\Big )|\nabla u_{N}^k|^2,\Delta ^2u_{N}^k\Big )\\&\quad \le \Vert f^{\prime }\Big (u_N^{k-1}\Big )\Vert _{L^4}\Vert \Delta u_{N}^k\Vert _{L^4}\Vert \Delta ^2u_{N}^k\Vert +\Vert f{^{\prime \prime }}\Big (u_N^{k-1}\Big )\Vert _{L^6}\Vert \nabla u_{N}^k\Vert _{L^6}^2\Vert \Delta ^2u_{N}^k\Vert \\&\quad \le C\Vert u_N^{k-1}\Vert _{L^6}^{\frac{3}{2}}\Vert u_N^k\Vert ^{\frac{3}{8}}\Vert \Delta ^2u_N^k \Vert ^{\frac{13}{8}}+C\Vert u_N^{k-1}\Vert _{L^3}^{\frac{1}{2}}\Vert u_N^k\Vert ^{\frac{1}{6}} \Vert \Delta ^2u_N^k\Vert ^{\frac{11}{6}}\\&\quad \le \frac{\gamma }{4}\Vert \Delta ^2u_{N}^k\Vert ^2+C(\rho _0,\rho _1). \end{aligned} \end{aligned}$$

and

$$\begin{aligned} \begin{aligned}&B(u_{N}^{k-1},u_N^k,\Delta ^2u_{N}^k)\\&\quad = \Big (2G\Big (u_N^{k-1}\Big )\nabla u_{N}^k+G^{\prime }\Big (u_N^{k-1}\Big )\nabla u_{N}^{k-1}u_N^k,\Delta ^2u_{N}^k\Big )\\&\quad = \Big (2g^{\prime }\Big (\theta u_N^{k-1}\Big )\nabla u_N^k+\left[ \frac{g^{\prime }\Big (u_N^{k-1}\Big )}{u_N^{k-1}}-\frac{g^{\prime }\Big (\theta u_N^{k-1}\Big )}{u_N^{k-1}}\right] u_N^k\nabla u_{N}^{k-1},\Delta ^2u_N^k\Big )\\&\quad \le C\Vert g^{\prime }\theta u_N^{k-1}\Vert _{L^4}\Vert \nabla u_N^k\Vert _{L^4}\Vert \Delta ^2u_N^k\Vert +C\Vert u_N^{k-1}\Vert _{L^6}\Vert \nabla u_N^{k-1}\Vert _{L^6}\Vert u_N^k\Vert _{L^6}\Vert \Delta ^2u_N^k\Vert \\&\quad \le C\Vert g^{\prime }\theta u_N^{k-1}\Vert _{L^4}\Vert \nabla u_N^k\Vert _{L^4}\Vert \Delta ^2u_N^k\Vert \\&\qquad +C\Vert u_N^{k-1}\Vert _{L^6}\Vert \Delta u_N^{k-1}\Vert ^{\frac{5}{6}}\Vert u_N^{k-1}\Vert ^{\frac{1}{6}}\Vert u_N^k\Vert _{L^6}\Vert \Delta ^2u_N^k\Vert \\&\quad \le \frac{\gamma }{4}\Vert \Delta ^2u_{N}^k\Vert ^2+C(\rho _0,\rho _1)\Big (\Vert \Delta u_N^{k-1}\Vert ^2+1\Big ). \end{aligned} \end{aligned}$$

Summing up, we derive that

$$\begin{aligned} \bar{\partial }_t\Vert \Delta u_{N}^k\Vert ^2+\tau \Vert \bar{\partial }_t\Delta u_{N}^k\Vert ^2+\gamma \Vert \Delta ^2u_{N}^k\Vert ^2\le C(\rho _0,\rho _1)\Big (\Vert \Delta u_N^{k-1}\Vert ^2+1\Big ). \end{aligned}$$

(2.6)

Note that

$$\begin{aligned} \begin{aligned} \tau \sum _{k=k_0+1}^{k_0+N_0}C(\rho _0,\rho _1) =\,&C(\rho _0,\rho _1)r\triangleq \alpha _i,\quad i=1,2,\\ \tau \sum _{k=k_0+1}^{k_0+N_0}\Vert \Delta u_{N}^k\Vert ^2=\,&\frac{2m_0^2r}{C(\Omega )}+\frac{r}{\gamma }\rho _0^2\triangleq \alpha _3. \end{aligned} \end{aligned}$$

Hence, using discrete uniform Gronwall’s inequality to (2.6), we deduce that

$$\begin{aligned} \Vert \Delta u_{N}^n\Vert ^2\le \Big (\frac{\alpha _3}{r}+\alpha _2\Big )e^{\alpha _1}\triangleq \rho _2^2,\quad \forall n\ge n_2=n_1+N_0. \end{aligned}$$

(2.7)

When $k\ge 1$: As in the proof of the inequality (2.6), we have

$$\begin{aligned} \begin{aligned} \bar{\partial }_t\Vert \Delta u_{N}^k\Vert ^2+\tau \Vert \bar{\partial }_t\Delta u_{N}^k\Vert ^2+\gamma \Vert \Delta ^2u_{N}^k\Vert ^2\le C(E_0,E_1)\Big (\Vert \Delta u_N^{k-1}\Vert ^2+1\Big ) \end{aligned} \end{aligned}$$

(2.8)

Using Discrete Gronwall’s inequality to (2.8), we have

$$\begin{aligned} \Vert \Delta u_N^2\Vert ^2\le \Big (\Vert \Delta u_0\Vert ^2+C(E_0,E_1)t_n\Big )e^{C(E_0,E_1)t_n}\triangleq E_2. \end{aligned}$$

(2.9)

Taking the sum of (2.8) for k from 1 to n, we deduce that

$$\begin{aligned} \Vert \Delta u_{N}^n\Vert ^2+\tau ^2\sum _{k=1}^n\Vert \bar{\partial }_t\Delta u_{N}^n\Vert ^2+\gamma \tau \sum _{k=1}^n\Vert \Delta ^2u_{N}^k\Vert ^2\le C(1+t_n),\quad \forall n\ge 1. \end{aligned}$$

(2.10)

Combining (2.7), (2.9), and (2.10), we complete the proof of this lemma. $\square$

Corollary 2.6

In addition to the conditions of Lemma 2.5, we have the following estimates

$$\begin{aligned} \begin{aligned}&\Vert u_N^n\Vert _{\infty }\le C(\rho _0,\rho _1,\rho _2),\quad \forall n\ge n_2,\\&\Vert u_N^n\Vert _{\infty }\le C(E_0,E_1,E_2),\quad \forall n\ge 1,\\&\Vert u_N^n\Vert _{W^{1,p}}\le C(\rho _0,\rho _1,\rho _2),\quad \forall n\ge n_2,~~1<p<\infty ,\\&\Vert u_N^n\Vert _{W^{1,p}}\le C(E_0,E_1,E_2),\quad \forall n\ge 1,~~1<p<\infty . \end{aligned} \end{aligned}$$

Now, we give the main result in this section.

