The Reduced-Dimension Method for Crank–Nicolson Mixed Finite Element Solution Coefficient Vectors of the Extended Fisher–Kolmogorov Equation

Abstract

A reduced-dimension (RD) method based on the proper orthogonal decomposition (POD) technology and the linearized Crank–Nicolson mixed finite element (CNMFE) scheme for solving the 2D nonlinear extended Fisher–Kolmogorov (EFK) equation is proposed. The method reduces CPU runtime and error accumulation by reducing the dimension of the unknown CNMFE solution coefficient vectors. For this purpose, the CNMFE scheme of the above EFK equation is established, and the uniqueness, stability and convergence of the CNMFE solutions are discussed. Subsequently, the matrix-based RDCNMFE scheme is derived by applying the POD method. Furthermore, the uniqueness, stability and error estimates of the linearized RDCNMFE solution are proved. Finally, numerical experiments are carried out to validate the theoretical findings. In addition, we contrast the RDCNMFE method with the CNMFE method, highlighting the advantages of the dimensionality reduction method.

1. Introduction

The primary goal of this research study is to apply the POD-based reduced-dimension method to solve the following nonlinear extended Fisher–Kolmogorov (EFK) equation:

$$\begin{array}{c}\hfill \left\{\begin{array}{cc}{u}_t+\gamma {\mathsf{\Delta }}^{2}u-\mathsf{\Delta }u+f(u)=0,\hfill & (\mathit{x},t)\in \Omega \times J,\hfill \\ u(\mathit{x},0)=u_0(\mathit{x}),\hfill & \mathit{x}\in \Omega ,\hfill \\ u(\mathit{x},t)=\mathsf{\Delta }u(\mathit{x},t)=0,\hfill & (\mathit{x},t)\in \partial \Omega \times \overline{J},\hfill \end{array}\right.\end{array}$$

(1)

where $\Omega \subset R^d(d=1,2,3)$ is a bounded convex polygonal domain with boundary $\partial \Omega$. $J=(0,T]$ is the time interval, where $0<T<\infty$. Coefficient $\gamma$ is a positive constant. $u_0(\mathit{x})$ is the known initial function. $f(u)=u^3-u$ is the Lipschitz continuous function of u, that is, there exists a positive constant C such that

$$|f(u_1)-f(u_2)|⩽C|u_1-u_2|.$$

(2)

For simplicity, we assume that $u_0(\mathit{x})=0$ in the following theoretical analysis.

When $\gamma =0$ in (1), the second-order diffusion equation is obtained, which is called the standard Fisher–Kolmogorov equation. Coullet et al. [1] added a stabilizing fourth-order derivative term to obtain Equation (1) and called it the extended Fisher–Kolmogorov equation. The EFK equation has a very important physical background. It has been applied to various kinds of physics and engineering problems. Because of the complexity of the equation, it is difficult to obtain its analytical solution. Therefore, it is very important to solve the equation numerically.

There are many numerical methods for solving PEDs, for instance, the direct local boundary integral equation method [2], the Fourier pseudo-spectral method [3], the modified cubic B-spline and polynomial-based differential quadrature method [4,5], the explicit finite difference method (EFDM) [6,7], the radial basis function (RBF) method [8], physics-informed neural networks (PINNs) [6,9,10] and so on. These numerical methods are also used to solve EFK equations. He [11] used a three-level linearly implicit finite difference method to solve 1D and 2D EFK equations, and the method is second-order convergent in both the time and space directions. Kumar et al. [12] proposed two RBF-based mesh-free schemes for the numerical simulation of nonlinear EFK models. Tarbiyati and Nemati [13] presented a method for initializing weights in PINNs by using FD to study the one-dimensional diffusion (heat) equation, the FK equation and the nonlinear Burgers equation. Sultan and Zhang [14] used the moving mesh finite difference method (MMFDM) in conjunction with PINNs to solve the FK equation.

In addition, various fourth-order finite difference schemes [15,16,17,18], which have fourth-order and second-order accuracy in space and time variables, respectively, have been developed to solve EFK equations. Sweilam et al. [19] studied the stochastic EFK equation by combining the compact finite difference scheme and the semi-implicit Euler–Maruyama approach. Danumjaya and Pani [20] applied mixed finite element methods to solve EFK equations with different types of boundary conditions. Doss and Nandini [21] employed a splitting technique to derive an ${\mathrm{H}}^1$-Galerkin mixed finite element scheme for EFK equations. Wang et al. [22] used a new linearized CNMFE method to study EFK equations, in which $\nabla u$ belongs to the weaker ${(L^2(\Omega ))}^2$ space instead of the classical $H(\mathrm{div};\Omega )$ space. In this paper, we establish a CNMFE scheme for the 2D EFK equation and analyze the uniqueness, stability and convergence of the CNMFE solutions.

However, in practical problems, using the CNMFE method to solve EFK equations is a problem with a large number of unknowns, which makes running extremely slow, leads to an increase in rounding errors, and makes it difficult to obtain accurate numerical solutions. Therefore, it is crucial to lessen the number of unknowns in the CNMFE method for EFK equations to reduce the computational load, save CPU running time, and minimize the accumulation of rounding errors in the calculation process. The proper orthogonal decomposition (POD) technology, reducing the amount of unknowns in numerical schemes, is an efficient dimensionality reduction method for solving partial differential equations (PDEs). It reduces CPU running time and reduces the accumulation of round-off errors. The POD technology can be applied to the numerical methods for various PDEs, such as the finite difference method [23,24,25], the Schwarz domain decomposition method [26], the finite spectral element method [27,28], etc. For reduced-dimension FE and MFE models based on POD techniques, there are two dimensionality reduction methods. One method is that the optimized models are established by reducing the dimension of the FE or MFE subspace. The parabolic equation [29], the Sobolev equation [30], the Burgers equation [31] and some other PDEs [32,33,34,35] are solved by the reduced-dimension method of constructing the POD subspace. The other is a new reduced-dimension method for the CNFE [36,37] and CNMFE [38,39,40,41] solution coefficient vectors proposed by Luo et al. in 2020. In [42], Luo and Yang combined the unknown solution coefficient vector dimensionality reduction method and the continuous space–time finite element method to study 2D non-stationary incompressible Navier–Stokes equations.

There are existing order reduction methods for EFK equations, such as that in the [24], which combines the high-order compact finite difference scheme with the POD method to obtain the spatial sixth-order accuracy while greatly shortening the calculation time. The article in [40] uses the POD technique to reduce the dimensionality of two-grid CNMFE solution coefficient vectors, in which the matrix theory is used to deeply explore the stability and convergence of TGRDECNMFE solutions. However, the reduced-dimension method proposed in this paper is theoretically significantly different from the above research methods. This paper integrates POD technology with the linearized CNMFE method, and proposes a new reduced-dimension method for 2D nonlinear EFK equations. In this study, we adopt the CN fully discrete format in the temporal direction to achieve second-order time accuracy. Additionally, we linearize the nonlinear term to significantly enhance computational speed and efficiency while preserving model accuracy. Unlike [40], our work utilizes traditional finite element techniques to analyze uniqueness, stability, and convergence of the RDCNMFE solutions. Our proposed method combines the advantages of POD and linearized CNMFE methods for effective dimensionality reduction of the nonlinear EFK equation. This approach not only maintains accuracy but also greatly improves computational speed and efficiency, providing a novel insight and solution for solving the EFK equation.

The structure of this paper is as follows. In Section 2, we propose the CNMFE scheme and discuss the uniqueness, stability and convergence of the CNMFE solutions. In Section 3, we generate the POD basis to derive the reduced-dimension matrix model of the CNMFE solution coefficient vectors and analyze the uniqueness, stability and error estimate of the RDCNMFE solutions. In Section 4, we employ the numerical experiments of the 1D and 2D EFK equations to verify the effectiveness of the RDCNMFE method and correctness of theoretical analysis. In Section 5, we summarize the major conclusions and findings.

2. The Crank–Nicolson Mixed Finite Element Method of the EFK Equation

2.1. The CNMFE Schemes

The Sobolev spaces and their corresponding norms mentioned in this paper adhere to standard definitions (see [43]). In order to develop the CNMFE scheme of the EFK Equation (1), we initially introduce a diffusion term $\omega =-\mathsf{\Delta }u$, resulting in the following two lower-order equations:

$$\left\{\begin{array}{c}{u}_t-\gamma \mathsf{\Delta }\omega +\omega +f(u)=0,(\mathit{x},t)\in \Omega \times J,\hfill \\ -\mathsf{\Delta }u=\omega ,(\mathit{x},t)\in \Omega \times J.\hfill \end{array}\right.$$

(3)

Let $V=H_0^1(\Omega )$. With Green’s formula, the mixed functional form of (3) is as follows.

Problem 1. 

Find $\{u,\omega \}$: $[0,T]\mapsto V\times V$ satisfying

$$\begin{array}{c}\hfill \left\{\begin{array}{c}(u_t,v)+\gamma (\nabla \omega ,\nabla v)+(\omega ,v)+(f(u),v)=0,\forall v\in V,\hfill \\ (\omega ,g)-(\nabla u,\nabla g)=0,\forall g\in V,\hfill \\ u(\mathit{x},0)=0,\omega (\mathit{x},0)=0,\mathit{x}\in \Omega .\hfill \end{array}\right.\end{array}$$

(4)

Let ${\Im }_h$ be the quasi-uniform triangulation subdivision on $\overline{\Omega }$. The FE subspace $V_h$, spanned by the orthonormal basis ${\left\{{\zeta }_j(\mathit{x})\right\}}_{j=1}^M$ under the inner product in $H^1(\Omega )$, is defined as follows:

$$V_h=\{v_h\in V\cap C(\Omega );v_h{\mid }_K\in P_{M-1}(K),K\in \Im \}=\mathrm{span}\{{\zeta }_j(\mathit{x}):1⩽j⩽M\},$$

(5)

in which $P_{M-1}(K)$ is generated by $(M-1)$th degree polynomials on $K\in \Im$.

For given integer $N>0$, let $\mathsf{\Delta }t=T/N$, ${\varphi }^n=\varphi (t_n)$, ${\overline{\partial }}_t{\varphi }^n=({\varphi }^n-{\varphi }^{n-1})/\mathsf{\Delta }t$ and ${\varphi }^{n-{\displaystyle \frac{1}{2}}}=({\varphi }^n+{\varphi }^{n-1})/2$. Therefore, we can rewrite Problem 1 at time $t=t_{n-{\displaystyle \frac{1}{2}}}$ in the following form.

$$\begin{array}{c}\hfill \left\{\begin{array}{c}(u_t(t_{n-{\displaystyle \frac{1}{2}}}),v)+\gamma (\nabla \omega (t_{n-{\displaystyle \frac{1}{2}}}),\nabla v)+(\omega (t_{n-{\displaystyle \frac{1}{2}}}),v)+(f(u(t_{n-{\displaystyle \frac{1}{2}}})),v)=0,\forall v\in V,\hfill \\ (\omega (t_{n-{\displaystyle \frac{1}{2}}}),g)-(\nabla u(t_{n-{\displaystyle \frac{1}{2}}}),\nabla g)=0,\forall g\in V.\hfill \end{array}\right.\end{array}$$

(6)

Then, the equivalent expression of (6) is as follows.

$$\begin{array}{c}\hfill \left\{\begin{array}{c}{\displaystyle (\frac{u^n-u^{n-1}}{\mathsf{\Delta }t},v)+\gamma (\nabla {\omega }^{n-\frac{1}{2}},\nabla v)+({\omega }^{n-\frac{1}{2}},v)+(f({\widehat{\widehat{u}}}^{n-\frac{1}{2}}),v)}\hfill \\ =(R_1^{n-{\displaystyle \frac{1}{2}}},v)+(R_2^{n-{\displaystyle \frac{1}{2}}},v),\forall v\in V,\hfill \\ ({\omega }^{n-{\displaystyle \frac{1}{2}}},g)-(\nabla u^{n-{\displaystyle \frac{1}{2}}},\nabla g)=0,\forall g\in V,\hfill \end{array}\right.\end{array}$$

(7)

where

$$f({\widehat{\widehat{u}}}^{n-{\displaystyle \frac{1}{2}}})={\displaystyle \frac{3}{2}}f(u^{n-1})-{\displaystyle \frac{1}{2}}f(u^{n-2}),$$

(8)

$$R_1^{n-{\displaystyle \frac{1}{2}}}={\displaystyle \frac{u^n-u^{n-1}}{\mathsf{\Delta }t}}-u_t(t_{n-{\displaystyle \frac{1}{2}}}),$$

(9)

$$R_2^{n-{\displaystyle \frac{1}{2}}}=f({\widehat{\widehat{u}}}^{n-{\displaystyle \frac{1}{2}}})-f(u(t_{n-{\displaystyle \frac{1}{2}}})).$$

(10)

Let $\{u_h^n,{\omega }_h^n\}$ represent the CNMFE approximations of solutions $\{u,\omega \}$ for Problem 1 at $t_n=n\mathsf{\Delta }t$. Thus, the CNMFE model of Problem 1 can be constructed as follows.

Problem 2. 

Find $\{u_h^n,{\omega }_h^n\}\in V_h\times V_h(1⩽n⩽N)$ satisfying

$$\begin{array}{c}\hfill \left\{\begin{array}{c}{\displaystyle ({\overline{\partial }}_t{u}_h^n,v_h)+\gamma (\nabla {\omega }_h^{n-\frac{1}{2}},\nabla v_h)+({\omega }_h^{n-\frac{1}{2}},v_h)+(f({\widehat{\widehat{u}}}_h^{n-\frac{1}{2}}),v_h)=0,\forall v_h\in V_h,}\hfill \\ ({\omega }_h^{n-{\displaystyle \frac{1}{2}}},g_h)-(\nabla u_h^{n-{\displaystyle \frac{1}{2}}},\nabla g_h)=0,\forall g_h\in V_h,\hfill \\ u_h^0(\mathit{x})=0,{\omega }_h^0(\mathit{x})=0,\mathit{x}\in \Omega .\hfill \end{array}\right.\end{array}$$

(11)

Remark 1. 

By means of a linearized term

$$f({\widehat{\widehat{u}}}_h^{n-{\displaystyle \frac{1}{2}}})={\displaystyle \frac{3}{2}}f(u_h^{n-1})-{\displaystyle \frac{1}{2}}f(u_h^{n-2})={\displaystyle \frac{3}{2}}{(u_h^{n-1})}^3-{\displaystyle \frac{1}{2}}{(u_h^{n-2})}^3-{\displaystyle \frac{3}{2}}{u}_h^{n-1}+{\displaystyle \frac{1}{2}}{u}_h^{n-2}.$$

(12)

We can observe that the formulation (11) is a linear scheme.

2.2. The Uniqueness, Stability and Convergence of the CNMFE Solutions

By employing the base functions of the FE space $V_h$, the solutions $\{u_h^n,{\omega }_h^n\}$ of Problem 2 can be formulated in the form of

$$\begin{array}{c}{\displaystyle u_h^n=\sum _{j=1}^M{U}_{hj}^n{\zeta }_j=\mathit{\zeta }·{\mathit{U}}_h^n,{\omega }_h^n=\sum _{j=1}^M{W}_{hj}^n{\zeta }_j=\mathit{\zeta }·{\mathit{W}}_h^n,}\hfill \end{array}$$

(13)

where $\mathit{\zeta }=({\zeta }_1,{\zeta }_2,\cdots ,{\zeta }_M)$ is the orthonormal base, satisfying

$$({\zeta }_i,{\zeta }_j)={\delta }_{ij},i,j=1,2,\cdots ,M.$$

${\mathit{U}}_h^n={(U_{h1}^n,U_{h2}^n,\cdots ,U_{hM}^n)}^T$ and ${\mathit{W}}_h^n={(W_{h1}^n,W_{h2}^n,\cdots ,W_{hM}^n)}^T$ are the unknown coefficient vectors of CNMFE solutions. From the expression of the solutions $\{u_h^n,{\omega }_h^n\}$ of (13), Problem 2 can be represented equivalently in the following matrix model.

