Unconditional Stability and Optimal Error Estimates of Euler Implicit/Explicit-SAV Scheme for the Navier–Stokes Equations

Abstract

The unconditional stability and convergence analysis of the Euler implicit/explicit scheme with finite element discretization are studied for the incompressible time-dependent Navier–Stokes equations based on the scalar auxiliary variable approach. Firstly, a corresponding equivalent system of the Navier–Stokes equations with three variables is formulated, the stable finite element spaces are adopted to approximate these variables and the corresponding theoretical analysis results are provided. Secondly, a fully discrete scheme based on the backward Euler method is developed, the temporal treatment is based on the Euler implicit/explicit scheme, which is implicit for the linear terms and explicit for the nonlinear term. Hence, a constant coefficient algebraic system is formed and it can be solved efficiently. The discrete unconditional energy dissipation and stability of numerical solutions in various norms are established with any restriction on the time step, optimal error estimates are also provided. Finally, some numerical results are provided to illustrate the performances of the considered numerical scheme.

1 Introduction

Suppose that $\varOmega \in {\mathbb {R}}^2$ is a bounded open set with Lipschitz continuous boundary $\partial \varOmega$. Consider the following time-dependent incompressible Navier–Stokes equations in $\varOmega \times (0,T]$

$$\begin{aligned} \left\{ \begin{aligned}&u_t - \nu \varDelta u +\nabla p + (u \cdot \nabla ) u = f,\\&\nabla \cdot u =0, \end{aligned} \right. \end{aligned}$$

(1)

subject to the homogeneous Dirichlet boundary condition

$$\begin{aligned} u|_{\partial \varOmega }=0 \end{aligned}$$

(2)

and the initial condition

$$\begin{aligned} u|_{t=0}=u_0(x). \end{aligned}$$

(3)

In these equations $u=u(x,t)$ and $p=p(x,t)$ are the velocity and pressure of the fluid at the space-time location $(x,t)\in \varOmega \times (0,T]$, respectively. The parameter $\nu =\frac{1}{Re}>0$ is the kinematic viscosity, Re is the Reynold number, $f=f(x,t)$ is the body force, T is the final time and $u_0(x)$ is the initial data of the fluid.

As a classical incompressible fluid model, the Navier-Stokes equations (1)–(3) have been widely used in the field of the computational fluid dynamics [5, 6, 34]. Many important models are formed by coupling the Navier–Stokes equations with other equations, for example, with the Maxwell equation form the MHD equations, with the nonlinear heat equation form the Boussinesq equations and so on. Due to the nonlinear and incompressible properties, to find the exact solutions of the Navier–Stokes equations becomes a difficult work. Therefore, numerous works have devoted to the developments of efficient numerical schemes for the Navier–Stokes equations (1)–(3), such as the nonlinear Galerkin method [1, 2], the projection method [27,28,29], two grid method [12, 17] and so on. In these numerical schemes, the treatment of nonlinear term is one of the key points. Generally speaking, the implicit and semi-implicit schemes are unconditional stable, but we need to treat a variable coefficient algebraic discrete problem, for example, [5, 16, 35] for the Navier–Stokes equations, [10, 23] for the MHD equations, [18, 25] for nonlinear parabolic problems. The implicit/explicit scheme is an attractive approach to deal with the nonlinear problem, because we just need to treat the constant coefficient algebraic discrete system. Furthermore, the considered problem can be split into a series of linear subproblems, both computational size and storage requirements are reduced. We only mention [7] for the dissipative evolution equations, [13, 24, 33] for the Navier–Stokes equations, [32] for the Cahn–Hilliard equations and the references therein. However, the stability of numerical solutions in implicit/explicit scheme holds under some restrictions on the time steps [9, 11, 33]. Namely, the following condition must be satisfied

$$\begin{aligned} \varDelta t\le C, \end{aligned}$$

(4)

where $\varDelta t$ is the time step, $C>0$ is a general constant, independents of $\varDelta t$ and mesh size h.

The scalar auxiliary variable (SAV) method was developed by Shen and his co-authors [20, 21, 30, 31] for the gradient flow. This method can be considered as an extension and improvement of the invariant energy quadratization (IEQ) method given in [36, 37, 39]. The main advantages of the SAV scheme can be list as follows: (I) unconditional energy dissipation law holds, (II) decoupled equations with constant coefficients need to be solved at each time step, (III) numerical schemes up to second order are accurate. Hence, the SAV method has been used to treat the gradient flow [30, 31], phase field model [38] and the references therein, some important and interesting stability and convergence results were established. In recently, the SAV method was extended to solve the incompressible Navier–Stokes equations [22], unconditional energy dissipation of the backward Euler and BDF2 schemes were developed, with a series of numerical examples illustrating the performances of the considered numerical method combining the spectral method. However, the stability and convergence results of numerical solutions in these schemes were not given. Later, Li et al presented the error analysis of the SAV approach for the Navier–Stokes equations based on the finite difference method in [19,20,21].

The aim of this paper is to establish the rigorous unconditional stability and optimal error estimates of the Euler implicit/explicit-SAV finite element method for the Navier–Stokes equations. We firstly develop the equivalent formulation of the Navier–Stokes equations by introducing the scalar auxiliary variable, the convergence results with finite element discretization are recalled. Then, a fully discrete implicit/explicit SAV finite element scheme is developed, the energy dissipation of numerical scheme, unconditional stability and the optimal error estimates of numerical solutions are provided. Compared with [9, 11, 13, 19, 24, 33], the main features of this work contain: (I) The unconditional energy dissipation of numerical scheme is presented. (II) Unconditional stability results of numerical solutions in Euler implicit/explicit scheme are developed. (III) Optimal error estimates of numerical approximations are established.

The outline of this paper can be list as follows. Section 2 is devoted to recall some basic results of the Navier–Stokes equations and present the corresponding equivalent form in SAV approach with finite element discretization. Section 3 gives the fully discrete implicit/explicit SAV FEM for the Navier-Stokes equations, unconditional energy dissipation and stability results are established. Section 4 develops the optimal error estimates of numerical solutions. Some numerical results are presented in Sect. 5 to confirm the established theoretical findings, and illustrate the performances of the considered numerical schemes. Finally, a conclusion is given in Sect. 6.

2 Function Setting and the Galerkin Finite Element Method

2.1 Preliminary

Assume that $\varOmega \subset {\mathbb {R}}^2$ satisfies the addition stated in (A1) below. Standard Sobolev spaces and the corresponding norms are used. Denote $H^i(\varOmega )$ the function with square integrable distribution derivatives up to order i ($i=1,2$) over the domain $\varOmega$, $H^1_0(\varOmega )$ is the closed subspace of $H^1(\varOmega )$ consisting of the functions with zero trace on $\varOmega$. We equip the spaces $H^i(\varOmega )(i=1,2)$ with the norm $\Vert \cdot \Vert _i$, $L^i(\varOmega )$ with the norm $\Vert \cdot \Vert _0$ and inner product $(\cdot ,\cdot )$, $H^1_0(\varOmega )$ with the scalar product $(\nabla u,\nabla v)$ and norm $\Vert u\Vert _1=(\nabla u,\nabla u)^{1/2}$. Set

$$\begin{aligned} X= & {} H_0^1(\varOmega )^2,\ \ V=\{v\in X;\nabla \cdot v=0\},\ \ \ M=L_0^2(\varOmega )=\{q\in L^2(\varOmega );\int _{\varOmega }qdx=0\}, \\ Y= & {} L^2(\varOmega )^2,\ \ H=\{v\in Y;\text{ div }v=0,v\cdot n|_{\partial \varOmega }=0\},\ \ {\mathbb {R}}=\{\text{ the } \text{ space } \text{ of } \text{ real } \text{ numbers }\}. \end{aligned}$$

We refer readers to [6, 14, 34] for details on these spaces. We denote the Stokes operator by $A=-P\varDelta$, where P is $L^2$-orthogonal projection of Y onto H and the domain of A by $D(A)=H^2(\varOmega )^2\cap V$. As mentioned above, an additional assumption about the domain $\varOmega$ is needed (see [1, 16, 34]).

(A1). Assume that $\varOmega$ is smooth so that the unique solution $(v,q)\in X\times M$ of the steady Stokes problem

$$\begin{aligned} -\nu \varDelta v+\nabla q=g,\ \ \text{ div }\ v=0\ \ \ \ x\in \varOmega ,\ \ \ v|_{\partial \varOmega }=0, \end{aligned}$$

for any prescribed $f\in Y$, exists and satisfies

$$\begin{aligned} \Vert v\Vert _2+\Vert q\Vert _1\le C_1\Vert g\Vert _0, \end{aligned}$$

where $C_1>0$ is a generic constant depending on the data $\nu$ and $\varOmega$.