Theorem 2.7

If $f\in C^2$, $g\in C^1$, $f^{\prime }(s)>0$, $|f^{\prime }(s)|\le A|s|^{\frac{3}{2}}$, $|f{^{\prime \prime }}(s)|\le A^{\prime }|s|^{\frac{1}{2}}$, $|g^{\prime }(s)|\le B|s|^2$, and $u_0\in H_p^2(\Omega )$, then the semigroup of operator $\{S_N^{\tau }(n)\}_{n\ge 0}$ generated by problem (1.4)–(1.5) has a compact global attractor ${\mathcal {A}}_N^{\tau }\subset H_p^2(\Omega )\bigcap S_N$.

Proof

Using Theorem I.1.1 of [30] and Lemmas 2.1, 2.3, 2.5, we complete the proof of this theorem. Since it is classical, we omit it. $\square$

3 Convergence of the Global Attractors

Let $G_N:L^2_p(\Omega )\rightarrow S_N$ be the integral projection operator, i.e., for any given $u\in L^2(\Omega )$, we have

$$\begin{aligned} \left( \nabla G_Nu,\nabla v\right) +(G_Nu,v)=(u,v),\quad \forall v\in S_N. \end{aligned}$$

(3.1)

Then for any u, $v\in L^2(\Omega )$, we have $(G_Nu,v)=(u,G_Nv)$.

Lemma 3.1

(see [12]) For the integral projection operator $G_N$, we have

$$\begin{aligned} \begin{aligned} (1)\quad&\Vert G_Nu\Vert _2~|P_Nu\Vert ,\quad \forall u\in L^2_p(\Omega );\\ (2)\quad&\Vert G_N\nabla u\Vert =\Vert \nabla G_Nu\Vert ,\quad \forall u\in H^1_p(\Omega );\\ (3)\quad&\Vert G_N\Delta u\Vert =\Vert \nabla [G_N\nabla u]\Vert =\Vert \Delta G_Nu\Vert ,\quad \forall u\in H^2_p(\Omega );\\ (4)\quad&\Vert G_N^2\Delta u\Vert =\Vert \nabla [G_N^2\nabla u]\Vert =\Vert \Delta (G_N^2u)\Vert ,\quad \forall u\in H^2_p(\Omega ). \end{aligned} \end{aligned}$$

Similar to Lemma 2.5, the following result can be proved easily.

Lemma 3.2

Under the hypotheses of Lemma 2.5, we have the estimates for the smooth solution u(x, t) of problem (1.1)–(1.3)

$$\begin{aligned} \begin{aligned}&\int _0^t\Vert \nabla u_{t}\Vert ^2ds\le C(1+t),\\&\qquad t\Vert u_t\Vert ^2+\int _0^ts\Vert \Delta u_{t}\Vert ^2ds\le C(1+t^2),\\&\qquad t^2\Vert \nabla u_{t}\Vert ^2+\int _0^ts^2\Vert \nabla \Delta u_{t}\Vert ^2ds\le C(1+t^3),\\&\qquad t^3\Vert \Delta u_{t}\Vert ^2+\int _0^ts^3(\Vert u_{tt}\Vert ^2+\Vert \Delta ^2u_{t}\Vert ^2)ds\le C(1+t^4), \end{aligned} \end{aligned}$$

where the constant t is independent of t.

Using the same method as [31], we can obtain the following result easily.

Lemma 3.3

If $f\in C^2$, $g\in C^1$, $f^{\prime }(s)>0$, $|f^{\prime }(s)|\le A|s|^4$, $|g^{\prime }(s)|\le B|s|^5$, and $u_0\in H_p^2(\Omega )$, then there exists an unique global solution $u(x,t)\in L^{\infty }({\mathbb {R}}^+;H_p^2(\Omega ))$ for the problem (1.1)–(1.3) such that

$$\begin{aligned} \int _0^t(\Vert u\Vert _{H^4}^2+\Vert u_t\Vert ^2)\mathrm{d}t\le C(t+1),\quad \forall t>0, \end{aligned}$$

where the constant c is independent of t. Furthermore, if $f\in C^3$, $g\in C^2$, then u(x, t) satisfies

$$\begin{aligned} t\Vert \nabla \Delta u\Vert ^2\le c(t^2+1),\quad \forall t>0, \end{aligned}$$

and there exists a global attractor ${\mathcal {A}}\subset H_p^2(\Omega )$ of problem (1.1)–(1.3).

Now, we give the main result of this subsection.

Theorem 3.4

Suppose that the conditions of Theorem 2.7 hold, then

$$\begin{aligned} \hbox {dist}({\mathcal {A}}^{\tau }_N,{\mathcal {A}})\rightarrow 0,\quad \hbox {as}~\tau \rightarrow 0,~N\rightarrow +\infty . \end{aligned}$$

Proof

Let $\Vert u_0\Vert _{H^2}\le R_0$. On account of Lemma 3.3, this theorem will be proved by taking the error estimates of the solution $u_N^n$ of discrete problem (1.4)–(1.5). Now, we accomplish them through two steps.