Problem 3. 

Find $\{{\mathit{U}}_h^n,{\mathit{W}}_h^n\}\in R^M\times R^M$ and $\{u_h^n,{\omega }_h^n\}\in V_h\times V_h(1⩽n⩽N)$ satisfying

$$\begin{array}{c}\hfill \left\{\begin{array}{c}{\displaystyle {\overline{\partial }}_t{\mathit{U}}_h^n+\gamma \mathit{B}{\mathit{W}}_h^{n-\frac{1}{2}}+{\mathit{W}}_h^{n-\frac{1}{2}}=\mathit{F}({\widehat{\widehat{\mathit{U}}}}_h^{n-\frac{1}{2}}),}\hfill \\ {\mathit{W}}_h^{n-{\displaystyle \frac{1}{2}}}-{\mathit{BU}}_h^{n-{\displaystyle \frac{1}{2}}}=0,\hfill \\ {\displaystyle u_h^n=\sum _{j=1}^M{U}_{hj}^n{\zeta }_j={\mathit{U}}_h^n·\mathit{\zeta },{\omega }_h^n=\sum _{j=1}^M{W}_{hj}^n{\zeta }_j={\mathit{W}}_h^n·\mathit{\zeta },}\hfill \end{array}\right.\end{array}$$

(14)

where

$$\mathit{B}={(b_{ij})}_{1⩽i,j⩽M},b_{ij}=(\nabla {\zeta }_j,\nabla {\zeta }_i),$$

and

$$\begin{array}{cc}\hfill {\displaystyle \mathit{F}({\widehat{\widehat{\mathit{U}}}}_h^{n-\frac{1}{2}})}& {\displaystyle =-\left(\frac{3}{2}\mathit{F}({\mathit{U}}_h^{n-1})-\frac{1}{2}\mathit{F}({\mathit{U}}_h^{n-2})\right)}\hfill \\ \hfill & {\displaystyle ={\left(-\left(\frac{3}{2}\left(f\left(\sum _{j=1}^M{U}_{hj}^{n-1}{\zeta }_j\right),{\zeta }_i\right)-\frac{1}{2}\left(f\left(\sum _{j=1}^M{U}_{hj}^{n-2}{\zeta }_j\right),{\zeta }_i\right)\right)\right)}_{1⩽i⩽M}.}\hfill \end{array}$$

Theorem 1. 

Problem 3 has the unique solutions $\{u_h^n,{\omega }_h^n\}\in V_h\times V_h$.

Proof of Theorem 1. 

Problem 3 can be written as

$$\left(\begin{array}{cc}\mathit{I}& {\displaystyle \frac{\gamma \mathsf{\Delta }t}{2}}\mathit{B}+{\displaystyle \frac{\mathsf{\Delta }t}{2}}\mathit{I}\\ -\mathit{B}& \mathit{I}\end{array}\right)\left(\begin{array}{c}{\mathit{U}}_h^n\\ {\mathit{W}}_h^n\end{array}\right)=\left(\begin{array}{cc}\mathit{I}& -{\displaystyle \frac{\gamma \mathsf{\Delta }t}{2}}\mathit{B}-{\displaystyle \frac{\mathsf{\Delta }t}{2}}\mathit{I}\\ \mathit{B}& -\mathit{I}\end{array}\right)\left(\begin{array}{c}{\mathit{U}}_h^{n-1}\\ {\mathit{W}}_h^{n-1}\end{array}\right)-\mathsf{\Delta }t\left(\begin{array}{c}\mathit{F}({\widehat{\widehat{\mathit{U}}}}_h^{n-{\displaystyle \frac{1}{2}}})\\ \tilde{\mathit{O}}\end{array}\right),$$

(15)

where $\mathit{I}$ stands for the $M\times M$ unit matrix, and $\tilde{\mathit{O}}$ denotes zero column vector of $M\times 1$.

From the definitions of matrices $\mathit{I}$ and $\mathit{B}$, we know that they are positive definite and thus invertible. From (15), we can conclude that the coefficient matrix is invertible as well. Hence, there exists a unique set of solutions $(u_h^n,{\omega }_h^n)\in V_h\times V_h$ for Problem 3. □

In order to prove the stability of the CNMFE solutions, we need to introduce the properties of $\mathit{B}$ in Problem 3.

Lemma 1 

([44] (Lemma 1.19)). Matrix $\mathit{B}$ satisfies the following inequality:

$${\parallel \mathit{B}\parallel }_{\infty }⩽C,{\parallel {\mathit{B}}^{-1}\parallel }_{\infty }⩽C.$$

(16)

Theorem 2. 

The CNMFE solutions $\{u_h^n,{\omega }_h^n\}$ are unconditionally stable.

Proof of Theorem 2. 

We represent (14) as

$$\left\{\begin{array}{c}{\displaystyle {\mathit{B}}^{-1}{\overline{\partial }}_t{\mathit{U}}_h^n+\gamma {\mathit{W}}_h^{n-\frac{1}{2}}+{\mathit{B}}^{-1}{\mathit{W}}_h^{n-\frac{1}{2}}={\mathit{B}}^{-1}\mathit{F}({\widehat{\widehat{\mathit{U}}}}_h^{n-\frac{1}{2}}),}\hfill \\ {\mathit{W}}_h^{n-{\displaystyle \frac{1}{2}}}={\mathit{BU}}_h^{n-{\displaystyle \frac{1}{2}}}.\hfill \end{array}\right.$$

(17)

Substituting the second formula of (17) into the first formula, we obtain

$${\displaystyle {\mathit{B}}^{-1}{\overline{\partial }}_t{\mathit{U}}_h^n+\gamma {\mathit{BU}}_h^{n-\frac{1}{2}}+{\mathit{U}}_h^{n-\frac{1}{2}}={\mathit{B}}^{-1}\mathit{F}({\widehat{\widehat{\mathit{U}}}}_h^{n-\frac{1}{2}}).}$$

(18)

Taking the inner product of (18) and ${\overline{\partial }}_t{\mathit{U}}_h^n$, we have

$${\displaystyle ({\mathit{B}}^{-1}{\overline{\partial }}_t{\mathit{U}}_h^n,{\overline{\partial }}_t{\mathit{U}}_h^n)+\gamma ({\mathit{BU}}_h^{n-\frac{1}{2}},{\overline{\partial }}_t{\mathit{U}}_h^n)+({\mathit{U}}_h^{n-\frac{1}{2}},{\overline{\partial }}_t{\mathit{U}}_h^n)=({\mathit{B}}^{-1}\mathit{F}({\widehat{\widehat{\mathit{U}}}}_h^{n-\frac{1}{2}}),{\overline{\partial }}_t{\mathit{U}}_h^n).}$$

(19)

The left side of (19) is

$$\begin{array}{cc}\hfill & {\displaystyle ({\mathit{B}}^{-1}{\overline{\partial }}_t{\mathit{U}}_h^n,{\overline{\partial }}_t{\mathit{U}}_h^n)+\gamma ({\mathit{BU}}_h^{n-\frac{1}{2}},{\overline{\partial }}_t{\mathit{U}}_h^n)+({\mathit{U}}_h^{n-\frac{1}{2}},{\overline{\partial }}_t{\mathit{U}}_h^n)}\hfill \\ =\hfill & {\displaystyle \parallel {({\mathit{B}}^{-1})}^{\frac{1}{2}}{\overline{\partial }}_t{\mathit{U}}_h^n{\parallel }^2+\frac{\gamma }{2\mathsf{\Delta }t}(\parallel {\mathit{B}}^{\frac{1}{2}}{\mathit{U}}_h^n{\parallel }^2-\parallel {\mathit{B}}^{\frac{1}{2}}{\mathit{U}}_h^{n-1}{\parallel }^2)+\frac{1}{2\mathsf{\Delta }t}(\parallel {\mathit{U}}_h^n{\parallel }^2-\parallel {\mathit{U}}_h^{n-1}{\parallel }^2),}\hfill \end{array}$$

(20)

and the right side of (19) is

$${\displaystyle ({\mathit{B}}^{-1}\mathit{F}({\widehat{\widehat{\mathit{U}}}}_h^{n-\frac{1}{2}}),{\overline{\partial }}_t{\mathit{U}}_h^n)⩽C\parallel {({\mathit{B}}^{-1})}^{\frac{1}{2}}{\parallel }_{\infty }^2\parallel \mathit{F}({\widehat{\widehat{\mathit{U}}}}_h^{n-\frac{1}{2}}){\parallel }^2+{\parallel {({\mathit{B}}^{-1})}^{\frac{1}{2}}{\overline{\partial }}_t{\mathit{U}}_h^n\parallel }^2.}$$

(21)

Combining Lemma 1 with the global Lipschitz-continuity of $f(u)$, the first term of (21) can be estimated as

$$\begin{array}{cc}\hfill & {\displaystyle \parallel {({\mathit{B}}^{-1})}^{\frac{1}{2}}{\parallel }_{\infty }^2{\parallel \mathit{F}({\widehat{\widehat{\mathit{U}}}}_h^{n-\frac{1}{2}})\parallel }^2}\hfill \\ ⩽\hfill & {\displaystyle C\parallel \frac{3}{2}\mathit{F}({\mathit{U}}_h^{n-1})-\frac{1}{2}\mathit{F}({\mathit{U}}_h^{n-2}){\parallel }^2}\hfill \\ ⩽\hfill & {\displaystyle C\left(\parallel \frac{1}{2}\mathit{F}({\mathit{U}}_h^{n-1})-\frac{1}{2}\mathit{F}({\mathit{U}}_h^{n-2}){\parallel }^2+\parallel \mathit{F}({\mathit{U}}_h^{n-1})-\mathit{F}({\mathit{U}}_h^0){\parallel }^2+{\parallel \mathit{F}({\mathit{U}}_h^0)\parallel }^2\right)}\hfill \\ ⩽\hfill & {\displaystyle C(\parallel {\mathit{U}}_h^{n-1}-{\mathit{U}}_h^{n-2}{\parallel }^2+\parallel {\mathit{U}}_h^{n-1}-{\mathit{U}}_h^0{\parallel }^2+\parallel \mathit{F}({\mathit{U}}_h^0){\parallel }^2)}\hfill \\ ⩽\hfill & {\displaystyle C(\parallel {\mathit{U}}_h^0{\parallel }^2+\parallel \mathit{F}({\mathit{U}}_h^0){\parallel }^2+\parallel {\mathit{U}}_h^{n-1}{\parallel }^2+\parallel {\mathit{U}}_h^{n-2}{\parallel }^2)}\hfill \\ ⩽\hfill & {\displaystyle C+C(\parallel {\mathit{U}}_h^{n-1}{\parallel }^2+\parallel {\mathit{U}}_h^{n-2}{\parallel }^2),2⩽n⩽N,}\hfill \end{array}$$

(22)

where $\parallel {\mathit{U}}_h^0\parallel$ and $\parallel \mathit{F}({\mathit{U}}_h^0)\parallel$ are bounded. Combining (20), (21) and (22), we have

$$\begin{array}{c}{\displaystyle \frac{\gamma }{2\mathsf{\Delta }t}(\parallel {\mathit{B}}^{\frac{1}{2}}{\mathit{U}}_h^n{\parallel }^2-\parallel {\mathit{B}}^{\frac{1}{2}}{\mathit{U}}_h^{n-1}{\parallel }^2)+\frac{1}{2\mathsf{\Delta }t}(\parallel {\mathit{U}}_h^n{\parallel }^2-\parallel {\mathit{U}}_h^{n-1}{\parallel }^2)⩽C+C(\parallel {\mathit{U}}_h^{n-1}{\parallel }^2+\parallel {\mathit{U}}_h^{n-2}{\parallel }^2).}\hfill \end{array}$$

(23)

Multiplying both sides of the inequality of (23) by $2\mathsf{\Delta }t$ and summing from 2 to $n(n⩽N)$, we have

$$\begin{array}{cc}\parallel {\mathit{U}}_h^n{\parallel }^2\hfill & {\displaystyle ⩽\parallel {\mathit{U}}_h^1{\parallel }^2+\gamma \parallel {\mathit{B}}^{\frac{1}{2}}{\mathit{U}}_h^1{\parallel }^2+2\mathsf{\Delta }t\sum _{i=2}^{n}C+C\mathsf{\Delta }t\sum _{i=0}^{n-1}{\parallel {\mathit{U}}_h^i\parallel }^2}\hfill \\ \hfill & {\displaystyle ⩽(1+\gamma \parallel {\mathit{B}}^{\frac{1}{2}}{\parallel }_{\infty })\parallel {\mathit{U}}_h^1{\parallel }^2+CT+C\mathsf{\Delta }t\sum _{i=0}^{n-1}{\parallel {\mathit{U}}_h^i\parallel }^2,}\hfill \end{array}$$

(24)

noting that $\parallel {\mathit{U}}_h^1\parallel ⩽C$, then

$${\displaystyle \parallel {\mathit{U}}_h^n{\parallel }^2⩽C+C\mathsf{\Delta }t\sum _{i=0}^{n-1}{\parallel {\mathit{U}}_h^i\parallel }^2.}$$

(25)

Applying the Gronwall Lemma to (25), we obtain

$$\parallel {\mathit{U}}_h^n{\parallel }^2⩽C{e}^{Cn\mathsf{\Delta }t}⩽C,1⩽n⩽N.$$

(26)

And because

$$\parallel {\mathit{W}}_h^n\parallel =\parallel \mathit{B}{\mathit{U}}_h^n{\parallel ⩽\parallel \mathit{B}\parallel }_{\infty }\parallel {\mathit{U}}_h^n\parallel ⩽C\parallel {\mathit{U}}_h^n\parallel ⩽C,1⩽n⩽N.$$

(27)

From (26) and (27), we have

$$\parallel {\mathit{U}}_h^n\parallel +\parallel {\mathit{W}}_h^n\parallel ⩽C,1⩽n⩽N.$$

(28)

Due to $\parallel \zeta \parallel ⩽C$, we can easily obtain

$$\parallel u_h^n\parallel +\parallel {\omega }_h^n\parallel =\parallel {\mathit{U}}_h^n·\mathit{\zeta }\parallel +\parallel {\mathit{W}}_h^n·\mathit{\zeta }\parallel ⩽C\parallel {\mathit{U}}_h^n\parallel \parallel \mathit{\zeta }\parallel +C\parallel {\mathit{W}}_h^n\parallel \parallel \mathit{\zeta }\parallel ⩽C,1⩽n⩽N.$$

(29)

(28) and (29) indicate that the CNMFE solution coefficient vectors $\{{\mathit{U}}_h,{\mathit{W}}_h\}$ are bounded, so the CNMFE solutions $\{u_h,{\omega }_h\}$ are unconditionally stable. □

In order to analyze the convergence of the linearized CNMFE scheme, we first need to establish the projection operator $R_h$.

Lemma 2. 