We remark that the validity of assumption (A1) is known (see [1, 6, 14, 34]) if $\partial \varOmega$ is of $C^2$ or if $\varOmega$ is a convex polygon in 2D. Furthermore, it is well known that (see [11, 14])

$$\begin{aligned} \Vert v\Vert _{H^2}\le C_1\Vert Av\Vert _0,\ \ \ \ v\in D(A). \end{aligned}$$

The following Poincáre inequalities hold

$$\begin{aligned} \Vert v\Vert _0^2\le \gamma _0\Vert v\Vert _1^2,\ \ \forall v\in X,\ \ \Vert v\Vert _1^2\le \gamma _0\Vert v\Vert _2^2\le \Vert Av\Vert _0^2,\ \ \forall v\in D(A), \end{aligned}$$

where $\gamma _0$ is a positive constant depending only on $\varOmega$.

Some assumptions about the prescribed data for problem (1) are needed [1, 10, 12, 16].

(A2). The initial data $u_0(x)$ and the body force f satisfy, for the positive constant $C_2$,

$$\begin{aligned} u_0\in H^2(\varOmega )^2\cap V,\ f\in L^{\infty }(0,T;H^1(\varOmega )^2) \ \ \text{ with }\ \Vert Au_0\Vert _0+\Vert f\Vert _1\le C_2. \end{aligned}$$

The continuous bilinear forms $a(\cdot ,\cdot )$ on $X\times X$ and $d(\cdot ,\cdot )$ on $X\times M$ are defined by

$$\begin{aligned} a(u,v)=\nu (\nabla u,\nabla v),\ \ d(v,q)=-(v,\nabla q)=(q,\text{ div }v)\ \ \forall \ u,v\in X,\ q\in M. \end{aligned}$$

Define the trilinear form $b(\cdot ,\cdot ,\cdot )$ on $X\times X\times X$ with $\nabla \cdot u=0$ by

$$\begin{aligned} b(u,v,w) =((u\cdot \nabla )v,w)+\frac{1}{2}((\nabla \cdot u)v,w) =\frac{1}{2}((u\cdot \nabla )v,w)-\frac{1}{2}((u\cdot \nabla )w,v). \end{aligned}$$

The following important estimates of the trilinear form $b(\cdot ,\cdot ,\cdot )$ can be found in [6, 9, 11, 13, 34] with $C_3$ is a positive constant depending on $\varOmega$

$$\begin{aligned}&b(u,v,v)=0, \ \ \ \ \ \ b(u,v,w)=-b(u,w,v), \ \ \ \ \ \ \ \forall \ u\in V,\ v,w\in X, \end{aligned}$$

(5)

$$\begin{aligned}&|b(u,v,w)|+|b(v,u,w)|+|b(w,u,v)| \nonumber \\&\quad \le \frac{C_3}{2}(\Vert u\Vert ^{1/2}_0\Vert u\Vert ^{1/2}_1\Vert v\Vert _1 +\Vert u\Vert _1\Vert v\Vert ^{1/2}_0\Vert v\Vert ^{1/2}_1)\Vert w\Vert _0^{1/2}\Vert w\Vert _1^{1/2},\ \ \forall \ u,v,w\in V, \end{aligned}$$

(6)

$$\begin{aligned}&|b(u,v,w)|+|b(v,u,w)|+|b(w,u,v)| \nonumber \\&\quad \le \frac{C_3}{2}(\Vert Av\Vert ^{1/2}_0\Vert v\Vert ^{1/2}_1\Vert u\Vert ^{1/2}_0\Vert u\Vert ^{1/2}_1 {+}\Vert Av\Vert ^{1/2}_0\Vert v\Vert ^{1/2}_0\Vert u\Vert _1)\Vert w\Vert _0,\ \ \forall \ u,v\in V,w\in X,\nonumber \\ \end{aligned}$$

(7)

$$\begin{aligned}&|b(u,v,w)|+|b(v,u,w)|+|b(w,u,v)| \nonumber \\&\quad \le \frac{C_3}{3}(\Vert u\Vert ^{1/2}_0\Vert Av\Vert ^{1/2}_0\nonumber \\&\qquad +\Vert v\Vert ^{1/2}_0\Vert Au\Vert ^{1/2}_0 +\Vert u\Vert ^{1/2}_1\Vert v\Vert ^{1/2}_1)\Vert Au\Vert ^{1/2}_0\Vert Av\Vert ^{1/2}_0\Vert w\Vert _{-1}, \ \ \forall \ u,v,w\in V.\ \ \ \ \ \ \end{aligned}$$

(8)

With above notations, the variational formulation of problem (1) reads as: for all $(v,q)\in X\times M$, find $(u,p)\in L^{\infty }(0,T;H)\cap L^2(0,T;X)\times L^2(0,T;M)$ such that

$$\begin{aligned} \left\{ \begin{array}{lcl} (u_t,v)+a(u,v)-d(v,p)+d(u,q)+b(u,u,v)=(f,v),\\ u(0)=u_0. \end{array} \right. \end{aligned}$$

(9)

The following regularity results can be obtained with simple modification to the argument given in [8, 14, 15] under the compatibility conditions.

Theorem 2.1

Under the assumptions (A1) and (A2), problem (9) admits a unique solution (u, p) satisfying the following estimates for all $t\in [0,T]$

$$\begin{aligned} \Vert \nabla u(t)\Vert _0+\Vert u_t\Vert _0+\Vert Au\Vert _0+\Vert p\Vert _1 + \Big (\int _0^t(\Vert u_{tt}\Vert _0^2+\Vert Au_t\Vert _0^2+\Vert p_t\Vert _1^2)ds\Big )^{1/2}\le C_4, \end{aligned}$$

where $C_4$ is a generic positive constant depending on the data $\nu ,\varOmega , C_1,C_2,C_3$ and T, which may take different values at its different places.

2.2 Galerkin Finite Element Method

From now on, let $0<h<1$ be a real positive parameter. The finite element subspace $(X_h,M_h)$ of (X, M) is characterized by ${\mathcal {T}}_h={\mathcal {T}}_h(\varOmega )$, a partitioning of ${\overline{\varOmega }}$ into triangles K or quadrilaterals K, assumed to be uniformly regular as $h\rightarrow 0$. For further details, we can refer to [6, 34]. Define the subspace $V_h$ of $X_h$ given by

$$\begin{aligned} V_h=\{v_h\in X_h:d(v_h,q_h)=0,\ \ \forall q_h\in M_h\}. \end{aligned}$$

Set $P_h:Y\rightarrow V_h$ denotes the $L^2$-orthogonal projection, it can be defined by

$$\begin{aligned} (P_hu,v_h)=(u,v_h),\ \ \forall \ u\in Y,\ v_h\in V_h. \end{aligned}$$

With above statements, a discrete analogue $A_h=-P_h\varDelta _h$ of the Stokes operator $A=-P\varDelta$ is defined by $(-\varDelta _hu_h,v_h)=(\nabla u_h,\nabla v_h)$ for all $u_h,v_h\in X_h$. The restriction of $A_h$ to $V_h$ is invertible, with the inverse $A_h^{-1}$. The discrete operator $A_h$ was first introduced in [14] to analyze and obtain the optimal estimates for the transient Navier–Stokes equations.

We set that the finite element spaces $X_h$ and $M_h$ approximating the velocity and pressure are assumed to satisfy the following discrete inf-sup condition: There exists a positive constant $\beta >0$ independent of h, such that

$$\begin{aligned} d(v_h,q_h)\ge \beta \Vert v_h\Vert _1\Vert q_h\Vert _0,\ \ \ \ \ \ \forall \ v_h\in X_h, q_h\in M_h. \end{aligned}$$

(10)

We give an example of the spaces $X_h$ and $M_h$ such that the condition (10) is satisfied. For any nonnegative integer l, we denote by $P_l(K)$ the space of polynomials of degrees less than or equal to l on K.

Example (The MINI element + piecewise constant space)

$$\begin{aligned}&X_h=\{v_h\in C^0(\varOmega )^2\cap X; v_h|_K\in P_1(K)^2\oplus \text{ span }\{27\lambda _1\lambda _2\lambda _3\},\ \forall \ K\in {\mathcal {T}}_h\},\\&M_h=\{q_h\in C^0(\varOmega )\cap M; q_h|_K\in P_1(K),\ \forall \ K\in {\mathcal {T}}_h\},\\&{\mathbb {R}}_h=\{s_h\in C^0(\varOmega )\cap R; s_h|_K\in P_0(K),\ \forall \ K\in {\mathcal {T}}_h\}. \end{aligned}$$

where $\lambda _1,\lambda _2$ and $\lambda _3$ are the barycenter coordinates of the reference element. Other example, such as the Taylor-Hood element for $X_h$ and $M_h$, we can refer to [6, 34].