• Step 1 Take the error estimates of the solution $v^n$ of the linear scheme as follows:

  $$\begin{aligned}&\displaystyle \Big (\bar{\partial }_tv_N^k+\gamma \Delta ^2 v_{N}^k-2\gamma \Delta v_{N}^k+\gamma v_N^k-\Delta f(u_N)-\nabla \cdot g(u_N),\varphi \Big )\nonumber \\&\displaystyle \quad =\Big (-2\gamma \Delta u_{N}^k+\gamma u_N^k,\varphi \Big ), \end{aligned}$$

  (3.2)
  $$\begin{aligned}&\displaystyle v_N^0=P_Nu_0,\quad \forall \varphi \in S_N. \end{aligned}$$

  (3.3)

  Set $u^k-v_N^k=u^k-P_Nu^k+P_Nu^k-v_N^k=\rho ^k+\theta ^k$. Hence, $\theta ^k$ satisfies

  $$\begin{aligned}&\displaystyle \Big (\bar{\partial }_t\theta ^k+\gamma \Delta ^2\theta ^k-2\gamma \Delta \theta ^k +\gamma \theta ^k-\Big (\bar{\partial }_tu^k-u_t^k\Big ),\varphi \Big )=0,\quad \forall \varphi \in S_N, \qquad \quad \end{aligned}$$

  (3.4)
  $$\begin{aligned}&\displaystyle \theta ^0=0. \end{aligned}$$

  (3.5)

  Let $\varphi =\theta ^k$ in (3.4). Simple calculations show that

  $$\begin{aligned} \frac{1}{2}\bar{\partial }_t\Vert \theta ^k\Vert ^2+\frac{\tau }{2}\Vert \bar{\partial }_t \theta ^k\Vert ^2+\gamma \Big (\Vert \Delta \theta \Vert ^2+\Vert \nabla \theta ^k\Vert ^2+\Vert \theta ^k\Vert ^2_1\Big ) =\Big (\bar{\partial }_tu^k-u_t^k,\theta ^k\Big ). \end{aligned}$$

  Note that

  $$\begin{aligned} \begin{aligned} \Big (\bar{\partial }_tu^k-u_t^k,\theta ^k\Big )&=\Big (\nabla \Big [G_N\Big (\bar{\partial }_tu^k-u_t^k\Big )\Big ], \nabla \theta ^k\Big )+\Big (G_N\Big (\bar{\partial }_tu^k-u_t^k\Big ),\theta ^k\Big ) \\&\le \frac{\gamma }{2}\Vert \theta ^k\Vert _1^2+\frac{1}{2\gamma }\Vert G_N\Big (\bar{\partial }_tu^k-u_t^k\Big ) \Vert ^2_1 \\&\le \frac{\gamma }{2}\Vert \theta ^k\Vert _1^2+\frac{1}{2\gamma }\frac{1}{\tau ^2} \Vert \int _{t_{k-1}}^{t_k}(s-t_{k-1})G_Nu_{tt}ds\Vert ^2_1 \\&\le \frac{\gamma }{2}\Vert \theta ^k\Vert _1^2+\frac{1}{2\gamma }\frac{1}{\tau ^2} \int _{t_{k-1}}^{t_k}\frac{(s-t_{k-1})^2}{s^2}ds\int _{t_{k-1}}^{t_k}s^2\Vert G_Nu_{tt}\Vert _1^2ds \\&\le \frac{\gamma }{2}\Vert \theta ^k\Vert _1^2+\frac{\tau }{2\gamma t_k^2} \int _{t_{k-1}}^{t_k}s^2\Vert G_Nu_{tt}\Vert ^2_1ds. \end{aligned}\nonumber \end{aligned}$$

  Therefore

  $$\begin{aligned} \bar{\partial }_t\Vert \theta ^k\Vert ^2+\tau \Vert \bar{\partial }_t\theta ^k\Vert ^2+2\gamma \Vert \Delta \theta ^k\Vert ^2+\gamma \Vert \theta ^k\Vert ^2_1\le \frac{\tau }{\gamma t_k^2}\int _{t_{k-1}}^{t_k}s^2\Vert G_Nu_{tt}\Vert _1^2ds. \end{aligned}$$

  Multiplying the above inequality by $t_k^2$, taking the sum for k from 1 to n, using $\Vert G_Nu_{tt}\Vert _1\le C\Vert u_{t}\Vert _3$, we get

  $$\begin{aligned} \begin{aligned}&t_n^2\Vert \theta ^n\Vert ^2+\gamma \tau \sum _{k=1}^nt_k^2\Big (2\Vert \Delta \theta ^k\Vert ^2+\Vert \theta ^k\Vert _1^2\Big )+\tau ^2\sum _{k=1}^nt_k^2\Vert \bar{\partial }_t\theta ^k\Vert ^2\\&\quad \le 3\tau \sum _{k=1}^nt_k\Vert \theta ^k\Vert ^2+\frac{\tau ^2}{\gamma }\sum _{k=1}^n\int _{t_{k-1}}^{t_k}s^2\Vert G_Nu_{tt}\Vert _1^2ds\\&\quad \le 3\tau \sum _{k=1}^nt_k\Vert \theta ^k\Vert ^2+C\tau ^2\int _0^{t_n}s^2\Vert u_{t}\Vert ^2_3ds\\&\quad \le 3\tau \sum _{k=1}^nt_k\Vert \theta ^k\Vert ^2+C\tau ^2(1+t_n^3). \end{aligned} \end{aligned}$$

  (3.6)

  Setting $\varphi =G_N\theta ^k$ in (3.4), we have

  $$\begin{aligned} \Big (\bar{\partial }_t\theta ^k+\gamma \Delta ^2\theta ^k- 2\gamma \Delta \theta ^k+\gamma \theta ^k-\Big (\bar{\partial }_tu^k-u_t^k\Big ), G_N\theta ^k\Big )=0. \end{aligned}$$

  Note that

  $$\begin{aligned} \begin{aligned}&\Big (\bar{\partial }_t\theta ^k,G_N\theta ^k\Big )=\frac{1}{2}\bar{\partial }_t\Vert G_N\nabla \theta ^k\Vert ^2, \\&\quad \Big (\gamma \Delta ^2\theta ^k- 2\gamma \Delta \theta ^k+\gamma \theta ^k,G_N\theta ^k\Big ) =\gamma \Vert \nabla \theta ^k\Vert ^2+\gamma \Vert \theta ^k\Vert ^2,\\&\quad \Big (\bar{\partial }_tu^k-u_t^k, G_N\theta ^k\Big )\le \frac{\gamma }{2}\Vert \theta ^k\Vert ^2+ \frac{1}{2\gamma }\Vert G_N\Big (\bar{\partial }_tu^k-u_t^k\Big )\Vert ^2. \end{aligned} \end{aligned}$$