Let $R_h:V\to V_h$ satisfy the following relation:

$$(\nabla (\varphi -R_h\varphi ),\nabla \chi )=0,\forall \chi \in V_h,$$

(30)

and

$$\parallel \varphi -R_h{\varphi \parallel }_{L^2(\Omega )}+h\parallel \varphi -R_h{\varphi \parallel }_{H^1(\Omega )}⩽C{h}^2{\parallel \varphi \parallel }_{H^2(\Omega )},$$

(31)

$$\parallel {\varphi }_t-R_h{\varphi }_t{\parallel }_{L^2(\Omega )}⩽C{h}^2{(\parallel \varphi \parallel }_{H^2(\Omega )}+\parallel {\varphi }_t{\parallel }_{H^2(\Omega )}).$$

(32)

In order to facilitate theoretical analysis, the errors can be decomposed into

$$u(t_n)-u_h^n=u(t_n)-R_{h}u(t_n)+R_{h}u(t_n)-u_h^n={\tau }^n+{\theta }^n,$$

(33)

$$\omega (t_n)-{\omega }_h^n=\omega (t_n)-R_h\omega (t_n)+R_h\omega (t_n)-{\omega }_h^n={\pi }^n+{\kappa }^n.$$

(34)

Subtracting (11) from (7) and using (30) at $t=t_{n-{\displaystyle \frac{1}{2}}}$, we obtain the following error equations.

$$\begin{array}{c}{\displaystyle \left(\frac{{\theta }^n-{\theta }^{n-1}}{\mathsf{\Delta }t},v_h\right)+\gamma (\nabla {\kappa }^{n-\frac{1}{2}},\nabla v_h)+({\kappa }^{n-\frac{1}{2}},v_h)+(f({\widehat{\widehat{u}}}^{n-\frac{1}{2}})-f({\widehat{\widehat{u}}}_h^{n-\frac{1}{2}}),v_h)}\hfill \\ {\displaystyle =-\left(\frac{{\tau }^n-{\tau }^{n-1}}{\mathsf{\Delta }t},v_h\right)-({\pi }^{n-\frac{1}{2}},v_h)+(R_1^{n-\frac{1}{2}},v_h)+(R_2^{n-\frac{1}{2}},v_h),\forall v_h\in V_h,}\hfill \end{array}$$

(35)

$$({\kappa }^{n-{\displaystyle \frac{1}{2}}},g_h)-(\nabla {\theta }^{n-{\displaystyle \frac{1}{2}}},\nabla g_h)=-({\pi }^{n-{\displaystyle \frac{1}{2}}},g_h),\forall g_h\in V_h.$$

(36)

In order to obtain the error estimates, we introduce the following lemma, which is easily obtained by Taylor expansion.

Lemma 3. 

$R_1^{n-{\displaystyle \frac{1}{2}}}$ and $R_2^{n-{\displaystyle \frac{1}{2}}}$ satisfy the following estimates

$$\parallel R_1^{n-{\displaystyle \frac{1}{2}}}{\parallel }^2⩽C\mathsf{\Delta }{t}^3{\parallel u_{ttt}\parallel }_{L^{\infty }(L^2)},$$

(37)

$$\parallel R_2^{n-{\displaystyle \frac{1}{2}}}{\parallel }^2⩽C(u)\mathsf{\Delta }{t}^3{\parallel u_{ttt}\parallel }_{L^{\infty }(L^2)}.$$

(38)

Based on Lemmas 2 and 3, the following theorems for Crank–Nicolson fully discrete error estimates is obtained.

Theorem 3. 

With $u_h^1=R_{h}u(t_1)$, assume that the solution’s regular properties for (4) satisfy $u_t\in L^2(H^2)$, $u_{ttt}\in L^{\infty }(L^2)$, $u,\omega \in L^{\infty }(H^2)$. Then, there exists a positive constant C independent of h and $\mathsf{\Delta }t$ such that

$$\begin{array}{c}\parallel u(t_J)-u_h^J\parallel ⩽C(h^2\parallel |\ast |\parallel +{(\mathsf{\Delta }t)}^2\parallel u_{ttt}{\parallel }_{L^{\infty }(L^2)}),\hfill \end{array}$$

(39)

where $\parallel |\ast |\parallel \triangleq {\parallel u\parallel }_{L^{\infty }(H^2)}+{\parallel \omega \parallel }_{L^{\infty }(H^2)}+{\parallel u_t\parallel }_{L^2(H^2)}$.

Proof of Theorem 3. 

Taking $v_h={\theta }^{n-{\displaystyle \frac{1}{2}}}$ and $g_h={\kappa }^{n-{\displaystyle \frac{1}{2}}}$ in (35) and (36), respectively, we obtain

$$\begin{array}{c}{\displaystyle \frac{\parallel {\theta }^n{\parallel }^2-{\parallel {\theta }^{n-1}\parallel }^2}{2\mathsf{\Delta }t}+\gamma (\nabla {\kappa }^{n-\frac{1}{2}},\nabla {\theta }^{n-\frac{1}{2}})+({\kappa }^{n-\frac{1}{2}},{\theta }^{n-\frac{1}{2}})+(f({\widehat{\widehat{u}}}^{n-\frac{1}{2}})-f({\widehat{\widehat{u}}}_h^{n-\frac{1}{2}}),{\theta }^{n-\frac{1}{2}})}\hfill \\ {\displaystyle =-\left(\frac{{\tau }^n-{\tau }^{n-1}}{\mathsf{\Delta }t},{\theta }^{n-\frac{1}{2}}\right)-({\pi }^{n-\frac{1}{2}},{\theta }^{n-\frac{1}{2}})+(R_1^{n-\frac{1}{2}},{\theta }^{n-\frac{1}{2}})+(R_2^{n-\frac{1}{2}},{\theta }^{n-\frac{1}{2}}),}\hfill \end{array}$$

(40)

$$\parallel {\kappa }^{n-{\displaystyle \frac{1}{2}}}{\parallel }^2-(\nabla {\theta }^{n-{\displaystyle \frac{1}{2}}},\nabla {\kappa }^{n-{\displaystyle \frac{1}{2}}})=-({\pi }^{n-{\displaystyle \frac{1}{2}}},{\kappa }^{n-{\displaystyle \frac{1}{2}}}).$$

(41)

Multiplying (41) by $\gamma$ and adding it to (40) and using Cauchy–Schwarz inequality and Young inequality, we obtain

$$\begin{array}{cc}\hfill & {\displaystyle \gamma \parallel {\kappa }^{n-\frac{1}{2}}{\parallel }^2+\frac{1}{2\mathsf{\Delta }t}(\parallel {\theta }^n{\parallel }^2-\parallel {\theta }^{n-1}{\parallel }^2)}\hfill \\ =\hfill & {\displaystyle -({\kappa }^{n-\frac{1}{2}},{\theta }^{n-\frac{1}{2}})-(f({\widehat{\widehat{u}}}^{n-\frac{1}{2}})-f({\widehat{\widehat{u}}}_h^{n-\frac{1}{2}}),{\theta }^{n-\frac{1}{2}})-\left(\frac{{\tau }^n-{\tau }^{n-1}}{\mathsf{\Delta }t},{\theta }^{n-\frac{1}{2}}\right)-({\pi }^{n-\frac{1}{2}},{\theta }^{n-\frac{1}{2}})}\hfill \\ \hfill & +(R_1^{n-{\displaystyle \frac{1}{2}}},{\theta }^{n-{\displaystyle \frac{1}{2}}})+(R_2^{n-{\displaystyle \frac{1}{2}}},{\theta }^{n-{\displaystyle \frac{1}{2}}})-\gamma ({\pi }^{n-{\displaystyle \frac{1}{2}}},{\kappa }^{n-{\displaystyle \frac{1}{2}}})\hfill \\ ⩽\hfill & {\displaystyle \left(\parallel \frac{{\tau }^n-{\tau }^{n-1}}{\mathsf{\Delta }t}\parallel +\parallel R_1^{n-\frac{1}{2}}\parallel +\parallel R_2^{n-\frac{1}{2}}\parallel \right)\parallel {\theta }^{n-\frac{1}{2}}\parallel +\parallel {\kappa }^{n-\frac{1}{2}}\parallel \parallel {\theta }^{n-\frac{1}{2}}\parallel +\gamma \parallel {\pi }^{n-\frac{1}{2}}\parallel \parallel {\kappa }^{n-\frac{1}{2}}\parallel }\hfill \\ \hfill & +(\parallel f({\widehat{\widehat{u}}}^{n-{\displaystyle \frac{1}{2}}})-f({\widehat{\widehat{u}}}_h^{n-{\displaystyle \frac{1}{2}}})\parallel +\parallel {\pi }^{n-{\displaystyle \frac{1}{2}}}\parallel )\parallel {\theta }^{n-{\displaystyle \frac{1}{2}}}\parallel \hfill \\ ⩽\hfill & {\displaystyle C\left(\parallel \frac{{\tau }^n-{\tau }^{n-1}}{\mathsf{\Delta }t}{\parallel }^2+\parallel R_1^{n-\frac{1}{2}}{\parallel }^2+\parallel R_2^{n-\frac{1}{2}}{\parallel }^2+{\parallel {\pi }^{n-\frac{1}{2}}\parallel }^2\right)+\frac{\gamma }{2}{\parallel {\kappa }^{n-\frac{1}{2}}\parallel }^2}\hfill \\ \hfill & +C\parallel f({\widehat{\widehat{u}}}^{n-{\displaystyle \frac{1}{2}}})-f({\widehat{\widehat{u}}}_h^{n-{\displaystyle \frac{1}{2}}}){\parallel }^2+C{\parallel {\theta }^{n-{\displaystyle \frac{1}{2}}}\parallel }^2.\hfill \end{array}$$

(42)

For the nonlinear term $\parallel f({\widehat{\widehat{u}}}^{n-{\displaystyle \frac{1}{2}}})-f({\widehat{\widehat{u}}}_h^{n-{\displaystyle \frac{1}{2}}}){\parallel }^2$, using the Young inequality, we have

$$\begin{array}{cc}\hfill & \parallel f({\widehat{\widehat{u}}}^{n-{\displaystyle \frac{1}{2}}})-f({\widehat{\widehat{u}}}_h^{n-{\displaystyle \frac{1}{2}}}){\parallel }^2\hfill \\ =\hfill & {\displaystyle \parallel \frac{3}{2}f(u^{n-1})-\frac{1}{2}f(u^{n-2})-\frac{3}{2}f(u_h^{n-1})+\frac{1}{2}f(u_h^{n-2}){\parallel }^2}\hfill \\ =\hfill & {\displaystyle \parallel \frac{3}{2}{(u^{n-1})}^3-\frac{3}{2}{u}^{n-1}-\frac{1}{2}{(u^{n-2})}^3+\frac{1}{2}{u}^{n-2}-\frac{3}{2}{(u_h^{n-1})}^3+\frac{3}{2}{u}_h^{n-1}+\frac{1}{2}{(u_h^{n-2})}^3-\frac{1}{2}{u}_h^{n-2}{\parallel }^2}\hfill \\ =\hfill & {\displaystyle \parallel \frac{3}{2}[{(u^{n-1})}^3-{(u_h^{n-1})}^3]-\frac{1}{2}[{(u^{n-2})}^3-{(u_h^{n-2})}^3]-\frac{3}{2}(u^{n-1}-u_h^{n-1})+\frac{1}{2}(u^{n-2}-u_h^{n-2}){\parallel }^2}\hfill \\ ⩽\hfill & {\displaystyle \parallel \frac{3}{2}(u^{n-1}-u_h^{n-1})[{(u^{n-1})}^2+u^{n-1}{u}_h^{n-1}+{(u_h^{n-1})}^2]{\parallel }^2}\hfill \\ \hfill & {\displaystyle +\parallel \frac{1}{2}(u^{n-2}-u_h^{n-2})[{(u^{n-2})}^2+u^{n-2}{u}_h^{n-2}+{(u_h^{n-2})}^2]{\parallel }^2}\hfill \\ \hfill & {\displaystyle +\parallel \frac{3}{2}(u^{n-1}-u_h^{n-1}){\parallel }^2+\parallel \frac{1}{2}(u^{n-2}-u_h^{n-2}){\parallel }^2}\hfill \\ =\hfill & {\displaystyle \frac{9}{4}(C_u^2+1)\parallel {\theta }^{n-1}+{\tau }^{n-1}{\parallel }^2+\frac{1}{4}(C_u^2+1){\parallel {\theta }^{n-2}+{\tau }^{n-2}\parallel }^2}\hfill \\ ⩽\hfill & C(\parallel {\tau }^{n-1}{\parallel }^2+\parallel {\tau }^{n-2}{\parallel }^2+\parallel {\theta }^{n-1}{\parallel }^2+\parallel {\theta }^{n-2}{\parallel }^2).\hfill \end{array}$$

(43)

Noting that

$$\parallel {\displaystyle \frac{{\tau }^n-{\tau }^{n-1}}{\mathsf{\Delta }t}}{\parallel }^2⩽{\displaystyle \frac{1}{\mathsf{\Delta }t}}{\int }_{t_{n-1}}^{t_n}{\parallel {\tau }_t(s)\parallel }^{2}ds.$$

(44)

Substitute (43) and (44) into (42) to obtain

$$\begin{array}{cc}\hfill & {\displaystyle \frac{\gamma }{2}\parallel {\kappa }^{n-\frac{1}{2}}{\parallel }^2+\frac{1}{2\mathsf{\Delta }t}(\parallel {\theta }^n{\parallel }^2-\parallel {\theta }^{n-1}{\parallel }^2)}\hfill \\ ⩽\hfill & {\displaystyle C\left(\frac{1}{\mathsf{\Delta }t}{\int }_{t_{n-1}}^{t_n}\parallel {\tau }_t{(s)\parallel }^{2}ds+\parallel R_1^{n-\frac{1}{2}}{\parallel }^2+\parallel R_2^{n-\frac{1}{2}}{\parallel }^2+{\parallel {\pi }^{n-\frac{1}{2}}\parallel }^2\right)}\hfill \\ \hfill & +C(\parallel {\tau }^{n-1}{\parallel }^2+\parallel {\tau }^{n-2}{\parallel }^2+\parallel {\theta }^n{\parallel }^2+\parallel {\theta }^{n-1}{\parallel }^2+\parallel {\theta }^{n-2}{\parallel }^2).\hfill \end{array}$$

(45)

Summing from $n=2,\cdots ,J$, the resulting inequality becomes

$$\begin{array}{cc}\hfill & {\displaystyle \gamma \mathsf{\Delta }t\sum _{n=2}^J\parallel {\kappa }^{n-\frac{1}{2}}{\parallel }^2+(1-C\mathsf{\Delta }t){\parallel {\theta }^J\parallel }^2}\hfill \\ ⩽\hfill & {\displaystyle \parallel {\theta }^1{\parallel }^2+C\mathsf{\Delta }t\sum _{n=2}^J(\parallel R_1^{n-\frac{1}{2}}{\parallel }^2+\parallel R_2^{n-\frac{1}{2}}{\parallel }^2+\parallel {\pi }^{n-\frac{1}{2}}{\parallel }^2)}\hfill \\ \hfill & {\displaystyle +C\mathsf{\Delta }t\sum _{n=0}^{J-1}(\parallel {\tau }^n{\parallel }^2+\parallel {\theta }^n{\parallel }^2)+C{\int }_0^{t_J}{\parallel {\tau }_t\parallel }^{2}ds.}\hfill \end{array}$$

(46)

Choosing $\mathsf{\Delta }{t}_0$, when $0<\mathsf{\Delta }t<\mathsf{\Delta }{t}_0$, $1-C\mathsf{\Delta }t>0$, and using the discrete Gronwall Lemma and Lemma 3, we obtain

$$\begin{array}{cc}\hfill & {\displaystyle \gamma \mathsf{\Delta }t\sum _{n=2}^J\parallel {\kappa }^{n-\frac{1}{2}}{\parallel }^2+\frac{1}{2}{\parallel {\theta }^J\parallel }^2}\hfill \\ ⩽\hfill & {\displaystyle \parallel {\theta }^1{\parallel }^2+C\mathsf{\Delta }t\sum _{n=0}^{J-1}\parallel {\tau }^n{\parallel }^2+C\mathsf{\Delta }t\sum _{n=2}^J(\parallel R_1^{n-\frac{1}{2}}{\parallel }^2+\parallel R_2^{n-\frac{1}{2}}{\parallel }^2+\parallel {\pi }^{n-\frac{1}{2}}{\parallel }^2)+C{\int }_0^{t_J}{\parallel {\tau }_t\parallel }^{2}ds}\hfill \\ ⩽\hfill & {\displaystyle \parallel {\theta }^1{\parallel }^2+C(\parallel {\tau }^n{\parallel }_{L^{\infty }(L^2)}^2+\parallel {\pi }^{n-\frac{1}{2}}{\parallel }_{L^{\infty }(L^2)}^2+{\int }_0^{t_J}\parallel {\tau }_t{\parallel }^2{ds)+C(\mathsf{\Delta }t)}^4{\parallel u_{ttt}\parallel }_{L^{\infty }(L^2)}^2.}\hfill \end{array}$$

(47)

Combining Lemma 2, (47) and the triangle inequality, the proof of (39) is accomplished. □

Theorem 4. 