The Galerkin finite element method for problem (9) is defined as follows: Find $(u_h,p_h)\in L^2(0,T;X_h) \times L^2(0,T;M_h)$, for all $t\in (0,T]$ and $(v_h,q_h)\in X_h\times M_h$, such that

$$\begin{aligned} \left\{ \begin{array}{llll} (u_{ht},v_h)+a(u_h,v_h)-d(v_h,p_h)+d(u_h,q_h)+b(u_h,u_h,v_h)=(f,v_h),\\ u_h(0)=u_{0h}=P_hu_0. \end{array} \right. \end{aligned}$$

(11)

Theorem 2.2

(See [6, 8, 10]) Under the assumptions (A1)-(A2), problem (11) admits a unique solution $(u_h,p_h)$. Furthermore, for all $t\in (0,T]$ it holds that

$$\begin{aligned}&\Vert \nabla u_h\Vert _0+\Vert u_{ht}\Vert _0+\Vert A_hu_h\Vert _0 +\Vert \nabla u_{ht}\Vert _0 + \Big (\int _0^t(\Vert u_{htt}\Vert _0^2+\Vert A_hu_{ht}\Vert _0^2)ds\Big )^{1/2}\le C_4,\\&\Vert u-u_h\Vert _0+h(\Vert \nabla (u-u_h)\Vert _0+\Vert p-p_h\Vert _0)\le C_4h^2. \end{aligned}$$

2.3 The Scalar Auxiliary Variable Approach

This subsection develops the equivalent form of the Navier–Stokes equations based on the scalar auxiliary variable approach. The finite element discretization is considered and the corresponding stability and convergence analysis results are also presented.

Firstly, we introduce the following scaler energy variable

$$\begin{aligned} E(t)=C_0+\frac{1}{2}\Vert u(x,t)\Vert _0^2, \end{aligned}$$

(12)

where $C_0>0$ is a fixed constant. Denote

$$\begin{aligned} S(t)=\sqrt{E(t)}. \end{aligned}$$

(13)

It follows from (13) and $\nabla \cdot u=0$ that problem (1) can be transformed into

$$\begin{aligned} \left\{ \begin{array}{lcl} u_t - \nu \varDelta u +\nabla p +\frac{S(t)}{\sqrt{E(t)}}(u \cdot \nabla ) u= f,\\ \nabla \cdot u =0,\\ S_t =\frac{1}{2S(t)}\int _{\varOmega }(u_t+\frac{S(t)}{\sqrt{E(t)}}(u\cdot \nabla )u)\cdot udx. \end{array} \right. \end{aligned}$$

(14)

The Galerkin finite element method for (14) reads as: for all $(v_h,q_h,s_h)\in X_h\times M_h\times {\mathbb {R}}_h$, find $(u_h,p_h,S_h)\in X_h\times M_h\times {\mathbb {R}}_h$ with $u_h(0)=P_hu_0$ and $S_h(0)=\sqrt{C_0+\frac{1}{2}\Vert u_h(0)\Vert _0^2}$, such that

$$\begin{aligned} \left\{ \begin{array}{lcl} (u_{ht},v_h)+a(u_h,v_h)-d(v_h,p_h)+d(u_h,q_h)+\frac{S_h(t)}{\sqrt{E_h(t)}}b(u_h,u_h,v_h)=(f,v_h),\\ (S_{ht},s_h)= \frac{1}{2S_h(t)}\Big ((u_{ht},u_h)+\frac{S_h(t)}{\sqrt{E_h(t)}}b(u_h,u_h,u_h),s_h\Big ). \end{array} \right. \end{aligned}$$

(15)

From (15), we can find the numerical solutions $u_h,p_h$ and $S_h$, then the discrete scaler energy variable $E_h$ is obtained. Furthermore, from (12) it holds

$$\begin{aligned} E_h(t)=C_0+\frac{1}{2}\Vert u_h(t)\Vert _0^2. \end{aligned}$$

(16)

From the second equation of (15) and (5), one finds that

$$\begin{aligned} S_{ht}=\frac{1}{2S_h}(u_{ht},u_h)\ \ \ \ \text{ with }\ \ S_h(0)=\sqrt{C_0+\frac{1}{2}\Vert u_h(0)\Vert _0^2}. \end{aligned}$$

This is an ODE, it is easy to obtain that

$$\begin{aligned} S_h(t)=\sqrt{C_0+\frac{1}{2}\Vert u_h(t)\Vert _0^2}. \end{aligned}$$

(17)

Combining (16) and (17), problem (15) transforms into (11), by Theorem 2.2, we have

$$\begin{aligned} |S-S_h|= & {} \Big |\sqrt{C_0+\frac{1}{2}\Vert u\Vert _0^2}-\sqrt{C_0+\frac{1}{2}\Vert u_h\Vert _0^2}\Big | =\frac{1}{2}\frac{|\Vert u\Vert _0^2-\Vert u_h\Vert _0^2|}{\sqrt{C_0+\frac{1}{2}\Vert u\Vert _0^2} +\sqrt{C_0+\frac{1}{2}\Vert u_h\Vert _0^2}} \nonumber \\\le & {} \frac{1}{4\sqrt{C_0}}(\Vert u\Vert _0+\Vert u_h\Vert _0)\Vert u-u_h\Vert _0\le C_4 h^2, \end{aligned}$$

(18)

$$\begin{aligned} |S_{ht}|= & {} |\frac{1}{2S_h}(u_{ht},u_h)| =\frac{1}{2\sqrt{C_0+\frac{1}{2}\Vert u_h(t)\Vert _0^2}}|(u_{ht},u_h)|\nonumber \\\le & {} \frac{1}{2\sqrt{C_0}}\Vert u_{ht}\Vert _0\Vert u_h\Vert _0 \le C_4,\ \ \ \ \ \ \end{aligned}$$

(19)

$$\begin{aligned} \int _0^t|S_{htt}|^2ds\le & {} \frac{1}{4C_0}\int _0^t\Big (\Vert u_{htt}\Vert ^2_0\Vert u_h\Vert ^2_0+\Vert u_{ht}\Vert ^4_0 +\frac{1}{4C^2_0}\Vert u_{ht}\Vert ^4_0\Vert u_h\Vert ^4_0 \Big )ds \le C_4. \end{aligned}$$

(20)

3 Fully Discrete Euler Implicit/Explicit-SAV Method

In this section we consider the time discretization of the Galerkin finite element method with the scalar auxiliary variable. We choose an integer N and define the time step $\varDelta t=\frac{T}{N}$ and the discrete times $t_n=n\varDelta t,n=0,1,..,N$. The Euler implicit/explicit scheme applied to the spatially discrete problem (15) consists of determining functions $(u_h^{n+1},p_h^{n+1},S_h^{n+1})\in X_h\times M_h\times {\mathbb {R}}_h$ as solutions of the recursive linear equations

$$\begin{aligned} \left\{ \begin{array}{lcl} (d_tu^{n+1}_h,v_h)+a(u^{n+1}_h,v_h)-d(v_h,p^{n+1}_h)+\frac{S^n_h}{\sqrt{E^n_h}} b(u^n_h,u^n_h,v_h)=(f^{n+1},v_h),\\ d(u^{n+1}_h,q_h)=0,\\ (d_tS^{n+1}_h,s_h)= \frac{1}{2S^{n+1}_h}\Big ((d_tu^{n+1}_h,u^{n+1}_h)+\frac{S^n_h}{\sqrt{E^n_h}} b(u^n_h,u^n_h,u^{n+1}_h),s_h\Big ), \end{array} \right. \end{aligned}$$

(21)

with $d_t\varphi _h^{n+1}=\frac{\varphi _h^{n+1}-\varphi _h^n}{\varDelta t},\ \varphi ^n_h$ takes $u^n_h$ or $S^n_h$, $u_h^0=P_hu_0,f^{n+1}=f(t_{n+1}), S_h^0=\sqrt{C_0+\frac{1}{2}\Vert u^0_h\Vert _0^2}$.

Based on the definition (16) of $E_h$, we have

$$\begin{aligned} E_h^n=C_0+\frac{1}{2}\Vert u_h^n\Vert _0^2\ge C_0. \end{aligned}$$

(22)

Theorem 3.1

With $f=0$, scheme (21) is unconditional energy dissipation in the sense that

$$\begin{aligned} |S_h^{n+1}|^2+|S_h^{n+1}-S_h^n|^2\le |S_h^n|^2. \end{aligned}$$

Proof

Choosing $v_h=u_h^{n+1}\varDelta t,\ q_h=p_h^{n+1}\varDelta t$ and $s_h=2S_h^{n+1}\varDelta t$ in (21), adding them together, using the fact $2(a-b,a)=|a|^2-|b|^2+|a-b|^2$, we have

$$\begin{aligned}&|S_h^{n+1}|^2-|S_h^n|^2+|S_h^{n+1}-S_h^n|^2 \\&\quad =\varDelta t(d_tu_h^{n+1}+\frac{S_h^n}{\sqrt{E_h^n}}(u_h^n\cdot \nabla )u_h^n, u_h^{n+1}) =-\varDelta t\nu \Vert \nabla u_h^{n+1}\Vert _0^2\le 0. \end{aligned}$$

Then we finish the proof. $\square$

Thanks to Theorem 3.1, we know that the total discrete energy of the Navier–Stokes equations in Euler implicit/explicit-SAV scheme (21) is dissipative.

The following classical discrete Gronwall lemma can be found in [26, 27].