  Hence

  $$\begin{aligned} \bar{\partial }_t\Vert G_N\nabla \theta ^k\Vert ^2+\gamma \Big (\Vert \nabla \theta ^k\Vert ^2+\Vert \theta ^k\Vert ^2\Big ) \le \frac{1}{\gamma }\Vert G_N\Big (\bar{\partial }_tu^k-u_t^k\Big )\Vert ^2. \end{aligned}$$

  (3.7)

  Multiplying (3.7) by $\tau t_k$, taking the sum for k from 1 to n, using $\Vert G_Nu_{tt}\Vert \le C\Vert u_{t}\Vert _2$, we get

  $$\begin{aligned} \begin{aligned}&t_n\Vert G_N\nabla \theta ^n\Vert ^2+\gamma \tau \sum _{k=1}^nt_k\Big (\Vert \nabla \theta ^k\Vert ^2+\Vert \theta ^k\Vert ^2\Big )\\&\quad \le \frac{\tau }{\gamma }\sum _{k=1}^nt_k\Vert G_N\Big (\bar{\partial }_tu^k-u_t^k\Big )\Vert ^2 +\tau \sum _{k=1}^n\Vert G_N\nabla \theta ^k\Vert ^2\\&\quad \le \frac{\tau ^2}{\gamma }\int _0^{t_n}s\Vert G_Nu_{tt}\Vert ^2ds+\tau \sum _{k=1}^n\Vert G_N\nabla \theta ^k\Vert ^2\\&\quad \le C\tau ^2(1+t_n^2)+\tau \sum _{k=1}^n\Vert G_N\nabla \theta ^k\Vert ^2. \end{aligned} \end{aligned}$$

  (3.8)

  Set $\varphi =G_N^2\theta ^k$ in (3.4). Therefore

  $$\begin{aligned} \Big (\bar{\partial }_t\theta ^k+\gamma \Delta ^2\theta ^k- 2\gamma \Delta \theta ^k+\gamma \theta ^k-\Big (\bar{\partial }_tu^k-u_t^k\Big ), G_N^2\theta ^k\Big )=0. \end{aligned}$$

  Note that

  $$\begin{aligned} \begin{aligned}&\Big (\bar{\partial }_t\theta ^k,G_N^2\theta ^k\Big )=\frac{1}{2}\bar{\partial }_t\Vert G_N\theta ^k\Vert ^2+\frac{\tau }{2}\Vert G_N\nabla \theta ^k\Vert ^2,\\&\quad \Big (\gamma \Delta ^2\theta ^k- 2\gamma \Delta \theta ^k+\gamma \theta ^k,G_N^2\theta ^k\Big )=\gamma \Vert \Delta G_N\theta ^k\Vert ^2+\gamma \Vert G_N\theta ^k\Vert ^2_1,\\&\quad \Big (\bar{\partial }_tu^k-u_t^k,G_N^2\theta ^k\Big )\le \frac{\gamma }{2}\Vert G_N\nabla \theta ^k\Vert ^2+\frac{1}{2\gamma }\Vert G_N^2\Big (\bar{\partial }_tu^k-u_t^k\Big )\Vert _1^2. \end{aligned} \end{aligned}$$

  Thus

  $$\begin{aligned} \bar{\partial }_t\Vert G_N\theta ^k\Vert ^2+\gamma \Vert G_N\theta ^k\Vert _1^{2} \le \frac{1}{\gamma }\Vert G_N^2\Big (\bar{\partial }_tu^k-u_t^k\Big )\Vert _1^2. \end{aligned}$$

  (3.9)

  Taking the sum of (3.9) for k form 1 to n, applying $\Vert G_N^2\nabla u_{tt}\Vert \le C\Vert \nabla u_{t}\Vert$, we obtain

  $$\begin{aligned} \begin{aligned}&\Vert G_N\theta ^k\Vert ^2+\gamma \tau \sum _{k=1}^n\Vert G_N\theta ^k\Vert ^2_1\\&\quad \le \frac{\tau }{\gamma }\sum _{k=1}^n\Vert G_N^2\Big (\bar{\partial }_tu^k-u_t^k\Big )\Vert _1^{2} \le \frac{\tau ^2}{\gamma }\int _0^{t_n}\Vert G_N^2\Big (\bar{\partial }_tu^k-u_t^k\Big )\Vert ^2_1ds\\&\quad \le C\tau ^2(1+t_n). \end{aligned} \end{aligned}$$

  (3.10)

  Adding (3.6), (3.8), and (3.10) together gives

  $$\begin{aligned} \begin{aligned}&t_n^2\Vert \theta ^k\Vert ^2+\tau ^2\sum _{k=1}^nt_k^2\Vert \bar{\partial }_t\theta ^k \Vert ^2+\gamma \tau \sum _{k=1}^nt_k^2\Vert \theta ^k\Vert _2^2 +\gamma \tau \sum _{k=1}^nt_k\Vert \theta ^k\Vert _2^2\\&\quad +\gamma \tau \sum _{k=1}^n\Vert G_N\theta ^k\Vert _1^2\le C\tau ^2(1+t_n^3). \end{aligned} \end{aligned}$$

  (3.11)

  Multiplying (3.11) by $\frac{1}{\gamma \tau }$, we derive that

  $$\begin{aligned} t_n^2\Big (\Vert \Delta \theta ^k\Vert ^2+\Vert \theta ^k\Vert ^2\Big )\le C\tau (1+t_n^3). \end{aligned}$$

  (3.12)

• Step 2 Take the error estimates of solution $u_N^n$ of problem (1.4)–(1.5). Set $v_N^k-u_N^k=e^k$. Hence, $e^k$ satisfies

  $$\begin{aligned}&\displaystyle \Big (\bar{\partial }_te^k+\gamma \Delta ^2e^k+\gamma \Delta \theta ^k-\gamma \theta ^k,\varphi \Big )+\Big (\nabla f(u^k)-f^{\prime }\Big (u_N^{k-1}\Big )\nabla u_{N}^k,\nabla \varphi \Big )\nonumber \\&\displaystyle \quad =\Big (\nabla \cdot g(u^k),\varphi \Big )-B\Big (u_N^{k-1},u_N^k,\varphi \Big ),\quad \forall \varphi \in S_N, k=1,2,\ldots ,\end{aligned}$$