With $u_h^1=R_{h}u(t_1)$ and ${\omega }_h^1=R_h\omega (t_1)$, assume that the solution’s regular properties for (4) satisfy $u_t,{\omega }_t\in L^2(H^2)$, $u_{ttt}\in L^{\infty }(L^2)$, $u,\omega \in L^{\infty }(H^2)$. Then, there exists a positive constant C independent of h and $\mathsf{\Delta }t$ such that

$$\begin{array}{c}\parallel \omega (t_J)-{\omega }_h^J\parallel ⩽C(h^2\parallel |★|\parallel +{(\mathsf{\Delta }t)}^2\parallel u_{ttt}{\parallel }_{L^{\infty }(L^2)}),\hfill \end{array}$$

(48)

where $\parallel |★|\parallel \triangleq {\parallel u\parallel }_{L^{\infty }(H^2)}+{\parallel \omega \parallel }_{L^{\infty }(H^2)}+\parallel u_t{\parallel }_{L^2(H^2)}+{\parallel {\omega }_t\parallel }_{L^2(H^2)}$.

Proof of Theorem 4. 

From (36), we can obtain

$${\displaystyle \left(\frac{{\kappa }^n-{\kappa }^{n-1}}{\mathsf{\Delta }t},g_h\right)-\left(\frac{\nabla {\theta }^n-\nabla {\theta }^{n-1}}{\mathsf{\Delta }t},\nabla g_h\right)=-\left(\frac{{\pi }^n-{\pi }^{n-1}}{\mathsf{\Delta }t},g_h\right),\forall g_h\in V_h.}$$

(49)

Taking ${\displaystyle v_h=\frac{{\theta }^n-{\theta }^{n-1}}{\mathsf{\Delta }t}}$ and $g_h={\kappa }^{n-{\displaystyle \frac{1}{2}}}$ in (35) and (49), respectively, we obtain

$$\begin{array}{cc}\hfill & {\displaystyle \parallel \frac{{\theta }^n-{\theta }^{n-1}}{\mathsf{\Delta }t}{\parallel }^2+\gamma \left(\nabla {\kappa }^{n-\frac{1}{2}},\frac{{\theta }^n-{\theta }^{n-1}}{\mathsf{\Delta }t}\right)+\left({\kappa }^{n-\frac{1}{2}},\frac{{\theta }^n-{\theta }^{n-1}}{\mathsf{\Delta }t}\right)}\hfill \\ \hfill & {\displaystyle +\left(f({\widehat{\widehat{u}}}^{n-\frac{1}{2}})-f({\widehat{\widehat{u}}}_h^{n-\frac{1}{2}}),\frac{{\theta }^n-{\theta }^{n-1}}{\mathsf{\Delta }t}\right)}\hfill \\ =\hfill & {\displaystyle -\left(\frac{{\tau }^n-{\tau }^{n-1}}{\mathsf{\Delta }t},\frac{{\theta }^n-{\theta }^{n-1}}{\mathsf{\Delta }t}\right)-\left({\pi }^{n-\frac{1}{2}},\frac{{\theta }^n-{\theta }^{n-1}}{\mathsf{\Delta }t}\right)}\hfill \\ \hfill & {\displaystyle +\left(R_1^{n-\frac{1}{2}},\frac{{\theta }^n-{\theta }^{n-1}}{\mathsf{\Delta }t}\right)+\left(R_2^{n-\frac{1}{2}},\frac{{\theta }^n-{\theta }^{n-1}}{\mathsf{\Delta }t}\right),}\hfill \end{array}$$

(50)

$${\displaystyle \frac{\parallel {\kappa }^n{\parallel }^2-{\parallel {\kappa }^{n-1}\parallel }^2}{2\mathsf{\Delta }t}-\left(\frac{\nabla {\theta }^n-\nabla {\theta }^{n-1}}{\mathsf{\Delta }t},\nabla {\kappa }^{n-\frac{1}{2}}\right)=-\left(\frac{{\pi }^n-{\pi }^{n-1}}{\mathsf{\Delta }t},{\kappa }^{n-\frac{1}{2}}\right).}$$

(51)

Multiplying (51) by $\gamma$ and adding it to (50) and using the Cauchy–Schwarz inequality and Young inequality, we have

$$\begin{array}{cc}\hfill & {\displaystyle \frac{\gamma }{2\mathsf{\Delta }t}(\parallel {\kappa }^n{\parallel }^2-\parallel {\kappa }^{n-1}{\parallel }^2)+\parallel \frac{{\theta }^n-{\theta }^{n-1}}{\mathsf{\Delta }t}{\parallel }^2}\hfill \\ =\hfill & {\displaystyle -\left({\kappa }^{n-\frac{1}{2}},\frac{{\theta }^n-{\theta }^{n-1}}{\mathsf{\Delta }t}\right)-\left(f({\widehat{\widehat{u}}}^{n-\frac{1}{2}})-f({\widehat{\widehat{u}}}_h^{n-\frac{1}{2}}),\frac{{\theta }^n-{\theta }^{n-1}}{\mathsf{\Delta }t}\right)}\hfill \\ \hfill & {\displaystyle -\left(\frac{{\tau }^n-{\tau }^{n-1}}{\mathsf{\Delta }t},\frac{{\theta }^n-{\theta }^{n-1}}{\mathsf{\Delta }t}\right)-\left({\pi }^{n-\frac{1}{2}},\frac{{\theta }^n-{\theta }^{n-1}}{\mathsf{\Delta }t}\right)}\hfill \\ \hfill & {\displaystyle +\left(R_1^{n-\frac{1}{2}},\frac{{\theta }^n-{\theta }^{n-1}}{\mathsf{\Delta }t}\right)+\left(R_2^{n-\frac{1}{2}},\frac{{\theta }^n-{\theta }^{n-1}}{\mathsf{\Delta }t}\right)-\gamma \left(\frac{{\pi }^n-{\pi }^{n-1}}{\mathsf{\Delta }t},{\kappa }^{n-\frac{1}{2}}\right)}\hfill \\ ⩽\hfill & {\displaystyle C(\parallel f({\widehat{\widehat{u}}}^{n-\frac{1}{2}})-f({\widehat{\widehat{u}}}_h^{n-\frac{1}{2}}){\parallel }^2+\parallel R_1^{n-\frac{1}{2}}{\parallel }^2+\parallel R_2^{n-\frac{1}{2}}{\parallel }^2+\parallel \frac{{\tau }^n-{\tau }^{n-1}}{\mathsf{\Delta }t}{\parallel }^2+\parallel \frac{{\pi }^n-{\pi }^{n-1}}{\mathsf{\Delta }t}{\parallel }^2)}\hfill \\ \hfill & {\displaystyle +C(\parallel {\pi }^{n-\frac{1}{2}}{\parallel }^2+\parallel {\kappa }^{n-\frac{1}{2}}{\parallel }^2)+\frac{1}{2}\parallel \frac{{\theta }^n-{\theta }^{n-1}}{\mathsf{\Delta }t}{\parallel }^2.}\hfill \end{array}$$

(52)

Similar to (44),

$$\parallel {\displaystyle \frac{{\pi }^n-{\pi }^{n-1}}{\mathsf{\Delta }t}}{\parallel }^2⩽{\displaystyle \frac{1}{\mathsf{\Delta }t}}{\int }_{t_{n-1}}^{t_n}{\parallel {\pi }_t(s)\parallel }^{2}ds.$$

(53)

Multiplying (52) by $2\mathsf{\Delta }t$, summing from $n=2,\cdots ,J$ and using (43), (44) and (53) to obtain

$$\begin{array}{cc}\hfill & {\displaystyle \gamma \parallel {\kappa }^J{\parallel }^2+\mathsf{\Delta }t\sum _{n=2}^J\parallel \frac{{\theta }^n-{\theta }^{n-1}}{\mathsf{\Delta }t}{\parallel }^2}\hfill \\ ⩽\hfill & {\displaystyle \gamma \parallel {\kappa }^1{\parallel }^2+C\mathsf{\Delta }t\sum _{n=2}^J(\parallel {\tau }^{n-1}{\parallel }^2+\parallel {\tau }^{n-2}{\parallel }^2+\parallel {\theta }^{n-1}{\parallel }^2+\parallel {\theta }^{n-2}{\parallel }^2+\parallel R_1^{n-\frac{1}{2}}{\parallel }^2+\parallel R_2^{n-\frac{1}{2}}{\parallel }^2)}\hfill \\ \hfill & {\displaystyle +C\mathsf{\Delta }t\sum _{n=2}^J\parallel {\pi }^{n-\frac{1}{2}}{\parallel }^2+C{\int }_0^{t_J}(\parallel {\tau }_t{(s)\parallel }^2+\parallel {\pi }_t{(s)\parallel }^2)ds+C\mathsf{\Delta }t\sum _{n=2}^J{\parallel {\kappa }^{n-\frac{1}{2}}\parallel }^2.}\hfill \end{array}$$

(54)

Using the Gronwall Lemma, we have

$$\begin{array}{cc}\hfill & {\displaystyle \gamma \parallel {\kappa }^J{\parallel }^2+\mathsf{\Delta }t\sum _{n=2}^J\parallel \frac{{\theta }^n-{\theta }^{n-1}}{\mathsf{\Delta }t}{\parallel }^2}\hfill \\ ⩽\hfill & {\displaystyle \gamma \parallel {\kappa }^1{\parallel }^2+C\mathsf{\Delta }t\sum _{n=0}^{J-1}(\parallel {\tau }^n{\parallel }^2+\parallel {\theta }^n{\parallel }^2)+C\mathsf{\Delta }t\sum _{n=2}^J(\parallel R_1^{n-\frac{1}{2}}{\parallel }^2+\parallel R_2^{n-\frac{1}{2}}{\parallel }^2+\parallel {\pi }^{n-\frac{1}{2}}{\parallel }^2)}\hfill \\ \hfill & {\displaystyle +C{\int }_0^{t_J}(\parallel {\tau }_t{(s)\parallel }^2+\parallel {\pi }_t(s){\parallel }^2)ds.}\hfill \end{array}$$

(55)

Substituting (47) and Lemma 3 into (55), we obtain

$$\begin{array}{cc}\hfill & {\displaystyle \gamma \parallel {\kappa }^J{\parallel }^2+\mathsf{\Delta }t\sum _{n=2}^J\parallel \frac{{\theta }^n-{\theta }^{n-1}}{\mathsf{\Delta }t}{\parallel }^2}\hfill \\ ⩽\hfill & {\displaystyle \gamma \parallel {\kappa }^1{\parallel }^2+C{\int }_0^{t_J}(\parallel {\tau }_t{(s)\parallel }^2+\parallel {\pi }_t{(s)\parallel }^2)ds+C(\parallel {\tau }^n{\parallel }_{L^{\infty }(L^2)}^2+\parallel {\pi }^{n-\frac{1}{2}}{\parallel }_{L^{\infty }(L^2)}^2)}\hfill \\ \hfill & {\displaystyle +C{(\mathsf{\Delta }t)}^4{\parallel u_{ttt}\parallel }_{L^{\infty }(L^2)}^2}\hfill \end{array}$$

(56)

Combining Lemma 2, (56) with the triangle inequality, we complete the proof. □

3. The Reduced-Dimension Method of the Crank–Nicolson Mixed Finite Element Solution Coefficient Vectors of the EFK Equation

3.1. Construction of POD Basis Vectors

Firstly, we need to compute the initial K-step solution coefficient vectors ${\{{\mathit{U}}_h^n,{\mathit{W}}_h^n\}}_{n=1}^K$ of Problem 3, which will yield two $M\times K$ snapshot matrices ${\mathit{A}}_1=({\mathit{U}}_h^1,{\mathit{U}}_h^2,\cdots ,{\mathit{U}}_h^K)$ and ${\mathit{A}}_2=({\mathit{W}}_h^1,{\mathit{W}}_h^2,\cdots ,{\mathit{W}}_h^K)$. Secondly, we need to compute the positive eigenvalues and arrange them in descending order ${\mu }_{i,1}⩾{\mu }_{i,2}⩾\cdots ⩾{\mu }_{i,r_i}>0(r_i=\mathrm{rank}({\mathit{A}}_i))$ of ${\mathit{A}}_i{\mathit{A}}_i^T(i=1,2)$, and the associated eigenvectors to form the matrix ${\tilde{\mathbf{\Phi }}}_i=({\mathit{\phi }}_{i,1},{\mathit{\phi }}_{i,2},\cdots ,{\mathit{\phi }}_{i,r_i})\in R^{M\times r_i}$. Finally, choosing the first d column vectors of ${\tilde{\mathbf{\Phi }}}_i$ as a set of POD bases ${\mathbf{\Phi }}_i=({\mathit{\phi }}_{i,1},{\mathit{\phi }}_{i,2},\cdots ,{\mathit{\phi }}_{i,d})(d⩽r_i,i=1,2)$, they satisfy the following equalities:

$$\parallel {\mathit{A}}_i-{\mathbf{\Phi }}_i{\mathbf{\Phi }}_i^T{\mathit{A}}_i{\parallel }_2=\sqrt{{\mu }_{i,d+1}},i=1,2,$$

(57)

where ${\displaystyle {\parallel \mathit{A}\parallel }_2=\underset{\mathit{v}\in R^M}{sup}\frac{\parallel \mathit{A}\mathit{v}\parallel }{\parallel \mathit{v}\parallel }}$ and ${\displaystyle \parallel \mathit{v}\parallel =\sqrt{\sum _{i=1}^M{\left|v_i\right|}^2}}$ of vector $\mathit{v}={(v_1,v_2,\cdots ,v_M)}^T$. When $n=1,2,\cdots ,K$, it follows that

$$\parallel {\mathit{U}}_h^n-{\mathbf{\Phi }}_1{\mathbf{\Phi }}_1^T{\mathit{U}}_h^n\parallel =\parallel ({\mathit{A}}_1-{\mathbf{\Phi }}_1{\mathbf{\Phi }}_1^T{\mathit{A}}_1){\mathit{e}}^n\parallel ⩽\parallel ({\mathit{A}}_1-{\mathbf{\Phi }}_1{\mathbf{\Phi }}_1^T{\mathit{A}}_1){\parallel }_2\parallel {\mathit{e}}^n\parallel ⩽\sqrt{{\mu }_{1,d+1}},$$

(58)

$$\parallel {\mathit{W}}_h^n-{\mathbf{\Phi }}_2{\mathbf{\Phi }}_2^T{\mathit{W}}_h^n\parallel =\parallel ({\mathit{A}}_2-{\mathbf{\Phi }}_2{\mathbf{\Phi }}_2^T{\mathit{A}}_2){\mathit{e}}^n\parallel ⩽\parallel ({\mathit{A}}_2-{\mathbf{\Phi }}_2{\mathbf{\Phi }}_2^T{\mathit{A}}_2){\parallel }_2\parallel {\mathit{e}}^n\parallel ⩽\sqrt{{\mu }_{2,d+1}},$$

(59)

Here, ${\mathit{e}}^n(1⩽n⩽K)$ denotes the unit vectors whose nth component is 1. Therefore, ${\mathbf{\Phi }}_i=({\mathit{\phi }}_{i,1},{\mathit{\phi }}_{i,2},\cdots ,{\mathit{\phi }}_{i,d})$$(d⩽r_i,i=1,2)$ is a set of optimal POD bases.