Lemma 3.2

Let c and $a_k,b_k,c_k,d_k$, for integers $k\ge 0$, be non-negative numbers such that

$$\begin{aligned} a_n+\varDelta t\sum _{k=0}^{n}b_k\le \varDelta t\sum _{k=0}^{n-1}d_ka_k+\varDelta t\sum _{k=0}^{n-1}c_k+c,\ \ \forall \ n\ge 1. \end{aligned}$$

Then

$$\begin{aligned} a_n+\varDelta t\sum _{k=0}^{n}b_k\le \text{ exp }(\varDelta t\sum _{k=0}^{n-1}d_k)(\varDelta t\sum _{k=0}^{n-1}c_k+c),\ \ \forall \ n\ge 1. \end{aligned}$$

Theorem 3.3

Under the Assumptions (A1)-(A2), $\forall 0\le m\le N$, for problem (21) it holds

$$\begin{aligned} |S_h^{m+1}|^2+\sum _{n=0}^{m}|S_h^{n+1}-S_h^n|^2+\frac{1}{2}\varDelta t\nu \sum _{n=0}^m\Vert \nabla u_h^{n+1}\Vert _0^2 \le M_0, \end{aligned}$$

(23)

$$\begin{aligned} \Vert u_h^{m+1}\Vert _0^2+\sum _{n=0}^m\Vert u_h^{n+1}-u^n\Vert _0^2+\varDelta t\nu \sum _{n=0}^m\Vert \nabla u_h^{n+1}\Vert _0^2 \le M_1, \end{aligned}$$

(24)

where $M_0{=}|S_h^0|^2{+}\frac{\varDelta t}{2\nu }\sum _{n=0}^m\Vert f^{n+1}\Vert _0^2, M_1{=}\Big (\Vert u_h^0\Vert _0^2{+}\frac{2\gamma _0}{\nu }\varDelta t\sum _{n=0}^m\Vert f^{n+1}\Vert _0^2\Big ) \exp \Big (\frac{4C_3^2M_0^2}{\nu ^2C_0}\Big )$.

Proof

Taking $v_h=u_h^{n+1}\varDelta t,q_h=p_h^{n+1}\varDelta t$ and $s_h=2S_h^{n+1}\varDelta t$ in problem (21), adding the resulting equations together, we obtain

$$\begin{aligned} |S_h^{n+1}|^2-|S_h^n|^2+|S_h^{n+1}-S_h^n|^2+\varDelta t\nu \Vert \nabla u_h^{n+1}\Vert _0^2= & {} (f^{n+1},u_h^{n+1})\varDelta t \\\le & {} \frac{\varDelta t}{2\nu }\Vert f^{n+1}\Vert ^2_0+\frac{\nu }{2}\varDelta t\Vert \nabla u_h^{n+1}\Vert _0^2. \end{aligned}$$

Eliminating the last term and summing from $n=0$ to m, we obtain (23).

Choosing $v_h=2u_h^{n+1}\varDelta t,q_h=2p_h^{n+1}\varDelta t$ in problem (21), one finds

$$\begin{aligned}&\Vert u_h^{n+1}\Vert _0^2-\Vert u_h^n\Vert _0^2 +\Vert u_h^{n+1}-u_h^n\Vert _0^2 +2\varDelta t\nu \Vert \nabla u_h^{n+1}\Vert _0^2 +2\varDelta t\frac{S_h^n}{\sqrt{E_h^n}}b(u_h^n,u_h^n,u_h^{n+1}) \nonumber \\&\quad =2\varDelta t(f^{n+1},u_h^{n+1}). \end{aligned}$$

(25)

We can treat the trilinear term and right-hand side term as follows

$$\begin{aligned} |2\varDelta t\frac{S_h^n}{\sqrt{E_h^n}}b(u_h^n,u_h^n,u_h^{n+1})|\le & {} 2C_3\varDelta t\frac{|S_h^n|}{\sqrt{E_h^n}}\Vert u_h^n\Vert _{L^4}\Vert \nabla u_h^{n+1}\Vert _{L^2}\Vert u_h^n\Vert _{L^4} \\\le & {} 2C_3\varDelta t\frac{|S_h^n|}{\sqrt{E_h^n}}\Vert u_h^n\Vert _0\Vert \nabla u_h^n\Vert _0\Vert \nabla u_h^{n+1}\Vert _0 \\\le & {} \frac{\nu }{2}\varDelta t\Vert \nabla u_h^{n+1}\Vert _0^2+\frac{2C_3^2\varDelta t}{C_0\nu }|S_h^n|^2\Vert u_h^n\Vert ^2_0\Vert \nabla u_h^n\Vert ^2_0, \\ |2\varDelta t(f^{n+1},u_h^{n+1})|\le & {} 2\varDelta t\Vert f^{n+1}\Vert _0\Vert u_h^{n+1}\Vert _0 \le \frac{\nu }{2}\varDelta t\Vert \nabla u_h^{n+1}\Vert _0^2+\frac{2\gamma _0}{\nu }\varDelta t\Vert f^{n+1}\Vert _0^2. \end{aligned}$$

Combining above inequalities with (25), summing from $n=0$ to m and using (22), one finds

$$\begin{aligned}&\Vert u_h^{m+1}\Vert _0^2+\sum _{n=0}^m\Vert u_h^{n+1}-u_h^n\Vert _0^2 +\varDelta t\nu \sum _{n=0}^m\Vert \nabla u_h^{n+1}\Vert _0^2 \\&\quad \le \Vert u_h^0\Vert _0^2+\frac{2\gamma _0}{\nu }\varDelta t\sum _{n=0}^m\Vert f^{n+1}\Vert _0^2+ \frac{2C_3^2}{C_0\nu }\varDelta t\sum _{n=0}^m|S_h^n|^2\Vert \nabla u_h^n\Vert ^2_0\Vert u_h^n\Vert ^2_0. \end{aligned}$$

With the help of (23) and Lemma 3.2, we complete the proof (24). $\square$

Theorem 3.4

Under the Assumptions (A1)-(A2), $\forall 0\le m\le N$, for problem (21) it holds

$$\begin{aligned} \Vert \nabla u_h^{m+1}\Vert _0^2+\frac{\nu }{2}\varDelta t\Vert A_hu_h^{m+1}\Vert _0^2 +\sum _{n=0}^m\Vert \nabla (u_h^{n+1}-u_h^n)\Vert _0^2+\frac{\nu }{2}\varDelta t\sum _{n=0}^m\Vert A_hu_h^{n+1}\Vert _0^2 \le M_2, \end{aligned}$$

where $M_2=(\Vert \nabla u_h^0\Vert _0^2+\frac{\nu }{2}\varDelta t\Vert A_hu_h^0\Vert _0^2 +\frac{2\varDelta t}{\nu }\sum _{n=0}^m\Vert f^{n+1}\Vert _0^2) \exp \Big (\frac{4C_3^4M_0^3M_1}{\nu ^4C^2_0}\Big )$.

Proof

It follows from $v_h=-2A_hu_h^{n+1}\varDelta t\in V_h,q_h=0$ in the first equation of (21) that

$$\begin{aligned}&\Vert \nabla u_h^{n+1}\Vert _0^2-\Vert \nabla u_h^n\Vert _0^2+\Vert \nabla (u_h^{n+1}-u_h^n)\Vert _0^2 +2\varDelta t\nu \Vert A_hu_h^{n+1}\Vert _0^2 \nonumber \\&\quad =2\varDelta t\frac{S_h^n}{\sqrt{E_h^n}}b(u_h^n,u_h^n,A_hu_h^{n+1}) -2\varDelta t(f^{n+1},A_hu_h^{n+1}). \end{aligned}$$

(26)

For the right-hand side terms, thanks to (22) and the Cauchy inequality, we have

$$\begin{aligned}&\left| 2\varDelta t\frac{S_h^n}{\sqrt{E_h^n}}b(u_h^n,u_h^n,A_hu_h^{n+1})\right| \\&\quad \le 2C_3\varDelta t\frac{|S_h^n|}{\sqrt{E_h^n}}\Vert u_h^n\Vert ^{1/2}_0\Vert \nabla u_h^n\Vert _0\Vert A_hu_h^n\Vert ^{1/2}_0\Vert A_hu_h^{n+1}\Vert _0 \\&\quad \le \frac{\nu }{2}\varDelta t\Vert A_hu_h^{n+1}\Vert _0^2+\frac{2C_3^2}{\nu C_0}\varDelta t|S_h^n|^2\Vert u_h^n\Vert _0\Vert \nabla u_h^n\Vert ^2_0\Vert A_hu_h^n\Vert _0 \\&\quad \le \frac{\nu }{2}\varDelta t\Vert A_hu_h^{n+1}\Vert _0^2+\frac{\nu }{2}\varDelta t\Vert A_hu_h^n\Vert _0^2 +\frac{2C_3^4}{\nu ^3C_0^2}\varDelta t|S_h^n|^4\Vert u_h^n\Vert ^2_0\Vert \nabla u_h^n\Vert ^4_0, \\&|2\varDelta t(f^{n+1},A_hu_h^{n+1})| \\&\quad \le 2\varDelta t\Vert f^{n+1}\Vert _0\Vert A_hu_h^{n+1}\Vert _0 \le \frac{\nu }{2}\varDelta t\Vert A_hu_h^{n+1}\Vert _0^2+\frac{2\varDelta t}{\nu }\Vert f^{n+1}\Vert _0^2. \end{aligned}$$