  (3.13)
  $$\begin{aligned}&\displaystyle e^0=0. \end{aligned}$$

  (3.14)

  Setting $\varphi =e^k$ in (3.13), we derive that

  $$\begin{aligned}&\frac{1}{2}\bar{\partial }_t\Vert e^k\Vert ^2+\frac{\tau }{2} \Vert \bar{\partial }_te^k\Vert ^2+\gamma \Vert \Delta e^k\Vert ^2 =-2\gamma \Big (\Delta \theta ^k,e^k\Big )+\gamma \Big (\theta ^k,e^k\Big )\nonumber \\&\quad -\Big (\nabla f(u^k)-f^{\prime }\Big (u_N^{k-1}\Big )\nabla u_{N}^k,\nabla e^k\Big ) \quad +\Big (\nabla \cdot g(u^k), e^k\Big )-B\Big (u_N^{k-1},u_N^k,e^k\Big ). \end{aligned}$$

  (3.15)

  Note that

  $$\begin{aligned}&-\Big (\nabla f(u^k)-f^{\prime }\Big (u_N^{k-1}\Big )\nabla u_{N}^k,\nabla e^k\Big ) =-\Big (f^{\prime }(u^k)\nabla u^k-f^{\prime }\Big (u_N^{k-1}\Big )\nabla u_{N}^k,\nabla e^k\Big )\\&\quad \quad =-\Big (f^{\prime }(u^k)\nabla u^k-f^{\prime }(u^k)\nabla u_{N}^k+f^{\prime }(u^k)\nabla u_{N}^k-f^{\prime }(u^{k-1})\nabla u_{N}^k\\&\quad \qquad +f^{\prime }(u^{k-1})\nabla u_{N}^k-f^{\prime }\big (u_N^{k-1}\big )\nabla u_{N}^k,\nabla e^k\Big )\\&\quad \quad =-\Big (f^{\prime }(u^k)(\nabla u^k-\nabla u_{N}^k),\nabla e^k\Big )-\Big (\Big [f^{\prime }(u^k)-f^{\prime }(u^{k-1})\Big ]\nabla u_{N}^k,\nabla e^k\Big )\\&\quad \qquad -\Big (\Big [f^{\prime }(u^{k-1})-f^{\prime }\Big (u_N^{k-1}\Big )\Big ]\nabla u_{N}^k,\nabla e^k\Big )\\&\quad \triangleq I_1+I_2+I_3.\\&\quad I_1=\Big (f^{\prime }(u^k)\Delta e^k+f{^{\prime \prime }}(u^k)\nabla u^k\nabla e^k,u^k-u_N^k\Big )\\&\qquad \le \Big (\Vert f^{\prime }(u^k)\Vert _{\infty }\Vert \Delta e^k\Vert +\Vert f{^{\prime \prime }}(u^k)\Vert _{\infty }\Vert \nabla u^k\Vert _{L^4}\Vert \nabla e^k\Vert _{L^4}\Big )\Vert u^k-u_N^k\Vert \\&\quad \quad \le \Big (\Vert f^{\prime }(u^k)\Vert _{\infty }\Vert \Delta e^k\Vert +C\Vert f{^{\prime \prime }}(u^k)\Vert _{\infty }\Vert \nabla u^k\Vert _{L^4}\Vert \Delta e^k\Vert \Big )\Vert u^k-u_N^k\Vert \\&\quad \quad \le C\Vert u^k-u_N^k\Vert \Big (\Vert e^k\Vert ^2+\Vert \Delta e^k\Vert ^2\Big )\\&\quad \quad \le C\Big (\Vert u^k-u^{k-1}\Vert +\Vert u^{k-1}-u_N^{k-1}\Vert +\Vert u_N^{k-1}-u_N^k\Vert \Big )\Big (\Vert e^k\Vert ^2+\Vert \Delta e^k\Vert ^2\Big )\\&\quad \quad \le C\left( \Vert e^{k-1}\Vert ^2+\Vert \rho ^{k-1}\Vert ^2+\Vert \theta ^{k-1}\Vert ^2+\tau \int _{t_{k-1}}^{t_k}\Vert u_t\Vert ^2ds+\tau ^2\Vert \bar{\partial }_tu_N^k\Vert ^2\right) \\&\quad \qquad +\varepsilon \Big (\Vert e^k\Vert ^2+\Vert \Delta e^k\Vert ^2\Big ),\\&\quad I_2= -\Big (f{^{\prime \prime }}(\phi _1u^k+(1-\phi _1)u^{k-1})\nabla u_{N}^k(u^k-u^{k-1}),\nabla e^k\Big )\\&\qquad \le \Vert f{^{\prime \prime }}\Big (\phi _1u^k+(1-\phi _1)u^{k-1}\Big )\Vert _{\infty }\Vert \nabla u_{N}^k\Vert _{L^4}\Vert u^k-u^{k-1}\Vert \Vert \nabla e^k\Vert _{L^4}\\&\qquad \le C\Vert f{^{\prime \prime }}\Big (\phi _1u^k+(1-\phi _1)u^{k-1}\Big )\Vert _{\infty }\Vert \nabla u_{N}^k\Vert _{L^4}\Vert u^k-u^{k-1}\Vert \Big (\Vert e^k\Vert ^2+\Vert \Delta e^k\Vert ^2\Big )\\&\qquad \le C\Big (\Vert e^k\Vert ^2+\Vert \Delta e^k\Vert ^2\Big )\Vert u^k-u^{k-1}\Vert \le \varepsilon \Big (\Vert e^k\Vert ^2+\Vert \Delta e^k\Vert ^2\Big )+C\tau \int _{t_{k-1}}^{t_k}\Vert u_t\Vert ^2\mathrm{d}s,\\&\qquad I_3=-\Big (f{^{\prime \prime }}(\phi _2u^{k-1}+(1-\phi _2)u_N^{k-1}\Big )\nabla u_{N}^k(u^{k-1}-u_N^{k-1}),\nabla e^k)\\&\le \Vert f{^{\prime \prime }}\Big (\phi _2u^{k-1}+(1-\phi _2)u_N^{k-1}\Big )\Vert _{\infty }\Vert \nabla u_{N}^k\Vert _{L^4}\Vert u^{k-1}-u_N^{k-1}\Vert \Vert \nabla e^k\Vert _{L^4}\\&\qquad \le C\Big (\Vert e^k\Vert ^2+\Vert \Delta e^k\Vert ^2\Big )\Vert u^{k-1}-u_N^{k-1}\Vert \\&\qquad \le \varepsilon \Big (\Vert e^k\Vert ^2+\Vert \Delta e^k\Vert ^2\Big )+C\Big (\Vert \rho ^{k-1}\Vert ^2+\Vert \theta ^{k-1}\Vert ^2+\Vert e^{k-1}\Vert ^2\Big ), \end{aligned}$$

