1. Introduction
In this paper, we consider the following two-dimensional time-dependent incompressible Navier–Stokes equations: 
      where 
 is a rectangular domain with boundary 
, 
 represents the velocity vector, 
 denotes the pressure, and 
 is the prescribed body force, 
. Moreover, 
 is the initial velocity and 
 is the final time.
Navier–Stokes equations are a classical incompressible fluid model and have been widely applied in the mathematical physics and the computational fluid dynamics fields [
1,
2,
3]. It is an unrealistic thing to find the the exact solutions of the Navier–Stokes equations due to their nonlinear and incompressible properties. Therefore, numerous works have been devoted to the development of efficient numerical approximations for Navier–Stokes Equations (1)–(4), including finite difference methods [
4,
5,
6,
7], Galerkin finite element methods [
8,
9,
10,
11,
12,
13,
14,
15,
16,
17,
18,
19,
20,
21] and other methods [
22,
23,
24,
25,
26]. In particular, a new second-order accurate finite difference scheme for the incompressible Navier–Stokes equations was discussed in [
4] by the primitive variable formulation. Based on the vorticity stream-function formulation and a fast Poisson solver defined on a general domain using the immersed interface method, a fast finite difference method was proposed and studied in [
5] for the incompressible Navier–Stokes equations. A new fully discrete finite element nonlinear Galerkin method was established in [
8] for the Navier–Stokes equations with two-grid finite element discretization in the spatial direction and Euler explicit scheme with variable time step size in the temporal direction. The boundedness, convergence and stability condition of the presented method were discussed under certain time-step constraints dependent on the coarse grid parameter. A Lagrange–Galerkin mixed finite element approximation was presented for the Navier–Stokes equations in [
13], and optimal error estimates were obtained with the mesh restriction 
, where 
, and 
, 
h and 
d denote the time step size, the mesh size and the dimension of the domain, respectively. In [
14], in terms of the special properties of a low-order nonconforming mixed finite element pair on the rectangular mesh, the superconvergence error estimates were derived for Navier–Stokes equations with the time-step constraints 
, 
 due to the inverse inequality used in the error analysis. In [
18], in terms of the error splitting technique developed in [
27,
28], the unconditional stability and convergence of a typical modified characteristics finite element method were studied for the time-dependent Navier–Stokes equations by introducing an iterated characteristic time-discrete system. Optimal error estimates were obtained under the boundedness of the numerical solution in 
-norm. Subsequently, the unconditionally optimal error estimates were derived in [
20] under the boundedness of the numerical solution in 
-norm, which is weaker than that in [
18].
It should be pointed out that optimal error estimates were obtained in [
20] due to roughly handling trilinear terms. Moreover, the superconvergence error estimates were derived in [
14] with a certain time-step restriction. To the best of our knowledge, there are few contributions on the unconditionally superclose and superconvergence error estimates for problems (1)–(4). The purpose of this paper is to consider the unconditionally superclose and superconvergence error estimates for incompressible time-dependent Navier–Stokes Equations (1)–(4). It should be pointed out that the difficulties come from the trilinear term (i.e., the convection term) 
 in the superclose and superconvergence error analysis. Therefore, we should deal with them carefully and skillfully.
In the present work, we focus on a low-order conforming finite element approximation, which is called the bilinear-constant scheme [
4,
29,
30], for problems (1)–(4). The key to our analysis is to employ the special properties (high-accuracy error estimation; see Lemmas 1–2 below) on the rectangular mesh and to treat the convection term 
 rigorously and skillfully. The superclose error estimates are obtained firstly for the velocity in 
-norm and for the pressure in 
-norm. Then, in terms of an efficient interpolation post-processing approach, the global superconvergence results are derived for the velocity in the 
-norm and for the pressure in the 
-norm. In addition, some numerical experiments are presented and tested.
The remainder of this paper is organized as follows. In 
Section 2, some preliminaries and lemmas are introduced, and the semi-implicit Euler fully discrete finite element approximation of the problem (1)–(4) is presented. In 
Section 3, the detailed superclose error estimates are studied with the help of the special properties of the bilinear-constant finite element pair combined with skillfully dealing with the trilinear term. Then, the global superconvergence error estimates are established by the interpolation post-processing technique. In 
Section 4, some numerical results are provided to verify the theoretical findings.
  2. Preliminaries
We denote by 
 the Sobolev spaces with the norm 
 and semi-norm 
 defined by
      
      where 
 for the multi-index 
, 
, 
 and 
. For 
, we let 
 denote 
. We omit the subscript when 
 and write 
 and 
 as 
 and 
 for simplicity. Moreover, we use 
 to represent the 
 inner product, i.e.,
      