Combining above inequalities with (26) and summing from $n=0$ to m, we get

$$\begin{aligned}&\Vert \nabla u_h^{m+1}\Vert _0^2+\frac{\nu }{2}\varDelta t\Vert A_hu_h^{m+1}\Vert _0^2 +\sum _{n=0}^m\Vert \nabla (u_h^{n+1}-u_h^n)\Vert _0^2 +\frac{\nu }{2}\varDelta t\sum _{n=0}^m\Vert A_hu_h^{n+1}\Vert _0^2 \\&\quad \le \Vert \nabla u_h^0\Vert _0^2+\frac{\nu }{2}\varDelta t\Vert A_hu_h^0\Vert _0^2 +\frac{2\varDelta t}{\nu }\sum _{n=0}^m\Vert f^{n+1}\Vert _0^2 \\&\qquad +\varDelta t\sum _{n=0}^m\frac{2C_3^4}{\nu ^3C_0^2}|S_h^n|^4\Vert u_h^n\Vert ^2_0\Vert \nabla u_h^n\Vert ^2_0\Vert \nabla u_h^n\Vert ^2_0. \end{aligned}$$

With the help of (23), (24) and Lemma 3.2, we obtain the desired results. $\square$

Theorem 3.5

Under the Assumptions (A1)-(A2), $\forall 0\le m\le N$, for problem (21) it holds

$$\begin{aligned} \Vert A_hu_h^{m+1}\Vert _0^2+\sum _{n=0}^m\Vert A_h(u_h^{n+1}-u_h^n)\Vert _0^2+\varDelta t\nu \sum _{n=0}^m\Vert A_h^{\frac{3}{2}}u_h^{n+1}\Vert _0^2 \le M_3, \end{aligned}$$

where $M_3=(\Vert A_hu_h^0\Vert _0^2 +\frac{2\varDelta t}{\nu }\sum _{n=0}^m\Vert f^{n+1}\Vert _1^2) \exp \Big (\frac{4C_3^2M_0M_2}{\nu ^2C_0}\Big )$.

Proof

Taking $v_h=2A_h^2u_h^{n+1}\varDelta t\in V_h,q_h=0$ in the first equation of (21), one finds that

$$\begin{aligned}&\Vert A_hu_h^{n+1}\Vert _0^2-\Vert A_hu_h^n\Vert _0^2+\Vert A_h(u_h^{n+1}-u_h^n)\Vert _0^2 +2\varDelta t\nu \Vert A_h^{\frac{3}{2}}u_h^{n+1}\Vert _0^2 \nonumber \\&\quad =2\varDelta t(f^{n+1},A_h^2u_h^{n+1}) -2\varDelta t\frac{S_h^n}{\sqrt{E_h^n}}b(u_h^n,u_h^n,A_h^2u_h^{n+1}). \end{aligned}$$

(27)

For the right-hand side terms, by (8), (22) and the Cauchy inequality, we have

$$\begin{aligned}&\left| 2\varDelta t\frac{S_h^n}{\sqrt{E_h^n}}b(u_h^n,u_h^n,A_h^2u_h^{n+1})\right| \\&\quad \le \frac{4C_3}{3}\varDelta t\frac{|S_h^n|}{\sqrt{E_h^n}} \Big (\Vert u_h^n\Vert ^{1/2}_0\Vert A_hu_h^n\Vert ^{3/2}_0 +\Vert \nabla u_h^n\Vert _0\Vert A_hu_h^n\Vert _0 \Big )\Vert A_h^2u_h^{n+1}\Vert _{-1} \\&\quad \le \frac{\nu }{2}\varDelta t\Vert A_h^{\frac{3}{2}}u_h^{n+1}\Vert _0^2 +\frac{C_3^2}{\nu C_0}\varDelta t|S_h^n|^2 (\Vert u_h^n\Vert _0\Vert A_hu_h^n\Vert _0+\Vert \nabla u_h^n\Vert _0^2) \Vert A_hu_h^n\Vert _0^2, \\&|2\varDelta t(f^{n+1},A_h^2u_h^{n+1})| \\&\quad \le 2\varDelta t\Vert f^{n+1}\Vert _1\Vert A_h^2u_h^{n+1}\Vert _{-1} \le \frac{\nu }{2}\varDelta t\Vert A_h^{\frac{3}{2}}u_h^{n+1}\Vert _0^2+\frac{2\varDelta t}{\nu }\Vert f^{n+1}\Vert _1^2. \end{aligned}$$

Combining above inequalities with (27) and summing from $n=0$ to m we obtain

$$\begin{aligned}&\Vert A_hu_h^{m+1}\Vert _0^2+\sum _{n=0}^m\Vert A_h(u_h^{n+1}-u_h^n)\Vert _0^2 +\varDelta t\nu \sum _{n=0}^m\Vert A_h^{\frac{3}{2}}u_h^{n+1}\Vert _0^2 \\&\quad \le \Vert A_hu_h^0\Vert _0^2+\frac{2\varDelta t}{\nu }\sum _{n=0}^m\Vert f^{n+1}\Vert _1^2 \\&\qquad +\frac{C_3^2}{\nu C_0}\varDelta t\sum _{n=0}^m|S_h^n|^2(\Vert u_h^n\Vert _0\Vert A_hu_h^n\Vert _0+\Vert \nabla u_h^n\Vert ^2_0)\Vert A_hu_h^n\Vert _0^2. \end{aligned}$$

With the help of (23), Theorem 3.4 and Lemma 3.2, we finish the proof. $\square$

4 Error Estimates of the Euler Implicit/Explicit-SAV Scheme

This section is devoted to establish the convergence results of fully discrete implicit/explicit-SAV finite element scheme (21). Firstly, we discrete the Navier–Stokes equations (15) on $n+1$th time level to obtain

$$\begin{aligned} \left\{ \begin{array}{lllll} (d_tu_h(t_{n+1}),v_h){+}a(u_h(t_{n+1}),v_h){-}d(p_h(t_{n+1}),v_h) {+}\frac{S_h(t_{n+1})}{\sqrt{E_h(t_{n+1})}}b(u_h(t_{n+1}),u_h(t_{n+1}),v_h) \\ +d(u_h(t_{n+1}),q_h) = (f^{n+1},v_h)+(\frac{1}{\varDelta t}\int _{t_n}^{t_{n+1}}(t_n-t)u_{htt}dt,v_h), \\ (d_tS_h(t_{n+1}),s_h) =\Big (\frac{1}{2S_h(t_{n+1})}(d_tu_h(t_{n+1}) +\frac{1}{\varDelta t}\int _{t_n}^{t_{n+1}}(t-t_n)u_{htt}dt,u_h(t_{n+1})),s_h \Big ) \\ +(\frac{1}{2\sqrt{E_h(t_{n+1})}}b(u_h(t_{n+1}),u_h(t_{n+1}),u_h(t_{n+1})),s_h) +(\frac{1}{\varDelta t}\int _{t_n}^{t_{n+1}}(t_n-t)S_{htt}dt,s_h). \end{array} \right. \end{aligned}$$

(28)

Denote the errors

$$\begin{aligned} e_u^n=u_h(t_n)-u_h^n,\ \ \ e_p^n=p_h(t_n)-p_h^n,\ \ \ \ e_S^n=S_h(t_n)-S_h^n. \end{aligned}$$

The following error equations can be obtained by combining (21) with (28)

$$\begin{aligned} \left\{ \begin{array}{lllll} (d_te_u^{n+1},v_h)+a(e_u^{n+1},v_h)-d(e_p^{n+1},v_h) +\frac{S_h(t_{n+1})}{\sqrt{E_h(t_{n+1})}}b(u_h(t_{n+1}),u_h(t_{n+1}),v_h) \\ -\frac{S_h^n}{\sqrt{E_h^n}}b(u_h^n,u_h^n,v_h) +d(e_u^{n+1},q_h) = (\frac{1}{\varDelta t}\int _{t_n}^{t_{n+1}}(t_n-t)u_{htt}dt,v_h), \\ (d_te_S^{n+1},s_h) = \Big ( \frac{1}{2S_h(t_{n+1})} ( d_tu_h(t_{n+1}), u_h(t_{n+1}) ) -\frac{1}{2S_h^{n+1}} ( d_tu_h^{n+1}, u_h^{n+1} ),s_h \Big ) \\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ +\Big ( \frac{1}{2\sqrt{E_h(t_{n+1})}}b(u_h(t_{n+1}),u_h(t_{n+1}),u_h(t_{n+1})) -\frac{S_h^n}{2S_h^{n+1}\sqrt{E_h^n}} b(u_h^n,u_h^n,u_h^{n+1}),s_h \Big ) \\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ +\frac{1}{\varDelta t}\Big ( \int _{t_n}^{t_{n+1}}(t_n-t)S_{htt}dt +\frac{1}{2S_h(t_{n+1})} ( \int _{t_n}^{t_{n+1}}(t-t_n)u_{htt}dt, u_h(t_{n+1}) ), s_h\Big ), \end{array} \right. \end{aligned}$$

(29)

Lemma 4.1

Under the Assumptions (A1)–A2) and $e_u^0=0$, for all $m\ge 1$, we have

$$\begin{aligned}&\Vert e_u^{m+1}\Vert _0^2+\sum _{n=0}^m\Vert e_u^{n+1}-e_u^n\Vert _0^2 +\frac{\nu }{2}\varDelta t\Vert \nabla e_u^{m+1}\Vert _0^2 +\frac{\nu }{2}\varDelta t\sum _{n=0}^m\Vert \nabla e_u^{n+1}\Vert _0^2 \\&\quad \le C_5\left( \varDelta t^2+\varDelta t\sum _{n=0}^m|e_S^n|^2\right) , \end{aligned}$$

where $C_5>0$ is a constant depending on the data $\nu ,\varOmega , C_1,C_2,C_3,C_4$ and T, which may take different values at its different places.