  where $\phi _1,\phi _2\in (0,1)$. Hence

  $$\begin{aligned}&\Big (\nabla f(u^k)-f^{\prime }\big (u_N^{k-1}\big )\nabla u_{N}^k,\nabla e^k\Big )\le 3\varepsilon \Big (\Vert e^k\Vert ^2+\Vert \Delta e^k\Vert ^2\Big )\\&\quad +C\left( \Vert \rho ^{k-1}\Vert ^2+\Vert \theta ^{k-1}\Vert ^2+\Vert e^{k-1}\Vert ^2+\tau \int _{t_{k-1}}^{t_k} \Vert u_t\Vert ^2ds+\tau ^2\Vert \bar{\partial }_tu_N^k\Vert ^2\right) . \end{aligned}$$

  On the other hand, we have

  $$\begin{aligned}&\Big (\nabla \cdot g(u^k), e^k\Big )-B\Big (u_N^{k-1},u_N^k,e^k\Big )\nonumber \\&\quad = \Big (G(u^k)\nabla \cdot u^k-G\big (u_N^{k-1}\big )\nabla \cdot u_{N}^k,e^k\Big )-\Big (G(u^k)u^k-G\big (u_N^{k-1}\big )u_N^k,\nabla \cdot e^k\Big )\nonumber \\&\quad =\Big (G(u^k)\nabla \cdot \big (u^k-u_{N}^k\big ),e^k\Big )+\Big (G(u^k)-G(u^{k-1}), \nabla \cdot u_{N}^ke^k\Big )\nonumber \\&\qquad +\Big (G(u^{k-1}-G\big (u_N^{k-1}\big ),\nabla \cdot u_{N}^ke^k\Big )+\Big (u^k(G(u^k)-G(u^{k-1}),\nabla \cdot e^k\Big )\nonumber \\&\qquad +\Big (u^k G(u^{k-1})-G\big (u_N^{k-1}\big ),\nabla \cdot e^k\Big )+\Big (G(u^k)\big (u^k-u_N^k\big ),\nabla \cdot e^k\Big )\nonumber \\&\quad \le C\left( \Vert e^{k-1}\Vert ^2+\Vert \rho ^{k-1}\Vert ^{2} +\Vert \theta ^{k-1}\Vert ^2+\tau ^2\Vert \bar{\partial }_tu_N^k \Vert ^2+\tau \int _{t_{k-1}}^{t_k}\Vert u_t\Vert ^2\mathrm{d}s\right) \nonumber \\&\qquad +\varepsilon \Big (\Vert \Delta e^k\Vert ^2+\Vert e^k\Vert ^2\Big ). \end{aligned}$$

  (3.16)

  We also have

  $$\begin{aligned} -2\gamma \Big (\Delta \theta ^k,e^k\Big )+\gamma \Big (\theta ^k,e^k\Big ) \le \varepsilon \Big (\Vert \Delta e^k\Vert ^2+\Vert e^k\Vert ^2\Big )+C\Vert \theta ^k\Vert ^2. \end{aligned}$$

  Based on Poincaré’s inequality, we have

  $$\begin{aligned} \Vert e^k\Vert ^2\le [C(\Omega )]^2\Vert \Delta e^k\Vert ^2. \end{aligned}$$

  Then, a simple calculation shows that

  $$\begin{aligned} \begin{aligned}&\bar{\partial }_t\Vert e^k\Vert ^2+2\Big [\gamma -5\varepsilon (1+[C(\Omega )]^2)\Big ]\Vert \Delta e^k\Vert ^2+\tau \Vert \bar{\partial }_te^k\Vert ^2 \\&\quad \le C\left( \Vert e^{k-1}\Vert ^2+\Vert \rho ^{k-1}\Vert ^2+\Vert \theta ^k\Vert ^2+\Vert \theta ^{k-1} \Vert ^2+\tau ^2\Vert \bar{\partial }_tu_N^k\Vert ^2+\tau \int _{t_{k-1}}^{t_k}\Vert u_t\Vert ^2\mathrm{d}s\right) , \end{aligned} \end{aligned}$$

  (3.17)

  where $\varepsilon$ is small enough, it satisfies $\gamma -5\varepsilon (1+[C(\Omega )]^2)>0$. Using discrete Gronwall’s inequality to (3.17), we have

  $$\begin{aligned} \begin{aligned} \Vert e^n\Vert ^2&\le Ce^{Ct_n}\tau \sum _{k=1}^n\left( \Vert \rho ^{k-1}\Vert ^2+\Vert \theta ^k\Vert ^2+\tau ^2 \Vert \bar{\partial }_tu_N^k\Vert ^2+\tau \int _{t_{k-1}}^{t_k}\Vert u_t\Vert ^2\mathrm{d}s\right) \\&\le Ce^{Ct_n}(N^{-4}+\tau ). \end{aligned} \end{aligned}$$

  (3.18)

  Taking the sum of (3.17) for k from 1 to n and using (3.18), we have

  $$\begin{aligned} \tau \sum _{k=1}^n\Vert \Delta e^k\Vert ^2+\tau ^2\sum _{k=1}^n\Vert \bar{\partial }_te^k\Vert ^2\le C(N^{-4}+\tau ). \end{aligned}$$

  (3.19)