Remark 2. 

Since the order $M\times M$ of matrix ${\mathit{A}}_i{\mathit{A}}_i^T(i=1,2)$ is much larger than the order $K\times K$ of matrix ${\mathit{A}}_i^T{\mathit{A}}_i$, and the eigenvalues of ${\mathit{A}}_i{\mathit{A}}_i^T$ and ${\mathit{A}}_i^T{\mathit{A}}_i$ are equal, we first calculate the initial d eigenvalues ${\mu }_{i,j}(1⩽j⩽d)$ of matrix ${\mathit{A}}_i^T{\mathit{A}}_i$ and their corresponding eigenvectors ${\mathit{\varphi }}_{i,j}(1⩽j⩽d)$. Then, we use the relation ${\mathit{\phi }}_{i,j}={\mathit{A}}_i{\mathit{\varphi }}_{i,j}/\sqrt{{\mu }_{i,j}}(1⩽j⩽d)$ to obtain the eigenvectors of ${\mathit{A}}_i{\mathit{A}}_i^T$, and generate a set of POD basis vectors ${\mathbf{\Phi }}_i=({\mathit{\phi }}_{i,1},{\mathit{\phi }}_{i,2},\cdots ,{\mathit{\phi }}_{i,d})(d⩽r_i)$.

3.2. Formulation of the RDCNMFE Scheme

Firstly, letting ${\mathit{\alpha }}_d^n={({\alpha }_1^n,{\alpha }_2^n,\cdots ,{\alpha }_d^n)}^T$, ${\mathit{\beta }}_d^n={({\beta }_1^n,{\beta }_2^n,\cdots ,{\beta }_d^n)}^T$, and ${\mathit{U}}_d^n=(U_{d1}^n,U_{d2}^n,\cdots ,U_{dM}^n{)}^T$ and ${\mathit{W}}_d^n={(W_{d1}^n,W_{d2}^n,\cdots ,W_{dM}^n)}^T$ represent the RDCNMFE solution coefficient vectors. Secondly, we can instantly obtain the first K RDCNMFE solution vectors ${\mathit{U}}_d^n={\mathbf{\Phi }}_1{\mathbf{\Phi }}_1^T{\mathit{U}}_h^n=:{\mathbf{\Phi }}_1{\mathit{\alpha }}_d^n$ and ${\mathit{W}}_d^n={\mathbf{\Phi }}_2{\mathbf{\Phi }}_2^T{\mathit{W}}_h^n=:{\mathbf{\Phi }}_2{\mathit{\beta }}_d^n(1⩽n⩽K)$ from Section 3.1. Finally, by using ${\mathit{U}}_d^n={\mathbf{\Phi }}_1{\mathit{\alpha }}_d^n,{\mathit{W}}_d^n={\mathbf{\Phi }}_2{\mathit{\beta }}_d^n(K+1⩽n⩽N)$ instead of $\{{\mathit{U}}_h^n,{\mathit{W}}_h^n\}$ in Problem 3, the following matrix-based RDCNMFE scheme can be established.

Problem 4. 

Find $\{{\mathit{U}}_d^n,{\mathit{W}}_d^n\}\in R^M\times R^M$ and $\{u_d^n,{\omega }_d^n\}\in V_h\times V_h(1⩽n⩽N)$ satisfying

$$\left\{\begin{array}{c}\begin{array}{c}{\mathit{\alpha }}_d^n={\mathbf{\Phi }}_1^T{\mathit{U}}_h^n,{\mathit{\beta }}_d^n={\mathbf{\Phi }}_2^T{\mathit{W}}_h^n,1⩽n⩽K,\hfill \\ {\mathbf{\Phi }}_1{\overline{\partial }}_t{\mathit{\alpha }}_d^n+\gamma \mathit{B}{\mathbf{\Phi }}_2{\mathit{\beta }}_d^{n-{\displaystyle \frac{1}{2}}}+{\mathbf{\Phi }}_2{\mathit{\beta }}_d^{n-{\displaystyle \frac{1}{2}}}=\mathit{F}({\mathbf{\Phi }}_1{\widehat{\widehat{\mathit{\alpha }}}}_d^{n-{\displaystyle \frac{1}{2}}}),K+1⩽n⩽N,\hfill \\ {\mathbf{\Phi }}_2{\mathit{\beta }}_d^{n-{\displaystyle \frac{1}{2}}}-\mathit{B}{\mathbf{\Phi }}_1{\mathit{\alpha }}_d^{n-{\displaystyle \frac{1}{2}}}=0,K+1⩽n⩽N,\hfill \\ {\displaystyle u_d^n=\sum _{j=1}^M{U}_{dj}^n{\zeta }_j={\mathit{U}}_d^n·\mathit{\zeta }={\mathbf{\Phi }}_1{\mathit{\alpha }}_d^n·\mathit{\zeta },{\omega }_d^n=\sum _{j=1}^M{W}_{dj}^n{\zeta }_j={\mathit{W}}_d^n·\mathit{\zeta }={\mathbf{\Phi }}_2{\mathit{\beta }}_d^n·\mathit{\zeta },1⩽n⩽N,}\hfill \end{array}\hfill \end{array}\right.$$

(60)

in which $\{{\mathit{U}}_h^n,{\mathit{W}}_h^n\}(n=1,2,\cdots ,K)$ are the first K solution coefficient vectors for Problem 3. The definitions of matrix $\mathit{B}$, vector $\mathit{F}$, as well as FE basis function vectors $\mathit{\zeta }=({\zeta }_1(x),{\zeta }_2(x),\cdots ,{\zeta }_M(x))$ are given in Section 2.2.

3.3. The Uniqueness, Stability and Error Estimate of the RDCNMFE Solutions

For the RDCNMFE solutions of Problem 4, the unique, stable and convergent properties are as follows.

Theorem 5. 

Under the same assumptions as Theorems 3 and 4, if $\{u^n,{\omega }^n\}\in V\times V$ are the weak solutions to Problem 1 and $\{u_d^n,{\omega }_d^n\}\in V_h\times V_h$ are the solutions to Problem 4, then for any $1⩽n⩽N$, the RDCNMFE solutions have uniqueness and unconditional stability, and the following error estimate is obtained.

$$\begin{array}{c}\hfill \parallel u^n-u_d^n\parallel +\parallel {\omega }^n-{\omega }_d^n\parallel ⩽C(h^2+\mathsf{\Delta }{t}^2+\sqrt{{\mu }_{1,d+1}}+\sqrt{{\mu }_{2,d+1}}).\end{array}$$

(61)

Proof of Theorem 5. 

(1) Discuss the uniqueness of the RDCNMFE solutions.

For $1⩽n⩽K$, Theorem 1 guarantees the uniqueness of solutions $\{u_h^n,{\omega }_h^n\}$ to Problem 3. Therefore, the solutions $\{u_d^n,{\omega }_d^n\}$ generated from first and fourth formulas in Problem 4 must also be unique.

For $K+1⩽n⩽N$, using ${\mathit{U}}_d^n={\mathbf{\Phi }}_1{\mathit{\alpha }}_d^n$, ${\mathit{W}}_d^n={\mathbf{\Phi }}_2{\mathit{\beta }}_d^n(K+1⩽n⩽N)$, the last three formulas of Problem 4 can be converted to

$$\begin{array}{c}{\overline{\partial }}_t{\mathit{U}}_d^n+\gamma \mathit{B}{\mathit{W}}_d^{n-{\displaystyle \frac{1}{2}}}+{\mathit{W}}_d^{n-{\displaystyle \frac{1}{2}}}=\mathit{F}({\widehat{\widehat{\mathit{U}}}}_d^{n-{\displaystyle \frac{1}{2}}}),K+1⩽n⩽N,\hfill \end{array}$$

(62)

$$\begin{array}{c}{\mathit{W}}_d^{n-{\displaystyle \frac{1}{2}}}-\mathit{B}{\mathit{U}}_d^{n-{\displaystyle \frac{1}{2}}}=0,K+1⩽n⩽N,\hfill \end{array}$$

(63)

$$\begin{array}{c}{\displaystyle u_d^n=\sum _{j=1}^M{U}_{dj}^n{\zeta }_j={\mathit{U}}_d^n·\mathit{\zeta },{\omega }_d^n=\sum _{j=1}^M{W}_{dj}^n{\zeta }_j={\mathit{W}}_d^n·\mathit{\zeta },K+1⩽n⩽N.}\hfill \end{array}$$

(64)

For $K+1⩽n⩽N$, the set of solutions ${\{u_h^n,{\omega }_h^n\}}_{n=K+1}^N$ for Problem 3 is unique. Since (62)–(64) follow the same formats as Problem 3, the set of solutions ${\{u_d^n,{\omega }_d^n\}}_{n=K+1}^N$ for (62)–(64) is also unique.

(2)

Demonstrate the stability of the RDCNMFE solutions.

When $1⩽n⩽K$, since the vectors in ${\mathbf{\Phi }}_1$ and ${\mathbf{\Phi }}_2$ are orthonormal, using Theorem 2, we have

$$\begin{array}{cc}\parallel u_d^n\parallel +\parallel {\omega }_d^n\parallel \hfill & =\parallel {\mathit{U}}_d^n·\zeta \parallel +\parallel {\mathit{W}}_d^n·\zeta \parallel \hfill \\ \hfill & =\parallel {\mathbf{\Phi }}_1{\mathbf{\Phi }}_1^T{\mathit{U}}_h^n·\zeta \parallel +\parallel {\mathbf{\Phi }}_2{\mathbf{\Phi }}_2^T{\mathit{W}}_h^n·\zeta \parallel \hfill \\ \hfill & ⩽C(\parallel u_h^n\parallel +\parallel {\omega }_h^n\parallel )\hfill \\ \hfill & ⩽C,1⩽n⩽L.\hfill \end{array}$$

(65)

When $K+1⩽n⩽N$, since $\mathit{B}$ is a positive definite symmetric matrix, we can rewrite (62) as

$${\mathit{B}}^{-1}{\overline{\partial }}_t{\mathit{U}}_d^n+\gamma {\mathit{W}}_d^{n-{\displaystyle \frac{1}{2}}}+{\mathit{B}}^{-1}{\mathit{W}}_d^{n-{\displaystyle \frac{1}{2}}}={\mathit{B}}^{-1}\mathit{F}({\widehat{\widehat{\mathit{U}}}}_d^{n-{\displaystyle \frac{1}{2}}}),K+1⩽n⩽N.$$

(66)

Putting (63) into (66), we have

$${\mathit{B}}^{-1}{\overline{\partial }}_t{\mathit{U}}_d^n+\gamma \mathit{B}{\mathit{U}}_d^{n-{\displaystyle \frac{1}{2}}}+{\mathit{U}}_d^{n-{\displaystyle \frac{1}{2}}}={\mathit{B}}^{-1}\mathit{F}({\widehat{\widehat{\mathit{U}}}}_d^{n-{\displaystyle \frac{1}{2}}}),K+1⩽n⩽N.$$

(67)

Taking the inner product of (67) and ${\overline{\partial }}_t{\mathit{U}}_d^n$,

$$\begin{array}{cc}\hfill & ({\mathit{B}}^{-1}{\overline{\partial }}_t{\mathit{U}}_d^n,{\overline{\partial }}_t{\mathit{U}}_d^n)+\gamma (\mathit{B}{\mathit{U}}_d^{n-{\displaystyle \frac{1}{2}}},{\overline{\partial }}_t{\mathit{U}}_d^n)+({\mathit{U}}_d^{n-{\displaystyle \frac{1}{2}}},{\overline{\partial }}_t{\mathit{U}}_d^n)\hfill \\ =\hfill & ({\mathit{B}}^{-1}\mathit{F}({\widehat{\widehat{\mathit{U}}}}_d^{n-{\displaystyle \frac{1}{2}}}),{\overline{\partial }}_t{\mathit{U}}_d^n),K+1⩽n⩽N.\hfill \end{array}$$

(68)

Then, the left side of (68) is

$$\begin{array}{cc}\hfill & {\displaystyle ({\mathit{B}}^{-1}{\overline{\partial }}_t{\mathit{U}}_d^n,{\overline{\partial }}_t{\mathit{U}}_d^n)+\gamma (\mathit{B}{\mathit{U}}_d^{n-\frac{1}{2}},{\overline{\partial }}_t{\mathit{U}}_d^n)+({\mathit{U}}_d^{n-\frac{1}{2}},{\overline{\partial }}_t{\mathit{U}}_d^n)}\hfill \\ =\hfill & {\displaystyle \parallel {({\mathit{B}}^{-1})}^{\frac{1}{2}}{\overline{\partial }}_t{\mathit{U}}_d^n{\parallel }^2+\frac{\gamma }{2\mathsf{\Delta }t}(\parallel {\mathit{B}}^{\frac{1}{2}}{\mathit{U}}_d^n{\parallel }^2-\parallel {\mathit{B}}^{\frac{1}{2}}{\mathit{U}}_d^{n-1}{\parallel }^2)}\hfill \\ \hfill & {\displaystyle +\frac{1}{2\mathsf{\Delta }t}(\parallel {\mathit{U}}_d^n{\parallel }^2-\parallel {\mathit{U}}_d^{n-1}{\parallel }^2),K+1⩽n⩽N,}\hfill \end{array}$$

(69)

and the right side of (68) is

$$({\mathit{B}}^{-1}\mathit{F}({\widehat{\widehat{\mathit{U}}}}_d^{n-{\displaystyle \frac{1}{2}}}),{\overline{\partial }}_t{\mathit{U}}_d^n)⩽C\parallel {({\mathit{B}}^{-1})}^{{\displaystyle \frac{1}{2}}}{\parallel }_{\infty }^2\parallel \mathit{F}({\widehat{\widehat{\mathit{U}}}}_d^{n-{\displaystyle \frac{1}{2}}}){\parallel }^2+{\parallel {({\mathit{B}}^{-1})}^{{\displaystyle \frac{1}{2}}}{\overline{\partial }}_t{\mathit{U}}_d^n\parallel }^2,K+1⩽n⩽N.$$

(70)

Using the same technique as (22), we have

$$\parallel {({\mathit{B}}^{-1})}^{{\displaystyle \frac{1}{2}}}{\parallel }_{\infty }^2\parallel \mathit{F}({\widehat{\widehat{\mathit{U}}}}_d^{n-{\displaystyle \frac{1}{2}}}){\parallel }^2⩽C+C(\parallel {\mathit{U}}_d^{n-1}{\parallel }^2+\parallel {\mathit{U}}_d^{n-2}{\parallel }^2),K+1⩽n⩽N.$$