Proof

Taking $v_h=2e_u^{n+1}\varDelta t,q_h=2e_p^{n+1}\varDelta t$ in problem (29), we have

$$\begin{aligned}&\Vert e_u^{n+1}\Vert _0^2{-}\Vert e_u^n\Vert _0^2 {+}2\varDelta t\frac{S_h(t_{n+1})}{\sqrt{E_h(t_{n+1}})}b(u_h(t_{n+1}),u_h(t_{n+1}),e_u^{n+1}) {-}2\varDelta t\frac{S_h^n}{\sqrt{E_h^n}}b(u_h^n,u_h^n,e_u^{n+1}) \nonumber \\&\quad +2\varDelta t\nu \Vert \nabla e_u^{n+1}\Vert _0^2 +\Vert e_u^{n+1}-e_u^n\Vert _0^2 = 2(\int _{t_n}^{t_{n+1}}(t_n-t)u_{htt}dt,e_u^{n+1}).\ \ \ \ \end{aligned}$$

(30)

For the right-hand side term, by the Hölder inequality, one finds

$$\begin{aligned} \left| 2\left( \int _{t_n}^{t_{n+1}}(t_n-t)u_{htt}dt,e_u^{n+1}\right) \right| \le \frac{\nu }{4}\varDelta t\Vert \nabla e_u^{n+1}\Vert _0^2+\frac{4\gamma _0}{\nu }\varDelta t^2\int _{t_n}^{t_{n+1}}\Vert u_{htt}\Vert ^2_0dt. \end{aligned}$$

For the trilinear terms, we have

$$\begin{aligned}&2\varDelta t \Big | \frac{S_h(t_{n+1})}{\sqrt{E_h(t_{n+1})}}b(u_h(t_{n+1}),u_h(t_{n+1}),e_u^{n+1}) -\frac{S_h^n}{\sqrt{E_h^n}}b(u_h^n,u_h^n,e_u^{n+1}) | \\&\quad =2\varDelta t \Big | b(u_h(t_{n+1}),u_h(t_{n+1}),e_u^{n+1})-b(u_h(t_n),u_h(t_n),e_u^{n+1}) +b(u_h(t_n),u_h(t_n),e_u^{n+1}) \\&\qquad -b(u_h^n,u_h^n,e_u^{n+1}) {+}\frac{e_S^n}{\sqrt{E_h(t_n)}}b(u_h^n,u_h^n,e_u^{n+1}) {+}S_h^n\Big (\frac{1}{\sqrt{E_h(t_n})}{-}\frac{1}{\sqrt{E_h^n}}\Big ) b(u_h^n,u_h^n,e_u^{n+1}) \Big |. \end{aligned}$$

By the Taylor expansion, (6), (7) and (22), we have

$$\begin{aligned}&2\varDelta t\Big |b(u_h(t_{n+1}),u_h(t_{n+1}),e_u^{n+1})-b(u_h(t_n),u_h(t_n),e_u^{n+1})\Big | \\&\quad = 2\varDelta t^2 \Big | b(u_{ht}(t_n),u_h(t_{n+1}),e_u^{n+1})+b(u_h(t_n),u_{ht}(t_n),e_u^{n+1}) \Big | \\&\quad \le 2C_3\varDelta t^2\Vert u_{ht}\Vert _0(\Vert A_hu_h(t_{n+1})\Vert _0+\Vert A_hu_h(t_n)\Vert _0)\Vert \nabla e_u^{n+1}\Vert _0 \\&\quad \le \frac{\nu }{4}\varDelta t\Vert \nabla e_u^{n+1}\Vert _0^2 +\frac{8C_3^2}{\nu }\varDelta t^3\Vert u_{ht}\Vert ^2_0(\Vert A_hu_h(t_{n+1})\Vert ^2_0+\Vert A_hu_h(t_n)\Vert ^2_0), \\&2\varDelta t\Big |b(u_h(t_n),u_h(t_n),e_u^{n+1}){-}b(u_h^n,u_h^n,e_u^{n+1})\Big | {=}2\varDelta t \Big | b(e_u^n,u_h(t_n),e_u^{n+1}){+}b(u_h^n,e_u^n,e_u^{n+1}) \Big | \\&\quad \le 2C_3\varDelta t \Vert \nabla e_u^{n+1}\Vert _0\Vert e_u^n\Vert ^{1/2}_0\Vert \nabla e_u^n\Vert _0^{1/2}(\Vert \nabla u_h^n\Vert _0+\Vert \nabla u_h(t_n)\Vert _0) \\&\quad \le \frac{\nu }{4}\varDelta t\Vert \nabla e_u^{n+1}\Vert ^2_0 +\frac{\nu }{2}\varDelta t\Vert \nabla e_u^n\Vert ^2_0 +\frac{8C_3^4}{\nu ^3}\varDelta t(\Vert \nabla u_h^n\Vert ^2_0+\Vert \nabla u_h(t_n)\Vert ^2_0)^2\Vert e_u^n\Vert _0^2, \\&2\varDelta t|\frac{e_S^n}{\sqrt{E_h(t_n)}}b(u_h^n,u_h^n,e_u^{n+1})| \le \frac{2C_3}{\sqrt{C_0}}\varDelta t|e_S^n|\Vert \nabla u_h^n\Vert _0\Vert \nabla e_u^{n+1}\Vert _0\Vert \nabla u_h^n\Vert _0 \\&\quad \le \frac{\nu }{8}\varDelta t\Vert \nabla e_u^{n+1}\Vert ^2_0 +\frac{8C_3^2}{\nu }\varDelta t\frac{|e_S^n|^2}{C_0}\Vert \nabla u_h^n\Vert _0^4, \\&2\varDelta t|S_h^n(\frac{1}{\sqrt{E_h(t_n})}-\frac{1}{\sqrt{E_h^n}}) b(u_h^n,u_h^n,e_u^{n+1})| \\&\quad = 2\varDelta t|S_h^n|\frac{|E_h^n-E_h(t_n)|}{\sqrt{E_h(t_n)}\sqrt{E_h^n}(\sqrt{E_h(t_n)}+\sqrt{E_h^n})}|b(u_h^n,u_h^n,e_u^{n+1})|. \end{aligned}$$

Based on the definitions of $E_h^n,E_h(t_n)$ and the following fact that

$$\begin{aligned} |E_h^n-E_h(t_n)|=\frac{1}{2}(\Vert u_h^n\Vert _0^2-\Vert u_h(t_n)\Vert _0^2) \le \Vert e_u^n\Vert _0\Big (\sqrt{E_h^n}+\sqrt{E_h(t_n)}\Big ), \end{aligned}$$

one finds

$$\begin{aligned} 2\varDelta t|S_h^n(\frac{1}{\sqrt{E_h(t_n})}-\frac{1}{\sqrt{E_h^n}}) b(u_h^n,u_h^n,e_u^{n+1})|\le & {} \frac{2C_3}{C_0} \varDelta t|S_h^n|\Vert e_u^n\Vert _0\Vert \nabla e_u^{n+1}\Vert _0\Vert \nabla u_h^n\Vert ^2_0\\\le & {} \frac{\nu }{8}\varDelta t\Vert \nabla e_u^{n+1}\Vert ^2_0 {+}\frac{8C_3^2}{C^2_0\nu }\varDelta t|S_h^n|^2\Vert \nabla u^n\Vert ^4_0\Vert e_u^n\Vert _0^2. \end{aligned}$$

Combining above inequalities with (30), summing from $n=0$ to m, using Lemma 3.2 and Theorems 3.3–3.4, we finish the proof. $\square$

Lemma 4.2

Under the Assumptions (A1)-(A2) and $e_u^0=0$, for all $m\ge 0$, it holds

$$\begin{aligned} \nu \Vert \nabla e_u^{m+1}\Vert _0^2 +\varDelta t\sum _{n=0}^m\Vert d_te_u^{n+1}\Vert _0^2 \le C_5\Big (\varDelta t^2+\varDelta t\sum _{n=0}^m|e_S^n|^2\Big ). \end{aligned}$$