  Setting $\varphi =\Delta e^k$ in (3.13), we derive that

  $$\begin{aligned} \begin{aligned}&\bar{\partial }_t\Vert \nabla e^k\Vert ^2+\gamma \Vert \nabla \Delta e^k\Vert ^2+\tau \Vert \bar{\partial }_t\nabla e^k\Vert ^2=\gamma \Big (\nabla \theta ^k,\nabla \Delta e^k\Big )+\gamma \Big (\theta ^k,\Delta e^k\Big )\\ {}&+\quad \Big (\nabla f(u^k)-f^{\prime }\big (u_N^{k-1}\big )\nabla u_{N}^k,\nabla \Delta e^k\Big ) -\Big (\nabla \cdot g(u^k),\Delta e^k\Big )+B\Big (u_N^{k-1},u_N^k,\Delta e^k\Big ).\nonumber \end{aligned} \end{aligned}$$

  Note that

  $$\begin{aligned} \begin{aligned}&\gamma \Big (\nabla \theta ^k,\nabla \Delta e^k\Big )+\gamma \Big (\theta ^k,\Delta e^k\Big )\le \frac{\gamma }{10}\Big (\Vert \nabla \Delta e^k\Vert ^2+\Vert e^k\Vert ^2\Big )+C\Vert \theta _x^k\Vert ^2,\\&\quad \Big (\nabla f(u^k)-f^{\prime }\big (u_N^{k-1}\big )\nabla u_{N}^k,\nabla \Delta e^k\Big )\\&\quad =\left( \Big [f^{\prime }(u^k)-f^{\prime }(u^{k-1})\Big ]\nabla u^k,\nabla \Delta e^k\right) +\Big (f^{\prime }(u^{k-1})\nabla \big (u^k-u_{N}^k\big ),\nabla \Delta e_{xxx}^k\Big )\\&\quad \quad +\left( \Big [f^{\prime }(u^{k-1})-f^{\prime }\big (u_N^{k-1}\big )\Big ]\nabla u_{N}^k,\nabla \Delta e^k\right) \\&\quad \le \frac{3\gamma }{10}\Vert \nabla \Delta e^k\Vert ^2+C\Big (\Vert e^k\Vert ^2+\Vert \rho ^{k}\Vert _1^2+\Vert \theta ^{k}\Vert ^2_1+\tau ^2\Vert \bar{\partial }_tu_{N}^k\Vert ^2_1\Big ),\\&\quad \quad -\Big (\nabla \cdot g(u^k),\Delta e^k\Big )+B\Big (u_N^{k-1},u_N^k,\Delta e^k\Big ) \\&\quad \le \frac{\gamma }{10}\Big (\Vert \nabla \Delta e^k\Vert ^2+\Vert \nabla e^k\Vert ^2\Big )+C\Big (\Vert e^k\Vert ^2+\Vert \rho ^{k}\Vert _1^2+\Vert \theta ^{k}\Vert ^2_1+\tau ^2\Vert \bar{\partial }_tu_{N}^k\Vert ^2_1\Big ). \end{aligned} \end{aligned}$$

  Summing up, we get

  $$\begin{aligned} \begin{aligned} \bar{\partial }_t\Vert \nabla e^k\Vert ^2+\gamma \Vert \nabla \Delta e^k\Vert ^2 \le C\Big (\Vert e^k\Vert ^2+\Vert \rho ^{k}\Vert _1^2+\Vert \theta ^{k}\Vert ^2_1+\tau ^2\Vert \bar{\partial }_tu_{N}^k\Vert ^2_1\Big ). \end{aligned} \end{aligned}$$

  Multiplying the above relation by $\tau t_k$, summing them for k form 1 to n, and using Lemma 2.3, (3.18), (3.19), we derive that

  $$\begin{aligned} \begin{aligned}&t_n\Vert \nabla e^k\Vert ^2+\gamma \tau \sum _{k=1}^nt_k\Vert \Delta e^k\Vert ^2\\&\quad \le \tau \sum _{k=1}^n\Vert \nabla e^k\Vert ^2+\tau \sum _{k=1}^nt_k\Big (\Vert e^k\Vert ^2+\Vert \rho ^k\Vert _1^{2} +\Vert \theta ^k\Vert _1^2+\tau ^2\Vert \bar{\partial }_tu_N^k\Vert _1^2\Big )\\&\quad \le C(1+t_n^4)e^{Ct_n}(N^{-2}+\tau ). \end{aligned} \end{aligned}$$

  (3.20)

  Setting $\varphi =\Delta ^2e^k$ in (3.13), we derive that

  $$\begin{aligned} \begin{aligned}&\bar{\partial }_t\Vert \Delta e^k\Vert ^2+\gamma \Vert \Delta ^2e^k\Vert ^2+\tau \Vert \bar{\partial }_t\Delta e^k\Vert ^2\\&\quad =\gamma \Big (-\Delta \theta ^k+\theta ^k,\Delta ^2e^k\Big ) \qquad +\left( \nabla \cdot \Big [\nabla \cdot f(u^k)-f^{\prime }\big (u_N^{k-1}\big )\nabla \cdot u_{N}^k\Big ],\Delta ^2e^k\right) \\&\qquad +B\Big (u^k,u^k,\Delta ^2e^k\Big )-B\Big (u_N^{k-1},u_N^k,\Delta ^2u^k\Big ). \end{aligned} \end{aligned}$$

  Note that

  $$\begin{aligned} \gamma \Big (-\Delta \theta ^k+\theta ^k,\Delta ^2e^k\Big ) \le \frac{\gamma }{6}\Vert \Delta ^2e^k\Vert ^2+C\Vert \theta ^k\Vert ^2_2. \end{aligned}$$