For the mathematical setting of the problem (1)–(4), we introduce the following Sobolev spaces 
 and 
M [
31], i.e.,
      
Moreover, for any Banach space 
X and 
, let 
 be the space of all measurable function 
 with the norm
      
Let 
 be a uniform rectangular mesh over 
 with mesh size 
h. For a given element 
, its four vertices are denoted by 
 in the counterclockwise order (see 
Figure 1 left). For the velocity, we choose 
 as the general bilinear finite element space. For the pressure, we assume that the subdivision 
 is obtained from 
 by dividing each element of 
 into four small congruent rectangles. Let 
 consist of piecewise constant functions with respect to 
 such that the local basis functions for 
 on a 
-patch of 
 (see 
Figure 1 right) are indicated in 
Figure 2. Then, the finite element space for pressure is defined by 
. In the following discussion, we always assume that 
 with 
 (see 
Figure 1 right). Thus, the finite element approximation spaces 
 and 
 for the bilinear-constant scheme are described by ([
1,
29,
30])
      
      where 
 denotes the space of all polynomials of degree 
 with respect to each of the two variables, 
x and 
y. It is shown in [
1,
29,
30] that the bilinear-constant scheme satisfies the Babuška–Brezzi condition, i.e.,
      
      where 
 is a constant, independent of 
h.
For the velocity, we use the Lagrange nodal interpolation operator 
 as the corresponding interpolation operator. For the pressure, we first introduce the local 
-projection 
 of 
p by
      
      and then define the operator 
 with respect to 
 by
      
      where 
 with the notations of 
, 
. We can check that
      
      which implies that 
 for 
.
Lemma 1 ([
29])
. Suppose that  and , then there hold Lemma 2. Supposing that  on element e, we have Furthermore, there holds for   Proof.  In order to obtain a second-order accuracy estimate, we adopt the high-accuracy integral technique developed in [
32]. We introduce the following error function on an element 
e (see 
Figure 3):
        
        where 
 denotes the barycentric coordinate of element 
e. One can check that 
, 
. Note that 
v is constant on an element 
e, and we have
        
        where we have that 
, 
 and 
 are constants in the above estimate.
In the same way, we also have
        
Therefore, by adding (
14) and (
15), the desired result (
11) is derived. Furthermore, summing up (
11) with respect to 
 gives the result (
12).    □
 We present the discrete Gronwall’s inequality, which is useful in the following error analysis.
Lemma 3 ([
33,
34])
. Let τ, B and , , , , for integers , be non-negative numbers such that suppose that , for all k, and set . Then The Young inequality will be frequently used in the following analysis, and we present it here.
      
The weak formulation of (1)–(4) is as follows: find 
, such that
      
Moreover, in order to give the fully discrete scheme, let 
 be a given uniform partition of the time interval with time step 
 and 
, 
. For a smooth function 
w defined on 
, denote
      
Then, the semi-implicit backward Euler low-order conforming mixed finite element scheme (bilinear-constant scheme) is as follows: for given 
, find 
 such that
      
      with the initial approximation 
.
The unconditionally optimal error estimate for scheme (18)–(19) is shown in [
20] by using the error splitting technique with the low-order conforming mixed finite element method, specifically, by introducing the following time-discrete system [
18,
20]: 
Then, the error between the exact solution and the numerical solution comprises two parts, where one is the temporal error and the other is the spatial error. Therefore, the numerical solution in the -norm can be bounded without any time-step restrictions. Here, we present a lemma to state the -independent boundedness of the numerical solution.
Lemma 4 ([
20])
. Suppose that  is the solution of (1)–(4) with suitable regularity and  and  are the solutions of (20)–(23) and (18)–(19). Then, there exist positive constants  and  such that when  and , we have the τ-independent boundedness of numerical solutions where K is a constant, which is not dependent on h, τ and n.   3. Superclose and Superconvergence Error Estimates
In this section, we firstly present the superclose error estimates for the velocity and pressure variables. Then, we give the superconvergence result in terms of the superclose error estimates as well as the interpolation technique.
Theorem 1. Suppose that  and  are the solutions of (16)–(19) and , ,  and . Then, we have the following superclose estimates Furthermore, we have the following optimal error estimates:  Proof.  For simplicity, we split the errors 
 and 
 as follows:
        