(71)

Combining (69), (70) and (71), we have

$$\begin{array}{cc}\hfill & {\displaystyle \frac{\gamma }{2\mathsf{\Delta }t}(\parallel {\mathit{B}}^{\frac{1}{2}}{\mathit{U}}_d^n{\parallel }^2-\parallel {\mathit{B}}^{\frac{1}{2}}{\mathit{U}}_d^{n-1}{\parallel }^2)+\frac{1}{2\mathsf{\Delta }t}(\parallel {\mathit{U}}_d^n{\parallel }^2-\parallel {\mathit{U}}_d^{n-1}{\parallel }^2)}\hfill \\ ⩽\hfill & {\displaystyle C+C(\parallel {\mathit{U}}_d^{n-1}{\parallel }^2+\parallel {\mathit{U}}_d^{n-2}{\parallel }^2),K+1⩽n⩽N.}\hfill \end{array}$$

(72)

Multiplying both sides of (72) by $2\mathsf{\Delta }t$, and summing from 2 to n, we obtain

$$\begin{array}{cc}{\displaystyle \parallel {\mathit{U}}_d^n{\parallel }^2}\hfill & {\displaystyle ⩽\parallel {\mathit{U}}_d^1{\parallel }^2+\gamma \parallel {\mathit{B}}^{\frac{1}{2}}{\mathit{U}}_d^1{\parallel }^2+2\mathsf{\Delta }t\sum _{i=2}^{n}C+C\mathsf{\Delta }t\sum _{i=0}^{n-1}{\parallel {\mathit{U}}_d^i\parallel }^2}\hfill \\ \hfill & {\displaystyle ⩽(1+\gamma \parallel {\mathit{B}}^{\frac{1}{2}}{\parallel }_{\infty }^2)\parallel {\mathit{U}}_d^1{\parallel }^2+CT+C\mathsf{\Delta }t\sum _{i=0}^{n-1}{\parallel {\mathit{U}}_d^i\parallel }^2,K+1⩽n⩽N.}\hfill \end{array}$$

(73)

Noting that

$$\parallel {\mathit{U}}_d^1{\parallel }^2=\parallel {\mathbf{\Phi }}_1{\mathbf{\Phi }}_1^T{\mathit{U}}_h^1{\parallel }^2⩽C{\parallel {\mathit{U}}_h^1\parallel }^2⩽C,$$

(74)

and putting (74) into (73), we have

$$\parallel {\mathit{U}}_d^n{\parallel }^2⩽C+C\mathsf{\Delta }t\sum _{i=0}^{n-1}{\parallel {\mathit{U}}_d^i\parallel }^2,K+1⩽n⩽N.$$

(75)

Using the Gronwall Lemma for (75),

$$\parallel {\mathit{U}}_d^n{\parallel }^2⩽C{e}^{Cn\mathsf{\Delta }t}⩽C,K+1⩽n⩽N.$$

(76)

And

$$\parallel {\mathit{W}}_d^n\parallel =\parallel \mathit{B}{\mathit{U}}_d^n{\parallel ⩽\parallel \mathit{B}\parallel }_{\infty }\parallel {\mathit{U}}_d^n\parallel ⩽C\parallel {\mathit{U}}_d^n\parallel ⩽C,K+1⩽n⩽N.$$

(77)

So we have

$$\parallel {\mathit{U}}_d^n\parallel +\parallel {\mathit{W}}_d^n\parallel ⩽C,K+1⩽n⩽N.$$

(78)

Thus, noting that $\parallel \zeta \parallel ⩽C$, we obtain

$$\parallel u_d^n\parallel +\parallel {\omega }_d^n\parallel =\parallel {\mathit{U}}_d^n·\zeta \parallel +\parallel {\mathit{W}}_d^n·\zeta \parallel ⩽C\parallel {\mathit{U}}_d^n\parallel ·\parallel \zeta \parallel +C\parallel {\mathit{W}}_d^n\parallel ·\parallel \zeta \parallel ⩽C,K+1⩽n⩽N.$$

(79)

From (65) and (79), the RDCNMFE solutions $\{u_d^n,{\omega }_d^n\}(1⩽n⩽N)$ are unconditionally stable.

(3)

Analyze the error estimate of the RDCNMFE solutions.

When $1⩽n⩽K$, from $\parallel \zeta \parallel ⩽C$, (58) and (59), we have

$$\begin{array}{cc}\parallel u_h^n-u_d^n\parallel +\parallel {\omega }_h^n-{\omega }_d^n\parallel \hfill & ⩽\parallel {\mathit{U}}_h^n-{\mathit{U}}_d^n{\parallel }_{\infty }\parallel \zeta \parallel +\parallel {\mathit{W}}_h^n-{\mathit{W}}_d^n{\parallel }_{\infty }\parallel \zeta \parallel \hfill \\ \hfill & ⩽C\parallel {\mathit{U}}_h^n-{\mathbf{\Phi }}_1{\mathbf{\Phi }}_1^T{\mathit{U}}_h^n\parallel +\parallel {\mathit{W}}_h^n-{\mathbf{\Phi }}_2{\mathbf{\Phi }}_2^T{\mathit{W}}_h^n\parallel \hfill \\ \hfill & ⩽C(\sqrt{{\mu }_{1,d+1}}+\sqrt{{\mu }_{2,d+1}}),1⩽n⩽K.\hfill \end{array}$$

(80)

When $K+1⩽n⩽N$, letting ${\mathit{\delta }}^n={\mathit{U}}_h^n-{\mathit{U}}_d^n$ and ${\mathit{\rho }}^n={\mathit{W}}_h^n-{\mathit{W}}_d^n$, and combining (17), (66) and (63), we obtain

$$\begin{array}{c}{\mathit{B}}^{-1}{\overline{\partial }}_t{\mathit{\delta }}^n+\gamma {\mathit{\rho }}^{n-{\displaystyle \frac{1}{2}}}+{\mathit{B}}^{-1}{\mathit{\rho }}^{n-{\displaystyle \frac{1}{2}}}={\mathit{B}}^{-1}\mathit{F}({\widehat{\widehat{\mathit{U}}}}_h^{n-{\displaystyle \frac{1}{2}}})-{\mathit{B}}^{-1}\mathit{F}({\widehat{\widehat{\mathit{U}}}}_d^{n-{\displaystyle \frac{1}{2}}}),K+1⩽n⩽N,\hfill \end{array}$$

(81)

$$\begin{array}{c}{\mathit{\rho }}^{n-{\displaystyle \frac{1}{2}}}=\mathit{B}{\mathit{\delta }}^{n-{\displaystyle \frac{1}{2}}},K+1⩽n⩽N.\hfill \end{array}$$

(82)

Putting (82) into (81), we have

$${\mathit{B}}^{-1}{\overline{\partial }}_t{\mathit{\delta }}^n+\gamma \mathit{B}{\mathit{\delta }}^{n-{\displaystyle \frac{1}{2}}}+{\mathit{\delta }}^{n-{\displaystyle \frac{1}{2}}}={\mathit{B}}^{-1}\mathit{F}({\widehat{\widehat{\mathit{U}}}}_h^{n-{\displaystyle \frac{1}{2}}})-{\mathit{B}}^{-1}\mathit{F}({\widehat{\widehat{\mathit{U}}}}_d^{n-{\displaystyle \frac{1}{2}}}),K+1⩽n⩽N.$$

(83)

Taking the inner product of (83) and ${\overline{\partial }}_t{\mathit{\delta }}^n$,

$$\begin{array}{cc}\hfill & ({\mathit{B}}^{-1}{\overline{\partial }}_t{\mathit{\delta }}^n,{\overline{\partial }}_t{\mathit{\delta }}^n)+\gamma (\mathit{B}{\mathit{\delta }}^{n-{\displaystyle \frac{1}{2}}},{\overline{\partial }}_t{\mathit{\delta }}^n)+({\mathit{\delta }}^{n-{\displaystyle \frac{1}{2}}},{\overline{\partial }}_t{\mathit{\delta }}^n)\hfill \\ =\hfill & ({\mathit{B}}^{-1}\mathit{F}({\widehat{\widehat{\mathit{U}}}}_h^{n-{\displaystyle \frac{1}{2}}})-{\mathit{B}}^{-1}\mathit{F}({\widehat{\widehat{\mathit{U}}}}_d^{n-{\displaystyle \frac{1}{2}}}),{\overline{\partial }}_t{\mathit{\delta }}^n),K+1⩽n⩽N.\hfill \end{array}$$

(84)

Then, the left side of (84) is

$$\begin{array}{cc}\hfill & {\displaystyle ({\mathit{B}}^{-1}{\overline{\partial }}_t{\mathit{\delta }}^n,{\overline{\partial }}_t{\mathit{\delta }}^n)+\gamma (\mathit{B}{\mathit{\delta }}^{n-\frac{1}{2}},{\overline{\partial }}_t{\mathit{\delta }}^n)+({\mathit{\delta }}^{n-\frac{1}{2}},{\overline{\partial }}_t{\mathit{\delta }}^n)}\hfill \\ =\hfill & {\displaystyle \parallel {({\mathit{B}}^{-1})}^{\frac{1}{2}}{\overline{\partial }}_t{\mathit{\delta }}^n{\parallel }^2+\frac{\gamma }{2\mathsf{\Delta }t}(\parallel {\mathit{B}}^{\frac{1}{2}}{\mathit{\delta }}^n{\parallel }^2-\parallel {\mathit{B}}^{\frac{1}{2}}{\mathit{\delta }}^{n-1}{\parallel }^2)}\hfill \\ \hfill & {\displaystyle +\frac{1}{2\mathsf{\Delta }t}(\parallel {\mathit{\delta }}^n{\parallel }^2-\parallel {\mathit{\delta }}^{n-1}{\parallel }^2),K+1⩽n⩽N,}\hfill \end{array}$$

(85)

and the right side of (84) is

$$\begin{array}{cc}\hfill & ({\mathit{B}}^{-1}\mathit{F}({\widehat{\widehat{\mathit{U}}}}_h^{n-{\displaystyle \frac{1}{2}}})-{\mathit{B}}^{-1}\mathit{F}({\widehat{\widehat{\mathit{U}}}}_d^{n-{\displaystyle \frac{1}{2}}}),{\overline{\partial }}_t{\mathit{\delta }}^n)\hfill \\ ⩽\hfill & C\parallel {({\mathit{B}}^{-1})}^{{\displaystyle \frac{1}{2}}}{\parallel }_{\infty }^2\parallel \mathit{F}({\widehat{\widehat{\mathit{U}}}}_h^{n-{\displaystyle \frac{1}{2}}})-\mathit{F}({\widehat{\widehat{\mathit{U}}}}_d^{n-{\displaystyle \frac{1}{2}}}){\parallel }^2+{\parallel {({\mathit{B}}^{-1})}^{{\displaystyle \frac{1}{2}}}{\overline{\partial }}_t{\mathit{\delta }}^n\parallel }^2,K+1⩽n⩽N.\hfill \end{array}$$

(86)

Combining Lemma 1 with the global Lipschitz-continuity of $f(u)$, the first term of (86) can be estimated as

$$\begin{array}{cc}\hfill & {\displaystyle \parallel {({\mathit{B}}^{-1})}^{\frac{1}{2}}{\parallel }_{\infty }^2{\parallel \mathit{F}({\widehat{\widehat{\mathit{U}}}}_h^{n-\frac{1}{2}})-\mathit{F}({\widehat{\widehat{\mathit{U}}}}_d^{n-\frac{1}{2}})\parallel }^2}\hfill \\ ⩽\hfill & {\displaystyle C\parallel \left(\frac{3}{2}\mathit{F}({\mathit{U}}_h^{n-1})-\frac{1}{2}\mathit{F}({\mathit{U}}_h^{n-2})\right)-\left(\frac{3}{2}\mathit{F}({\mathit{U}}_d^{n-1})-\frac{1}{2}\mathit{F}({\mathit{U}}_d^{n-2})\right){\parallel }^2}\hfill \\ ⩽\hfill & {\displaystyle C\parallel \frac{3}{2}({\mathit{U}}_h^{n-1}-{\mathit{U}}_d^{n-1}){\parallel }^2+\parallel \frac{1}{2}({\mathit{U}}_h^{n-2}-{\mathit{U}}_d^{n-2}){\parallel }^2}\hfill \\ ⩽\hfill & {\displaystyle C(\parallel {\mathit{\delta }}^{n-1}{\parallel }^2+\parallel {\mathit{\delta }}^{n-2}{\parallel }^2),K+1⩽n⩽N.}\hfill \end{array}$$

(87)

Combining (85), (86) and (87), we have

$$\begin{array}{cc}\hfill & {\displaystyle \frac{\gamma }{2\mathsf{\Delta }t}(\parallel {\mathit{B}}^{\frac{1}{2}}{\mathit{\delta }}^n{\parallel }^2-\parallel {\mathit{B}}^{\frac{1}{2}}{\mathit{\delta }}^{n-1}{\parallel }^2)+\frac{1}{2\mathsf{\Delta }t}(\parallel {\mathit{\delta }}^n{\parallel }^2-\parallel {\mathit{\delta }}^{n-1}{\parallel }^2)}\hfill \\ ⩽\hfill & C(\parallel {\mathit{\delta }}^{n-1}{\parallel }^2+\parallel {\mathit{\delta }}^{n-2}{\parallel }^2),K+1⩽n⩽N.\hfill \end{array}$$

(88)

Multiplying both sides of (88) by $2\mathsf{\Delta }t$, and summing from $K+1$ to $n(n⩽N)$, we obtain

$$\begin{array}{cc}\parallel {\mathit{\delta }}^n{\parallel }^2\hfill & {\displaystyle ⩽\parallel {\mathit{\delta }}^K{\parallel }^2+\parallel {\mathit{B}}^{\frac{1}{2}}{\mathit{\delta }}^K{\parallel }^2+C\mathsf{\Delta }t\sum _{i=K-1}^{n-1}{\parallel {\mathit{\delta }}^i\parallel }^2}\hfill \\ \hfill & {\displaystyle ⩽C\parallel {\mathit{\delta }}^K{\parallel }^2+C\mathsf{\Delta }t\sum _{i=K-1}^{n-1}{\parallel {\mathit{\delta }}^i\parallel }^2,K+1⩽n⩽N.}\hfill \end{array}$$

(89)

Noting that

$$\parallel {\mathit{\delta }}^K{\parallel }^2=\parallel {\mathit{U}}_h^K-{\mathit{U}}_d^K{\parallel }^2={\parallel {\mathit{U}}_h^K-{\mathbf{\Phi }}_1{\mathbf{\Phi }}_1^T{\mathit{U}}_h^K\parallel }^2.$$

(90)

Putting (90) into (89), from (58) and (59), we have

$${\displaystyle \parallel {\mathit{\delta }}^n{\parallel }^2⩽C{\mu }_{1,d+1}+C\mathsf{\Delta }t\sum _{i=K-1}^{n-1}{\parallel {\mathit{\delta }}^i\parallel }^2,K+1⩽n⩽N.}$$

(91)

Applying the Gronwall Lemma for (91),

$$\parallel {\mathit{\delta }}^n{\parallel }^2⩽C{\mu }_{1,d+1}{e}^{Cn\mathsf{\Delta }t}⩽C{\mu }_{1,d+1},K+1⩽n⩽N.$$

(92)