Proof

Taking $v_h=2d_te_u^{n+1}\varDelta t\in V_h,q_h=0$ in problem (29), one finds

$$\begin{aligned}&\nu (\Vert \nabla e_u^{n+1}\Vert _0^2-\Vert \nabla e_u^n\Vert _0^2 +\Vert \nabla (e_u^{n+1}-e_u^n)\Vert _0^2) +2\varDelta t\Vert d_te_u^{n+1}\Vert _0^2\nonumber \\&\qquad - 2\varDelta t\frac{S_h^n}{\sqrt{E_h^n}}b(u_h^n,u_h^n,d_te_u^{n+1}) +2\varDelta t\frac{S_h(t_{n+1})}{\sqrt{E_h(t_{n+1})}}b(u_h(t_{n+1}),u_h(t_{n+1}),d_te_u^{n+1})\nonumber \\&\quad =2\left( \int _{t_n}^{t_{n+1}}(t_n-t)u_{htt}dt,d_te_u^{n+1}\right) .\ \ \ \ \end{aligned}$$

(31)

For the trilinear terms and right-hand side term, we can treat them as follows

$$\begin{aligned}&2\varDelta t|\frac{S_h^n}{\sqrt{E_h^n}}b(u_h^n,u_h^n,d_te_u^{n+1}) -\frac{S_h(t_{n+1})}{\sqrt{E_h(t_{n+1})}}b(u_h(t_{n+1}),u_h(t_{n+1}),d_te_u^{n+1})| \\&\quad = 2\varDelta t\Big | \frac{S_h^n-S_h(t_n)}{\sqrt{E_h^n}}b(u_h^n,u_h^n,d_te_u^{n+1}) -b(e_u^n,u_h^n,d_te_u^{n+1})-b(u_h(t_n),e_u^n,d_te_u^{n+1}) \\&\qquad +(b(u_h(t_n)-u_h(t_{n+1}),u_h(t_n),d_te_u^{n+1})+b(u_h(t_{n+1}),u_h(t_n) -u_h(t_{n+1}),d_te_u^{n+1}) \\&\qquad +S_h(t_n)(\frac{1}{\sqrt{E_h^n}}-\frac{1}{\sqrt{E_h(t_n)}})b(u_h^n,u_h^n,d_te_u^{n+1}) \Big | \\&\quad \le 2\varDelta t\Big | \frac{e_S^n}{\sqrt{E_h^n}}b(u_h^n,u_h^n,d_te_u^{n+1}) -b(e_u^n,u_h^n,d_te_u^{n+1}) - b(u_h(t_n),e_u^n,d_te_u^{n+1}) \Big | \\&\qquad +2\varDelta t^2\Big |(b(u_{ht}(t_n),u_h(t_n),d_te_u^{n+1})+b(u_h(t_{n+1}),u_{ht}(t_n),d_te_u^{n+1}) \Big | \\&\qquad +2\varDelta t\frac{|S_h(t_n)|\Vert e_u^n\Vert _0}{\sqrt{E_h^n}\sqrt{E_h(t_n)}}|b(u_h^n,u_h^n,d_te_u^{n+1})| \\&\quad \le \frac{4}{5}\varDelta t\Vert d_te_u^{n+1}\Vert _0^2 +\frac{5C_3^2}{C_0}\Vert \nabla u_h^n\Vert _0^2\Vert A_hu_h^n\Vert _0^2\varDelta t|e_S^n|^2 \\&\qquad +\frac{5C_3^2}{C_0^2}|S_h(t_n)|^2\Vert \nabla u_h^n\Vert _0^2\Vert A_hu_h^n\Vert _0^2\varDelta t\Vert e_u^n\Vert _0^2 \\&\qquad +5C_3^2(\Vert A_hu_h^n\Vert _0^2+\Vert A_hu_h(t_n)\Vert _0^2)\varDelta t\Vert \nabla e_u^n\Vert _0^2 +5C_3^2\varDelta t^3\Vert \nabla u_{ht}\Vert _0^2(\Vert A_hu_h(t_n)\Vert _0^2\\&\qquad +\Vert A_hu_h(t_{n+1})\Vert _0^2), \\&\qquad 2\left| \left( \int _{t_n}^{t_{n+1}}(t_n-t)u_{htt}dt,d_te_u^{n+1}\right) \right| \le \frac{1}{5}\varDelta t\Vert d_te_u^{n+1}\Vert _0^2+5\varDelta t^2\int _{t_n}^{t_{n+1}}\Vert u_{htt}\Vert ^2_0dt. \end{aligned}$$

Combining above inequalities with (31), summing from $n=0$ to m, using Lemmas 3.2, 4.1 and Theorems 3.3–3.5, we complete the proof. $\square$

Lemma 4.3

Under the Assumptions (A1)-(A2), $e_S^0=0$ and $\varDelta t\le \frac{C_0}{C_4^2+4}$, for all $m\ge 0$, it holds

$$\begin{aligned} |e_S^{m+1}|^2+\frac{1}{C_4^2+4}\sum _{n=0}^m|e_S^{n+1}-e_S^n|^2 \le C_5 \varDelta t^2. \end{aligned}$$

Proof

Choosing $s_h=2e_S^{n+1}\varDelta t$ in the second equation of problem (29), we have

$$\begin{aligned}&|e_S^{n+1}|^2-|e_S^n|^2 +|e_S^{n+1}-e_S^n|^2 \nonumber \\&\quad = \varDelta te_S^{n+1}\Big (\frac{1}{S_h(t_{n+1})} (d_tu_h(t_{n+1}),u_h(t_{n+1})) -\frac{1}{S_h^{n+1}} (d_tu_h^{n+1},u_h^{n+1}) \Big ) \nonumber \\&\qquad +\varDelta te_S^{n+1}\Big ( \frac{1}{\sqrt{E_h(t_{n+1}})}b(u_h(t_{n+1}),u_h(t_{n+1}),u_h(t_{n+1})) -\frac{S_h^n}{S_h^{n+1}\sqrt{E_h^n}}b(u_h^n,u_h^n,u_h^{n+1}) \Big ) \nonumber \\&\qquad + 2e_S^{n+1}\int _{t_n}^{t_{n+1}}(t_n-t)S_{htt}dt + \frac{e_S^{n+1}}{S_h(t_{n+1})}(\int _{t_n}^{t_{n+1}}(t_n-t)u_{htt}dt,u_h(t_{n+1})). \end{aligned}$$

(32)