  On the other hand, we have

  $$\begin{aligned} \begin{aligned}&\left( \nabla \cdot \Big [\nabla \cdot f(u^k)-f^{\prime }\big (u_N^{k-1}\big )\nabla \cdot u_{N}^k\Big ],\Delta ^2e^k\right) \\&\quad =\left( f^{\prime }(u^k)\Delta u^k-f^{\prime }\big (u_N^{k-1}\big )\Delta u_{N}^k+f{^{\prime \prime }}(u^k)|\nabla u^k|^2-f{^{\prime \prime }}\big (u_N^{k-1}\big )\nabla u_{N}^{k-1}\nabla u_{N}^k,\Delta ^2e^k\right) \\&\quad =\Big (f^{\prime }(u^k)\Delta u^k-f^{\prime }(u^{k-1})\Delta u^k,\Delta ^2e^k\Big )+\Big (f^{\prime }(u^{k-1})\Delta u^k-f^{\prime }(u^{k-1})\Delta u_{N}^k,\Delta ^2e^k\Big )\\&\qquad +\Big (f^{\prime }(u^{k-1})\Delta u_{N}^k-f^{\prime }\big (u_N^{k-1}\big )\Delta u_{N}^k,\Delta ^2e^k\Big ) +\Big (\Big [f{^{\prime \prime }}(u^k)-f{^{\prime \prime }}(u^{k-1})\Big ]|\nabla u^k|^2,\Delta ^2e^k\Big )\\&\qquad +\Big (f{^{\prime \prime }}(u^{k-1})\nabla u^k\nabla (u^k-u_{N}^k,\Delta ^2e^k\Big )\\&\qquad +\Big (f^{\prime \prime }(u^{k-1})\nabla u^k\nabla u_{N}^k-f{^{\prime \prime }}(u^{k-1})\nabla u^{k-1}\nabla u_{N}^{k},\Delta ^2e^k\Big )\\&\qquad +\Big (f{^{\prime \prime }}(u^{k-1})\nabla u^{k-1}\nabla u_{N}^k-f{^{\prime \prime }}(u^{k-1})\nabla u_{N}^{k-1}\nabla u_{N}^k,\Delta ^2e^k\Big )\\&\qquad +\left( \Big [f{^{\prime \prime }}(u^{k-1})-f{^{\prime \prime }}(u_{N}^{k-1})\Big ]\nabla u_{N}^{k-1}\nabla u_{N}^k,\Delta ^2e^k\right) \\&\quad \le \frac{\gamma }{6}\Vert \Delta ^2e^k\Vert ^2+C\Big (\tau ^2\Vert \bar{\partial }_tu^k \Vert _2^2+\Vert \rho ^k\Vert _2^2+\Vert \theta ^k\Vert _2^2+\Vert e^k\Vert ^2\Big ). \end{aligned} \end{aligned}$$

  and

  $$\begin{aligned}&B(u^k,u^k,\Delta ^2e^k)-B(u_N^{k-1},u_N^k,\Delta ^2e^k)\\&\quad =2\Big (G(u^k)\nabla \cdot u^k-G(u^{k-1})\nabla \cdot u^k,\Delta ^2e^k\Big )\\&\qquad +2\Big (G(u^{k-1})\nabla \cdot u^k-G\big (u_N^{k-1}\big )\nabla \cdot u^k,\Delta ^2e^k\Big )\\&\qquad +2\Big (G\big (u_N^{k-1}\big )\nabla \cdot u^k-G\big (u_N^{k-1}\big )\nabla \cdot u_{N}^k,\Delta ^2e^k\Big )\\&\qquad +\Big (G^{\prime }(u^k)u^k\nabla \cdot u^k-G^{\prime }(u^{k-1}u^k)\nabla \cdot u^k,\Delta ^2e^k\Big )\\&\qquad +\Big (G^{\prime }(u^{k-1})u^k\nabla \cdot u^k-G^{\prime }\big (u_N^{k-1}\big )u^k\nabla \cdot u^k,\Delta ^2e^k\Big )\\&\qquad +\Big (G^{\prime }\big (u_N^{k-1}\big )u^k\nabla \cdot u^k-G^{\prime }\big (u_N^{k-1}\big )u_N^k\nabla \cdot u^k ,\Delta ^2e^k\Big )\\&\qquad +\Big (G^{\prime }\big (u_N^{k-1}\big )u_N^k\nabla \cdot u^k -G^{\prime }\big (u_N^{k-1}\big )u_N^k\nabla \cdot u^{k-1},\Delta ^2e^k\Big )\\&\qquad +\Big (G^{\prime }\big (u_N^{k-1}\big )u_N^k\nabla \cdot u^{k-1}-G^{\prime }\big (u_N^{k-1}\big )u_N^k\nabla \cdot u^{k-1}_{N} ,\Delta ^2e^k\Big )\\&\quad \le \frac{\gamma }{6}\Vert \Delta ^2e^k\Vert ^2+C\Big (\Vert \rho ^k\Vert _2^2+\Vert \theta ^k\Vert _2^2+\Vert e^k\Vert ^{2} +\tau ^2\Vert \bar{\partial }_tu_N^k\Vert _2^2\Big ). \end{aligned}$$

Summing up, we get

$$\begin{aligned} \bar{\partial }_t\Vert \Delta e^k\Vert ^2+\gamma \Vert \Delta ^2e^k\Vert ^2\le C\left( \Vert \rho ^k\Vert _2^2+\Vert \theta ^k\Vert _2^2+\Vert e^k\Vert ^2+\tau ^{2} \Vert \bar{\partial }_tu_N^k\Vert _2^2\right) . \end{aligned}$$

(3.21)

By Lemma 1.3 and Lemma 3.3, we have

$$\begin{aligned} \Vert \rho ^k\Vert _2^2\le CN^{-2}\Vert \nabla \Delta u^k\Vert ^2\le C\left( \frac{1}{t_k}+t_k\right) N^{-2}. \end{aligned}$$

Therefore, multiplying (3.21) by $\tau t_k^2$, summing up, we deduce that

$$\begin{aligned} \begin{aligned}&t_n^2\Vert \Delta e^n\Vert ^2+\gamma \tau \sum _{k=1}^nt_k^2\Vert \Delta ^2e^k\Vert ^2\\&\quad \le C\tau \sum _{k=1}^nt_k^2\left[ \Big (\frac{1}{t_k}+t_k\Big )N^{-2}+\Vert \theta ^k\Vert _2^2+\Vert e^k \Vert ^2+\tau ^2\Vert \bar{\partial }_tu_N^k\Vert _2^2\right] +2\tau \sum _{k=1}^nt_k\Vert \Delta e^k\Vert ^2\\&\quad \le Ce^{Ct_n}(N^{-2}+\tau ). \end{aligned} \end{aligned}$$

It then follows from Triangle’s inequality that

$$\begin{aligned} \Vert u^n-u_N^n\Vert _2^2\le 3\left( \Vert \rho ^n\Vert _2^2+\Vert \theta ^n\Vert _2^2+\Vert e^n\Vert _2^2\right) \le C(N^{-2}+\tau ),\quad \forall t\in (0,+\infty ). \end{aligned}$$

Then, we complete the proof of the theorem. $\square$