And

$$\parallel {\mathit{\rho }}^n\parallel =\parallel \mathit{B}{\mathit{\delta }}^n{\parallel ⩽C\parallel \mathit{B}\parallel }_{\infty }\parallel {\mathit{\delta }}^n\parallel ⩽C\parallel {\mathit{\delta }}^n\parallel ,K+1⩽n⩽N,$$

(93)

Thus, we obtain

$$\parallel {\mathit{\delta }}^n\parallel +\parallel {\mathit{\rho }}^n\parallel ⩽C(\sqrt{{\mu }_{1,d+1}}+\sqrt{{\mu }_{2,d+1}}),K+1⩽n⩽N.$$

(94)

Further, from $\parallel \mathit{\zeta }\parallel ⩽C$, we obtain

$$\begin{array}{cc}\parallel u_h^n-u_d^n\parallel +\parallel {\omega }_h^n-{\omega }_d^n\parallel \hfill & ⩽\parallel {\mathit{U}}_h^n-{\mathit{U}}_d^n{\parallel }_{\infty }·\parallel \mathit{\zeta }\parallel +\parallel {\mathit{W}}_h^n-{\mathit{W}}_d^n{\parallel }_{\infty }·\parallel \mathit{\zeta }\parallel \hfill \\ \hfill & ⩽\parallel {\mathit{U}}_h^n-{\mathit{U}}_d^n\parallel ·\parallel \mathit{\zeta }\parallel +\parallel {\mathit{W}}_h^n-{\mathit{W}}_d^n\parallel ·\parallel \mathit{\zeta }\parallel \hfill \\ \hfill & ⩽C(\sqrt{{\mu }_{1,d+1}}+\sqrt{{\mu }_{2,d+1}}),K+1⩽n⩽N.\hfill \end{array}$$

(95)

Using the triangle inequality, Theorem 3 and 4, (80) and (95), we obtain

$$\begin{array}{cc}\parallel u^n-u_d^n\parallel +\parallel {\omega }^n-{\omega }_d^n\parallel \hfill & ⩽\parallel u^n-u_h^n\parallel +\parallel u_h^n-u_d^n\parallel +\parallel {\omega }^n-{\omega }_h^n\parallel +\parallel {\omega }_h^n-{\omega }_d^n\parallel \hfill \\ \hfill & ⩽C(h^2+\mathsf{\Delta }{t}^2+\sqrt{{\mu }_{1,d+1}}+\sqrt{{\mu }_{2,d+1}}),1⩽n⩽N.\hfill \end{array}$$

(96)

□

4. The Numerical Examples for the EFK Equations

In order to verify the effectiveness of the RDCNMFE method, we can compare the errors and orders between the genuine, CNMFE and RDCNMFE solutions and CPU runtime on different time nodes. In order to calculate the error, we use the norm

$$\parallel u-\tilde{u}{\parallel }_0={\left({\int }_{\Omega }{(u-\tilde{u})}^{2}d\mathit{x}\right)}^{{\displaystyle \frac{1}{2}}},{\parallel \omega -\tilde{\omega }\parallel }_0={\left({\int }_{\Omega }{(\omega -\tilde{\omega })}^{2}d\mathit{x}\right)}^{{\displaystyle \frac{1}{2}}}$$

as the $L^2$ errors, where $\{u,\omega \}$ and $\{\tilde{u},\tilde{\omega }\}$ are the exact and approximate solutions respectively; 0 can be omitted here.

We here adopt the numerical experiment that the following nonlinear EFK equation has a genuine solution. Generally speaking, it has no genuine solution.

$$\left\{\begin{array}{cc}{u}_t+\gamma {\mathsf{\Delta }}^{2}u-\mathsf{\Delta }u+u^3-u=g(\mathit{x},t),\hfill & \hfill \mathit{x}\in \Omega \times (0,T],\\ u(\mathit{x},t)=\mathsf{\Delta }u(\mathit{x},t)=0,\hfill & \hfill \mathit{x}\in \partial \Omega ,t\in [0,T],\\ u(\mathit{x},0)=u_0(\mathit{x}),\hfill & \hfill \mathit{x}\in \Omega ,\end{array}\right.$$

(97)

Example 1. 

The 1D EFK model (97) problem is considered with a genuine solution given in the following form:

$$u(x,t)={\mathrm{e}}^{-t}\sin (2\pi x),x\in [0,1].$$

(98)

The genuine solution of the auxiliary variable is

$$\omega (x,t)=4{\pi }^2{\mathrm{e}}^{-t}\sin (2\pi x),$$

(99)

and the time-dependent source term is

$$g(x,t)={\mathrm{e}}^{-t}(16\gamma {\pi }^4+4{\pi }^2-2)\sin (2\pi x)+{\mathrm{e}}^{-3t}{\sin }^3(2\pi x).$$

(100)

Firstly, the initial 20 CNMFE solution vectors $\{{\mathit{U}}_h^n,{\mathit{W}}_h^n\}(n=1,2,\cdots ,20)$ are obtained by calculating Problem 3 to establish the snapshot matrixes ${\mathit{A}}_1=({\mathit{U}}_h^1,{\mathit{U}}_h^2,\cdots ,{\mathit{U}}_h^{20})$ and ${\mathit{A}}_2=({\mathit{W}}_h^1,{\mathit{W}}_h^2,\cdots ,{\mathit{W}}_h^{20})$. Secondly, we calculate the eigenvalues ${\mu }_{i,j}(i=1,2,j=1,2,\cdots ,20)$ (arranged decreasingly) and the associated eigenvectors ${\mathit{\varphi }}_{i,j}(i=1,2,j=1,2,\cdots ,20)$ of the matrixes ${\mathit{A}}_i^T{\mathit{A}}_i(i=1,2)$. Then, we find that $\sqrt{{\mu }_{i,7}}⩽h^2+\mathsf{\Delta }{t}^2$, so that we just need to extract the first 6 eigenvectors ${\mathit{\varphi }}_{i,j}$ of the matrix ${\mathit{A}}_i^T{\mathit{A}}_i(i=1,2)$ to construct a set of POD basis vectors ${\mathbf{\Phi }}_i=({\mathit{\phi }}_{i,1},{\mathit{\phi }}_{i,2},\cdots ,{\mathit{\phi }}_{i,6})$, where ${\mathit{\phi }}_{i,j}={\mathit{A}}_i{\mathit{\varphi }}_{i,j}/\sqrt{{\mu }_{i,j}}(i=1,2,j=1,2,\cdots ,6)$. Finally, we compute the RDCNMFE solutions $\{u_d^n,{\omega }_d^n\}$ of (97) by Problem 4 and the CNMFE solutions $\{u_h^n,{\omega }_h^n\}$ of (97) by Problem 3 when $h=1/100$ and $\mathsf{\Delta }t=1/100$ at $T=1$. And they are compared with the genuine solution, respectively, exhibited in Figure 1 and Figure 2. As evident from Figure 1 and Figure 2, the RDCNMFE and CNMFE solutions closely mirror the genuine solution.

To facilitate comparison, we present in

Table 1. $L^2$ errors and order between the genuine, CNMFE and RDCNMFE solutions of u when $\mathsf{\Delta }t$ = 1/10,000 at T = 1.

		CNMFE Method		RDCNMFE Method
	h	${\displaystyle \left|\right|{\mathit{u}}^{\mathit{n}}-{\mathit{u}}_{\mathit{h}}^{\mathit{n}}\left|\right|}$	Order	${\displaystyle \left|\right|{\mathit{u}}^{\mathit{n}}-{\mathit{u}}_{\mathit{d}}^{\mathit{n}}\left|\right|}$	Order
$\gamma =1$	$1/40$	1.0743 × ${10}^{-3}$	–	1.0743 × ${10}^{-3}$	–
	$1/80$	2.6293 × ${10}^{-4}$	2.0307	2.6293 × ${10}^{-4}$	2.0307
	$1/160$	5.9726 × ${10}^{-5}$	2.1382	5.9726 × ${10}^{-5}$	2.1382
	$1/320$	9.0547 × ${10}^{-6}$	2.7216	9.0547 × ${10}^{-6}$	2.7216
$\gamma =2$	$1/40$	1.0840 × ${10}^{-3}$	–	1.0840 × ${10}^{-3}$	–
	$1/80$	2.6835 × ${10}^{-4}$	2.0142	2.6835 × ${10}^{-4}$	2.0142
	$1/160$	6.4055 × ${10}^{-5}$	2.0667	6.4056 × ${10}^{-5}$	2.0667
	$1/320$	1.2982 × ${10}^{-5}$	2.3028	1.2983 × ${10}^{-5}$	2.3027

and

Table 2. $L^2$ errors and order between the genuine, CNMFE and RDCNMFE solutions of $\omega$ when $\mathsf{\Delta }t$ = 1/10,000 at T = 1.

		CNMFE Method		RDCNMFE Method
	h	${\displaystyle \left|\right|{\mathit{\omega }}^{\mathit{n}}-{\mathit{\omega }}_{\mathit{h}}^{\mathit{n}}\left|\right|}$	Order	${\displaystyle \left|\right|{\mathit{\omega }}^{\mathit{n}}-{\mathit{\omega }}_{\mathit{d}}^{\mathit{n}}\left|\right|}$	Order
$\gamma =1$	$1/40$	7.8197 × ${10}^{-2}$	–	7.8197 × ${10}^{-2}$	–
	$1/80$	1.9323 × ${10}^{-2}$	2.0168	1.9323 × ${10}^{-2}$	2.0168
	$1/160$	4.5886 × ${10}^{-3}$	2.0742	4.5886 × ${10}^{-3}$	2.0742
	$1/320$	9.0468 × ${10}^{-4}$	2.3426	9.0470 × ${10}^{-4}$	2.3425
$\gamma =2$	$1/40$	7.8588 × ${10}^{-2}$	–	7.8588 × ${10}^{-2}$	–
	$1/80$	1.9541 × ${10}^{-2}$	2.0078	1.9541 × ${10}^{-2}$	2.0078
	$1/160$	4.7634 × ${10}^{-3}$	2.0365	4.7633 × ${10}^{-3}$	2.0365
	$1/320$	1.0680 × ${10}^{-2}$	2.1570	1.0680 × ${10}^{-3}$	2.1571

the $L^2$ errors and convergence orders of the CNMFE and RDCNMFE solutions for u and $\omega$, respectively, under the conditions of $\mathsf{\Delta }t$ = 1/10,000 and $\gamma =1,2$ at $T=1$. Notably, the analysis of these tables reveals that the $L^2$ errors and convergence rates achieved by the RDCNMFE method closely mirror those of the CNMFE method, so this is consistent with the previous theoretical analysis.

To further substantiate the efficiency of the RDCNMFE method, we have documented the errors and the CPU runtime required to compute the solutions using CNMFE and RDCNMFE methods, with $h=1/1000$ and $\mathsf{\Delta }t$ = 1/10,000, at various time points $T=1.0,2.0,3.0,4.0,5.0$. These findings, derived from solving Problems 3 and 4, are summarized in

Table 3. Comparison of $L^2$ errors and CPU runtime of CNMFE and RDCNMFE solutions when $\mathsf{\Delta }t=1/10,000$.

	CNMFE Method		RDCNMFE Method
Real Time	${\displaystyle \left|\right|{\mathit{u}}^{\mathit{n}}-{\mathit{u}}_{\mathit{h}}^{\mathit{n}}\left|\right|}$	CPU Runtime (s)	${\displaystyle \left|\right|{\mathit{u}}^{\mathit{n}}-{\mathit{u}}_{\mathit{d}}^{\mathit{n}}\left|\right|}$	CPU Runtime (s)
$T=1.0$	6.5981 × ${10}^{-6}$	41.090	6.5995 × ${10}^{-6}$	6.153
$T=2.0$	1.6443 × ${10}^{-6}$	82.527	1.6443 × ${10}^{-6}$	18.609
$T=3.0$	2.2493 × ${10}^{-6}$	125.082	2.2493 × ${10}^{-6}$	39.335
$T=4.0$	2.2493 × ${10}^{-6}$	164.239	2.2493 × ${10}^{-6}$	66.781
$T=5.0$	2.5537 × ${10}^{-6}$	205.047	2.5537 × ${10}^{-6}$	100.456

.

As demonstrated in Table 3, the RDCNMFE method consumes much less CPU run time than the CNMFE method. This numerical example demonstrates that the RDCNMFE method significantly reduces CPU running time.

Example 2. 

The 2D EFK model (97) problem is considered with genuine solution given in the following form:

$$u(x_1,x_2,t)={\mathrm{e}}^{-t}\sin (\pi x_1)\sin (\pi x_2),(x_1,x_2)\in {[0,1]}^2,t\in [0,T].$$

(101)

The genuine solution of the auxiliary variable is

$$\omega (x_1,x_2,t)=2{\pi }^2{\mathrm{e}}^{-t}\sin (\pi x_1)\sin (\pi x_2),$$

(102)

and the time-dependent source term is

$$g(x_1,x_2,t)={\mathrm{e}}^{-t}(4\gamma {\pi }^4+2{\pi }^2-2)\sin (\pi x_1)\sin (\pi x_2)+{\mathrm{e}}^{-3t}{\sin }^3(\pi x_1){\sin }^3(\pi x_2).$$

(103)

We compute the RDCNMFE solutions $\{u_d^n,{\omega }_d^n\}$ of (97) by Problem 4 and the CNMFE solutions $\{u_h^n,{\omega }_h^n\}$ of (97) by Problem 3 when $h=\sqrt{2}/50$ and $\mathsf{\Delta }t=1/100$ at $T=1$. And they are compared with the genuine solution, respectively, exhibited in Figure 3 and Figure 4. As evident from Figure 3 and Figure 4, the RDCNMFE and CNMFE solutions closely mirror the genuine solution.

For comparison, we give the errors and convergence orders of the CNMFE and RDCNMFE solutions for u and $\omega$ when $\mathsf{\Delta }t=1/100,1/1000$ and $\gamma =1,2,$ at $T=1$ in

Table 4. $L^2$ errors and order between the genuine, CNMFE and RDCNMFE solutions of u when $\mathsf{\Delta }t=1/100$ at T = 1.

		CNMFE Method		RDCNMFE Method
	Grid	${\displaystyle \left|\right|{\mathit{u}}^{\mathit{n}}-{\mathit{u}}_{\mathit{h}}^{\mathit{n}}\left|\right|}$	Order	${\displaystyle \left|\right|{\mathit{u}}^{\mathit{n}}-{\mathit{u}}_{\mathit{d}}^{\mathit{n}}\left|\right|}$	Order
$\gamma =1$	$10\times 10$	3.0742 × ${10}^{-3}$	–	3.0742 × ${10}^{-3}$	–
	$20\times 20$	7.5577 × ${10}^{-4}$	2.0242	7.5577 × ${10}^{-4}$	2.0242
	$40\times 40$	1.7251 × ${10}^{-4}$	2.1313	1.7251 × ${10}^{-4}$	2.1313
	$80\times 80$	2.7256 × ${10}^{-5}$	2.6620	2.7274 × ${10}^{-5}$	2.6611
$\gamma =2$	$10\times 10$	3.1127 × ${10}^{-3}$	–	3.1127 × ${10}^{-3}$	–
	$20\times 20$	7.7452 × ${10}^{-4}$	2.0068	7.7452 × ${10}^{-4}$	2.0242
	$40\times 40$	1.8613 × ${10}^{-4}$	2.0570	1.8615 × ${10}^{-4}$	2.0568
	$80\times 80$	3.8899 × ${10}^{-5}$	2.2585	3.8994 × ${10}^{-5}$	2.2552

,

Table 5. $L^2$ errors and order between the genuine, CNMFE and RDCNMFE solutions of $\omega$ when $\mathsf{\Delta }t=1/100$ at T = 1.