We are now in the position of treating the right-hand side terms one by one

$$\begin{aligned}&\Big | \varDelta te_S^{n+1} \Big (\frac{1}{S_h(t_{n+1})} (d_tu_h(t_{n+1}),u_h(t_{n+1})) -\frac{1}{S_h^{n+1}} (d_tu_h^{n+1},u_h^{n+1}) \Big )\Big | \\&\quad = \varDelta t|e_S^{n+1}| \Big | (\frac{1}{S_h(t_{n+1})}-\frac{1}{S_h^{n+1}}) (d_tu_h(t_{n+1}),u_h(t_{n+1})) \\&\qquad +\frac{1}{S_h^{n+1}} \Big [(d_tu_h(t_{n+1}),e_u^{n+1}) + (d_te_u^{n+1},u_h^{n+1})\Big ] \Big | \\&\quad \le \varDelta t|e_S^{n+1}| \Big [\frac{|e_S^{n+1}|}{S_h(t_{n+1})S_h^{n+1}}\Vert u_{ht}\Vert _0\Vert u_h(t_{n+1})\Vert _0\\&\qquad +\frac{1}{S_h^{n+1}} \Big (\Vert u_{ht}\Vert _0\Vert e_u^{n+1}\Vert _0 +\Vert d_te_u^{n+1}\Vert _0\Vert u_h^{n+1}\Vert _0\Big ) \Big ] \\&\quad \le \frac{1}{C_0}\varDelta t|e_S^{n+1}|^2\Big (\Vert u_{ht}\Vert _0\Vert u_h(t_{n+1})\Vert _0+1\Big ) {+}\frac{\varDelta t}{2}\Vert u_{ht}\Vert _0^2\Vert e_u^{n+1}\Vert ^2_0 {+}\frac{\varDelta t}{2}\Vert d_te_u^{n+1}\Vert ^2_0\Vert u_h^{n+1}\Vert ^2_0, \\&\Big |\varDelta te_S^{n+1}\Big ( \frac{1}{\sqrt{E_h(t_{n+1}})}b(u_h(t_{n+1}),u_h(t_{n+1}),u_h(t_{n+1})) -\frac{S_h^n}{S_h^{n+1}\sqrt{E_h^n}}b(u_h^n,u_h^n,u_h^{n+1}) \Big )\Big | \\&\quad =\frac{\varDelta t|e_S^{n+1}|}{\sqrt{E^n_h}}|\frac{S_h^n}{S_h^{n+1}}| \Big | \varDelta tb(u_h(t_n),u_h(t_n),u_{ht}(t_n)) +b(e_u^n,u_h(t_n),u_h(t_{n+1})) \\&\qquad +b(u_h^n,e_u^n,u_h(t_{n+1})) +b(u_h^n,u_h^n,e_u^{n+1})\Big | \\&\quad \le \frac{C_3}{C_0}\varDelta t|S_h^n| |e_S^{n+1}|\Big ( \varDelta t\Vert \nabla u_h(t_n)\Vert _0\Vert A_hu_h(t_n)\Vert _0\Vert u_{ht}(t_n)\Vert _0 +\Vert \nabla u_h^n\Vert _0^2\Vert \nabla e_u^{n+1}\Vert _0 \\&\qquad +\Vert e_u^n\Vert _0(\Vert \nabla u_h(t_n)\Vert _0+\Vert \nabla u_h^n\Vert _0)\Vert A_hu_h(t_{n+1})\Vert _0 \Big ) \\&\quad \le \frac{3}{2C_0}\varDelta t|e_S^{n+1}|^2 +\frac{C_3^2}{2C_0}\varDelta t|S_h^n|^2 \Big ( \varDelta t^2\Vert \nabla u_h\Vert ^2_0\Vert A_hu_h\Vert ^2_0\Vert u_{ht}\Vert ^2_0 +\Vert \nabla u_h^n\Vert _0^4\Vert \nabla e_u^{n+1}\Vert ^2_0 \\&\qquad +\Vert e_u^n\Vert ^2_0(\Vert \nabla u_h(t_n)\Vert ^2_0+\Vert \nabla u_h^n\Vert ^2_0)\Vert A_hu_h\Vert ^2_0 \Big ), \\&\Big |\frac{e_S^{n+1}}{S_h(t_{n+1})}\Big (\int _{t_n}^{t_{n+1}}(t_n-t)u_{htt}dt,u_h(t_{n+1})\Big )\Big |\\&\quad \le \frac{\varDelta t}{4C_0}|e_S^{n+1}|^2+\varDelta t^2\int _{t_n}^{t_{n+1}}\Vert u_{htt}\Vert _0^2dt\Vert u_h(t_{n+1})\Vert ^2_0, \\&\Big |2e_S^{n+1}\int _{t_n}^{t_{n+1}}(t_n-t)S_{htt}dt\Big | \le 2|e_S^{n+1}|\varDelta t^\frac{3}{2}(\int _{t_n}^{t_{n+1}}|S_{htt}|^2dt)^{1/2}\\&\quad \le \frac{\varDelta t}{4C_0}|e_S^{n+1}|^2 +4C_0\varDelta t^2\int _{t_n}^{t_{n+1}}|S_{htt}|^2dt. \end{aligned}$$

Combining above inequalities with (32), summing from $n=0$ to m, using Theorems 2.2, 3.3–3.5, Lemmas 3.2, 4.1–4.2 and the condition $\varDelta t\le \frac{C_0}{C_4^2+4}$, we complete the proof. $\square$

Combining Theorem 2.2, (18) with Lemmas 4.1–4.3 and the inf-sup condition, we finally obtain the optimal error estimates of numerical solutions in Euler implicit/explicit-SAV scheme (29) for the Navier–Stokes equations.

Theorem 4.1

Under the Assumptions of Lemmas 4.2 and 4.3, for all $m\ge 0$ it holds

$$\begin{aligned} \Vert e_u^{m+1}\Vert _0^2+|e_S^{m+1}|^2 +h^2(\Vert \nabla e^{m+1}_u\Vert _0^2 +\varDelta t\sum _{n=0}^m\Vert e_p^{n+1}\Vert _0^2) \le C_5 (\varDelta t^2+h^4). \end{aligned}$$

(33)

5 Numerical Experiments

In this section, we present some numerical results to illustrate the performances of the fully discrete Euler implicit/explicit-SAV finite element scheme (21) for the Navier–Stokes equations. Due to we treat the nonlinear terms explicitly, so we can split the considered problems into a Stokes equations and a quadratic algebraic equation in one variable. It means that we can solve problem (21) as follows:

$$\begin{aligned}&(d_tu^{n+1}_h,v_h)+a(u^{n+1}_h,v_h)-d(v_h,p^{n+1}_h) +d(u^{n+1}_h,q_h)\nonumber \\&\quad =(f^{n+1},v_h)-\frac{S^n_h}{\sqrt{E^n_h}}b(u^n_h,u^n_h,v_h), \end{aligned}$$

(34)

and

$$\begin{aligned} (d_tS^{n+1}_h,s_h)= \frac{1}{2S^{n+1}_h}\Big ((d_tu^{n+1}_h,u^{n+1}_h)+\frac{S^n_h}{\sqrt{E^n_h}}b(u^n_h,u^n_h,u^{n+1}_h),s_h\Big ), \end{aligned}$$

(35)

with $u_h^0=P_hu_0,S_h^0=\sqrt{C_0+\frac{1}{2}\Vert u_h(0)\Vert _0^2}$.

Firstly, we solve the Stokes equations (34) with $u_h^n,S_h^n$ and for all $n\ge 0$.

Secondly, taking $v_h=u_h^{n+1}\varDelta t,q_h=p_h^{n+1}\varDelta t$ and $s_h=2S_h^{n+1}\varDelta t$ in (21), one gets

$$\begin{aligned} 2(S_h^{n+1})^2-2S_h^{n+1}S_h^n=\varDelta t(f^{n+1},u_h^{n+1}) -\nu \varDelta t\Vert \nabla u_h^{n+1}\Vert _0^2. \end{aligned}$$

(36)

Solving the equation (36) with the obtained $u_h^{n+1}$ and the quadratic formula.

Finally, we present some computational results to confirm the established theoretical results and show the performances of the considered numerical scheme (21). The partition of domain $\varOmega$ uses the triangle mesh with stable MINI element for the velocity and pressure. The mesh is obtained by dividing $\varOmega$ into squares and then drawing a diagonal in each square. Set $\varOmega =[0,1]\times [0,1]$, the viscosity parameter $\nu =1$ the final time $T=1$ and choose the following analytical solutions for the velocity $u(x,t)=(u_1(x,t),u_2(x,t))$ and pressure p(x, t)

$$\begin{aligned}&u_1(x,t)=10x^2(x-1)^2y(y-1)(2y-1)\exp (-t),\\&u_2(x,t)=-10x(x-1)(2x-1)y^2(y-1)^2\exp (-t),\\&p(x,t)=10(2x-1)(2y-1)\exp (-t). \end{aligned}$$

The computational results of Euler implicit/explicit-SAV scheme (21) are presented in Table 1 to verify the established results of Theorem 4.1. From these data, we can see that the convergence orders of velocity in $L^2$- and $H^1$-norms are 2 and 1, respectively, which confirm the provided theoretical findings (33) well.

Table 1 Numerical results of Euler implicit/explicit SAV scheme (21) with $C_0=1.0,\varDelta t=h^2$

The second example is a classical benchmark model: the lid-driven cavity problem. In this test, we consider the incompressible lid-driven cavity flow problem defined on the unit square. Setting $f=0$ and the boundary condition $u=0$ on $[\{0\}\times (0,1)]\cup [(0,1)\times \{0\}]\cup [\{1\}\times (0,1)]$ and $u=(1,0)^T$ on $(0,1)\times \{1\}$. The mesh consists of triangular element and the mesh size $h=\frac{1}{60},\ \varDelta t=0.001,\ C_0=10000$, the final time $T=500$ and the Taylor-Hood element is used to approximate the velocity and pressure. Figures 1 and 2 illustrate the velocity profiles of the lid-driven cavity problem along $x=0.5$ and $y=0.5$ in numerical scheme (21). Compared with the results provided by Erturk et al in [3] and Ghia et al in [4], we can see that the results obtained by the fully discrete implicit/explicit-SAV scheme are agree with Ghia’s and Erturk’s.

Fig. 1

The computed results of implicit/explicit-SAV scheme of vertical velocity profiles (x=0.5) with different Re. a Re = 1000, b Re = 2500, c Re = 5000

Fig. 2

The computed results of implicit/explicit-SAV scheme of horizontal velocity profiles (y = 0.5) with different Re. a Re = 1000, b Re = 2500, c Re = 5000

6 Conclusion

In this paper, a fully discrete implicit/explicit numerical scheme is considered for the incompressible Navier–Stokes equations. Compared with the published papers [8, 9, 11, 13, 33], the main feature of this work is developing the unconditional stability of numerical solutions by introducing the scalar auxiliary variable, which enriches and supplements the theoretical findings of finite element method. Some numerical results are also provided to show the performances of the considered numerical scheme. The constant $C_0$ in energy variable has an important influence on the computational results, it should be chosen carefully and experimentally, for example, one needs to choose $C_0\ge 10^4$ in the lid-driven cavity problem with high Reynold numbers. How to design a novel SAV factor independent of the constant $C_0$ is a meaningful topic, and is the goal of the following works.