		CNMFE Method		RDCNMFE Method
	Grid	${\displaystyle \left|\right|{\mathit{\omega }}^{\mathit{n}}-{\mathit{\omega }}_{\mathit{h}}^{\mathit{n}}\left|\right|}$	Order	${\displaystyle \left|\right|{\mathit{\omega }}^{\mathit{n}}-{\mathit{\omega }}_{\mathit{d}}^{\mathit{n}}\left|\right|}$	Order
$\gamma =1$	$10\times 10$	1.0990 × ${10}^{-1}$	–	1.0990 × ${10}^{-1}$	–
	$20\times 20$	2.7344 × ${10}^{-2}$	2.0068	2.7344 × ${10}^{-2}$	2.0068
	$40\times 40$	6.5061 × ${10}^{-3}$	2.0714	6.5069 × ${10}^{-3}$	2.0712
	$80\times 80$	1.2863 × ${10}^{-3}$	2.3385	1.2901 × ${10}^{-3}$	2.3345
$\gamma =2$	$10\times 10$	1.1069 × ${10}^{-1}$	–	1.1069 × ${10}^{-1}$	–
	$20\times 20$	2.7729 × ${10}^{-2}$	1.9971	2.7730 × ${10}^{-2}$	1.9970
	$40\times 40$	6.7869 × ${10}^{-3}$	2.0306	6.7930 × ${10}^{-3}$	2.0294
	$80\times 80$	1.5392 × ${10}^{-3}$	2.1406	1.5664 × ${10}^{-3}$	2.1166

,

Table 6. $L^2$ errors and order between the genuine, CNMFE and RDCNMFE solutions of u when $\mathsf{\Delta }t=1/1000$ at T = 1.

		CNMFE Method		RDCNMFE Method
	Grid	${\displaystyle \left|\right|{\mathit{u}}^{\mathit{n}}-{\mathit{u}}_{\mathit{h}}^{\mathit{n}}\left|\right|}$	Order	${\displaystyle \left|\right|{\mathit{u}}^{\mathit{n}}-{\mathit{u}}_{\mathit{d}}^{\mathit{n}}\left|\right|}$	Order
$\gamma =1$	$10\times 10$	3.0721 × ${10}^{-3}$	–	3.0721 × ${10}^{-3}$	–
	$20\times 20$	7.5360 × ${10}^{-4}$	2.0273	7.5360 × ${10}^{-4}$	2.0273
	$40\times 40$	1.7035 × ${10}^{-4}$	2.1453	1.7035 × ${10}^{-4}$	2.1453
	$80\times 80$	2.5365 × ${10}^{-5}$	2.7476	2.5365 × ${10}^{-5}$	2.7476
$\gamma =2$	$10\times 10$	3.1106 × ${10}^{-3}$	–	3.1106 × ${10}^{-3}$	–
	$20\times 20$	7.7235 × ${10}^{-4}$	2.0098	7.7235 × ${10}^{-4}$	2.0098
	$40\times 40$	1.8397 × ${10}^{-4}$	2.0698	1.8397 × ${10}^{-4}$	2.0698
	$80\times 80$	3.6797 × ${10}^{-5}$	2.3218	3.6797 × ${10}^{-5}$	2.3218

and

Table 7. $L^2$ errors and order between the genuine, CNMFE and RDCNMFE solutions of $\omega$ when $\mathsf{\Delta }t=1/1000$ at T = 1.

		CNMFE Method		RDCNMFE Method
	Grid	${\displaystyle \left|\right|{\mathit{\omega }}^{\mathit{n}}-{\mathit{\omega }}_{\mathit{h}}^{\mathit{n}}\left|\right|}$	Order	${\displaystyle \left|\right|{\mathit{\omega }}^{\mathit{n}}-{\mathit{\omega }}_{\mathit{d}}^{\mathit{n}}\left|\right|}$	Order
$\gamma =1$	$10\times 10$	1.0985 × ${10}^{-1}$	–	1.0985 × ${10}^{-1}$	–
	$20\times 20$	2.7300 × ${10}^{-2}$	2.0086	2.7300 × ${10}^{-2}$	2.0086
	$40\times 40$	6.4615 × ${10}^{-3}$	2.0789	6.4615 × ${10}^{-3}$	2.0789
	$80\times 80$	1.2424 × ${10}^{-3}$	2.3787	1.2424 × ${10}^{-3}$	2.3787
$\gamma =2$	$10\times 10$	1.1065 × ${10}^{-1}$	–	1.1065 × ${10}^{-1}$	–
	$20\times 20$	2.7684 × ${10}^{-2}$	1.9988	2.7685 × ${10}^{-2}$	1.9988
	$40\times 40$	6.7424 × ${10}^{-3}$	2.0377	6.7424 × ${10}^{-3}$	2.0377
	$80\times 80$	1.4948 × ${10}^{-3}$	2.1733	1.4977 × ${10}^{-3}$	2.1705

, respectively. As can be observed from Table 4, Table 5, Table 6 and Table 7, the errors and convergence rates of the RDCNMFE method are nearly identical to the CNMFE method, which aligns with the previous theoretical analysis. Figure 5 offers a more visual representation of the comparative results of the error convergence orders between $u_h^n$ and $u_d^n$ and between ${\omega }_h^n$ and ${\omega }_d^n$ when $\mathsf{\Delta }t=1/100$ and $\gamma =1$.

Additionally, we offer maximum absolute error $L_{\infty }$ when $\mathsf{\Delta }t=1/1000$ and $\gamma =1$ for u and $\omega$ in

Table 8. $L_{\infty }$ errors between the genuine, CNMFE and RDCNMFE solutions of u and $\omega$ when $\mathsf{\Delta }t=1/1000$ at T = 1.

		CNMFE Method		RDCNMFE Method
	Grid	${\displaystyle \left|\right|{\mathit{u}}^{\mathit{n}}-{\mathit{u}}_{\mathit{h}}^{\mathit{n}}{\left|\right|}_{{\mathit{L}}_{\infty }}}$	${\displaystyle \left|\right|{\mathit{\omega }}^{\mathit{n}}-{\mathit{\omega }}_{\mathit{h}}^{\mathit{n}}{\left|\right|}_{{\mathit{L}}_{\infty }}}$	${\displaystyle \left|\right|{\mathit{u}}^{\mathit{n}}-{\mathit{u}}_{\mathit{d}}^{\mathit{n}}{\left|\right|}_{{\mathit{L}}_{\infty }}}$	${\displaystyle \left|\right|{\mathit{\omega }}^{\mathit{n}}-{\mathit{\omega }}_{\mathit{d}}^{\mathit{n}}{\left|\right|}_{{\mathit{L}}_{\infty }}}$
$\gamma =1$	$10\times 10$	8.2452 × ${10}^{-5}$	2.3921 × ${10}^{-2}$	8.2452 × ${10}^{-5}$	2.3924 × ${10}^{-2}$
	$20\times 20$	6.0560 × ${10}^{-5}$	5.0950 × ${10}^{-3}$	6.0560 × ${10}^{-5}$	5.1000 × ${10}^{-3}$
	$40\times 40$	5.6503 × ${10}^{-5}$	1.6112 × ${10}^{-3}$	5.6503 × ${10}^{-5}$	1.6126 × ${10}^{-3}$
	$80\times 80$	5.5083 × ${10}^{-5}$	4.1443 × ${10}^{-4}$	5.5083 × ${10}^{-5}$	4.1564 × ${10}^{-4}$

. As can be seen from Table 8, the $L_{\infty }$ error obtained by the RDCNMFE method is not only quite close to that obtained by the CNMFE method, but even shows a slightly lower error level. This shows that the RDCNMFE method has some advantages over the CNMFE method in maintaining error control.

To provide additional evidence for the effectiveness of the RDCNMFE method, we recorded the errors and the CPU running time for computing the CNMFE and RDCNMFE solutions when $h=\sqrt{2}/80$ and $\mathsf{\Delta }t=1/100,1/1000$ at $T=1.0,1.5,2.0,2.5,3.0$, respectively. These results are obtained by solving Problems 3 and 4 and shown in

Table 9. Comparison of $L^2$ errors and CPU runtime of CNMFE and RDCNMFE solutions when $\mathsf{\Delta }t=1/100$.

	CNMFE Method			RDCNMFE Method
Real Time	${\displaystyle \left|\right|{\mathit{u}}^{\mathit{n}}-{\mathit{u}}_{\mathit{h}}^{\mathit{n}}\left|\right|}$	${\displaystyle \left|\right|{\mathit{\omega }}^{\mathit{n}}-{\mathit{\omega }}_{\mathit{h}}^{\mathit{n}}\left|\right|}$	CPU Runtime (s)	${\displaystyle \left|\right|{\mathit{u}}^{\mathit{n}}-{\mathit{u}}_{\mathit{d}}^{\mathit{n}}\left|\right|}$	${\displaystyle \left|\right|{\mathit{\omega }}^{\mathit{n}}-{\mathit{\omega }}_{\mathit{d}}^{\mathit{n}}\left|\right|}$	CPU Runtime (s)
$T=1.0$	2.7256 × ${10}^{-5}$	1.2863 × ${10}^{-3}$	224.368	2.7274 × ${10}^{-5}$	1.2901 × ${10}^{-3}$	4.410
$T=1.5$	2.5421 × ${10}^{-5}$	1.4650 × ${10}^{-3}$	260.357	4.7928 × ${10}^{-5}$	1.4653 × ${10}^{-3}$	4.545
$T=2.0$	1.7506 × ${10}^{-5}$	1.4287 × ${10}^{-3}$	285.453	1.7724 × ${10}^{-5}$	1.4523 × ${10}^{-3}$	4.635
$T=2.5$	1.1089 × ${10}^{-5}$	1.3743 × ${10}^{-3}$	311.536	1.5943 × ${10}^{-5}$	1.6888 × ${10}^{-3}$	4.678
$T=3.0$	6.8308 × ${10}^{-6}$	1.3343 × ${10}^{-3}$	342.801	6.8312 × ${10}^{-6}$	1.3403 × ${10}^{-3}$	4.919

and

Table 10. Comparison of $L^2$ errors and CPU runtime of CNMFE and RDCNMFE solutions when $\mathsf{\Delta }t=1/1000$.

	CNMFE Method			RDCNMFE Method
Real Time	${\displaystyle \left|\right|{\mathit{u}}^{\mathit{n}}-{\mathit{u}}_{\mathit{h}}^{\mathit{n}}\left|\right|}$	${\displaystyle \left|\right|{\mathit{\omega }}^{\mathit{n}}-{\mathit{\omega }}_{\mathit{h}}^{\mathit{n}}\left|\right|}$	CPU Runtime (s)	${\displaystyle \left|\right|{\mathit{u}}^{\mathit{n}}-{\mathit{u}}_{\mathit{d}}^{\mathit{n}}\left|\right|}$	${\displaystyle \left|\right|{\mathit{\omega }}^{\mathit{n}}-{\mathit{\omega }}_{\mathit{d}}^{\mathit{n}}\left|\right|}$	CPU Runtime (s)
$T=1.0$	2.5365 × ${10}^{-5}$	1.2424 × ${10}^{-3}$	774.149	2.5365 × ${10}^{-5}$	1.2424 × ${10}^{-3}$	5.136
$T=1.5$	2.4122 × ${10}^{-5}$	1.4376 × ${10}^{-3}$	1076.010	2.4122 × ${10}^{-5}$	1.4376 × ${10}^{-3}$	6.602
$T=2.0$	1.6704 × ${10}^{-5}$	1.4120 × ${10}^{-3}$	1378.655	1.6704 × ${10}^{-5}$	1.4120 × ${10}^{-3}$	7.932
$T=2.5$	1.0600 × ${10}^{-5}$	1.3642 × ${10}^{-3}$	1681.807	1.0600 × ${10}^{-5}$	1.3643 × ${10}^{-3}$	9.075
$T=3.0$	6.5339 × ${10}^{-6}$	1.3282 × ${10}^{-3}$	1969.729	6.5339 × ${10}^{-6}$	1.3303 × ${10}^{-3}$	11.476

.

By comparing the linear equation systems in Problems 3 and 4, it can be seen that the CNMFE scheme maintains the solution coefficient vectors $U_h$ and $W_h$ as M-dimensional column vectors at each time node, which directly leads to higher computational complexity. On the contrary, the RDCNMFE scheme greatly reduces the dimension of the solution coefficient vectors from M to $d(d\ll M)$ through POD technology. This method greatly reduces the amount of data processing, thereby significantly reducing the computational cost. As demonstrated in Table 9 and Table 10, with the increase in time, the CNMFE method with $2\times 6561$ degrees of freedom requires considerable CPU runtime at each time node. When $\mathsf{\Delta }t=1/100$, the running time increases by approximately 30 s for every additional $0.5$ s, and when $\mathsf{\Delta }t=1/1000$, the running time increases by approximately 300 s for every additional $0.5$ s. While performing at the same time node, the RDCNMFE method, with only $2\times 6$ degrees of freedom, requires very minimal CPU runtime. When $\mathsf{\Delta }t=1/100$, there is an increase in running time of only about $0.1$ s for every additional $0.5$ s, and when $\mathsf{\Delta }t=1/1000$, there is an increase in running time of only about 1 s for every additional $0.5$ s. The calculation time of the CNMFE method is approximately 70 times greater than that of the RDCNMFE method when $T=3.0$. This experiment demonstrates that the RDCNMFE method significantly reduces CPU running time. Furthermore, the results from Table 9 and Table 10 indicate that the RDCNMFE method produces slightly less $L^2$ errors compared to the CNMFE method.

According to the simulation results of the above two numerical examples, it is clear that the RDCNMFE method surpasses the CNMFE method, proving it as an effective numerical method for solving the nonlinear EFK equation.

5. Conclusions

In this study, we have focused on the reduced-dimension method of Crank–Nicolson mixed finite element solution coefficient vectors for 2D nonlinear EFK equations In order to give a full demonstration of the effectiveness of the reduced-dimension method, we have established the CNMFE scheme of the 2D nonlinear EFK equation and discussed the uniqueness, stability, and convergence of the CNMFE solution. Then, we have generated the POD basis based on the initial K CNMFE solutions, constructed the RDCNMFE matrix model, and employed the traditional finite element techniques to analyze the uniqueness, stability and error estimation of the RDCNMFE solution. Additionally, numerical experiments have been conducted to comparatively evaluate the performance of the two methods. Compared with the CNMFE method, the RDCNMFE method has fewer degrees of freedom at each time node. This characteristic enables the RDCNMFE method to greatly reduce computational load, running time, and error accumulation. In particular, this paper has simplified the actual calculation by linearizing the nonlinear term without iterative numerical calculation. Therefore, the RDCNMFE method constitutes a novel numerical technique for tackling nonlinear PDEs.

However, the methods presented in this paper exhibit certain limitations when applied to solving the EFK equation. The fourth-order derivative coefficient $\gamma$ significantly impacts the convergence and accuracy of the numerical solutions. Notably, the current methods are not suitable for situations where the fourth-order coefficient $\gamma$ takes on smaller values $(\gamma <1)$. In future research, we will endeavor to address this issue. And we intend to extend the scope of application of this method to PDEs with variable coefficients (in this paper, the coefficient $\gamma$ is a constant).

Practical problems usually have no exact solutions. In order to assess the viability of the proposed method, we initially discuss its convergence using problems that possess exact solutions. This step verifies the feasibility of the method. Once feasibility is confirmed, we can confidently apply the method to practical problems without exact solutions.