At , from (16)–(17), we have
Subtracting (18)–(19) from (27)–(28) gives the following error equations:
Taking 
 in (29) and 
 in (31) yields that
        
Now, we start to estimate 
, 
 term by term. By the Cauchy–Schwarz inequality and interpolation theory, we have
        
Applying the following equality
        
        we have
        
        where we used (8) of Lemma 1 in the above estimate.
With an application of (34) again, it follows that
        
        where we used (9) of Lemma 1 in the above estimate.
According to (10) of Lemma 1, 
 can be bounded by
        
To bound 
, we split 
 as follows:
        
In terms of the Cauchy–Schwarz inequality and interpolation theory, we have
        
Using inverse inequality and interpolation theory, we have
        
With the aid of (
35), one can check that
        
In order to obtain a high-accuracy error estimate, i.e., second-order accuracy, we introduce the local 
-projection defined as follows:
        
        where 
e is an element of the partition of 
 and 
 is the measure of 
e. Then, we have
        
Therefore, we have
        
        where we used Lemma 2 in the above estimate.
In the same way, we also have
        
By (24) of Lemma 4, the numerical solution 
 is bounded unconditionally in 
-norm, and we have
        
Based on the above estimates 
, 
, 
 reduces to
        
Moreover, by Taylor expansion, we have
        
With the estimates 
, 
, we obtain
        
On the other hand, from (29), it follows that
        
Then, one can check that by the Cauchy–Schwarz inequality and interpolation theory,
        
        and
        
Using a similar estimate process as 
, there holds
        
As an application of the Taylor expansion, we have
        
Therefore, we conclude that
        
Then, thanks to the discrete LBB condition (
 is a positive constant independent of mesh-size 
h), we have
        
Substituting (58) into (50) and using the Young inequality, (50) reduces to
        
Choosing 
 in (59) and summing up the resulting inequality, we have
        
        where we used 
, Lemma 1 and the same estimate process (44) for 
.
Hence, from (60), there holds
        
Then, an application of the Gronwall lemma (see Lemma 3) yields
        
Putting (62) into (58) gives that
        
The desired results are obtained, and the proof is complete.    □
 With the aid of the superclose error estimate in Theorem 1, we adopt the interpolation post-processing approach to improve the accuracy of the numerical solution 
 on the whole domain. Let 
 be the piecewise biquadratic nodal interpolation operator for the velocity associated with 
. Moreover, for the pressure, the postprocessing operator 
 is defined as
      
The following properties are shown in [
29]:
Moreover, we also have from [
29] that for 
,
      
Now, we state the global superonvergence result in the following theorem:
Theorem 2. Under the condition of Theorem 1, there holds  Proof.  By (64) and (65), we have
        
The proof is complete.    □
   4. Numerical Experiment
In this section, we present some numerical results to confirm the correctness of the theoretical analysis. The software we used is MATLAB 2018a with a 3.20 GHz Intel Core i5-6500 CPU processor and 8 GB memory.
Example 1. Let 
, and divide 
 into 
 uniform rectangles. Moreover, the function 
 and the initial and boundary conditions are chosen corresponding to the exact solution [
35]:
        
 We set the final time 
 in the computation. In order to confirm the error estimates in Theorems 1 and 2, we present the numerical errors in 
Table 1, 
Table 2, 
Table 3, 
Table 4, 
Table 5 and 
Table 6 at 
, respectively. Obviously, it can be seen that the numerical results are in agreement with the theoretical analysis, i.e., the errors 
, 
, 
 and 
 for the velocity 
 are of order 
, 
, 
 and 
, respectively. In addition, the errors 
, 
 and 
 for the pressure 
p are of orders 
, 
 and 
, respectively. At the same time, we also give the graphics of the exact solutions (
) and finite element solutions (
) at 
 on mesh 
 (see 
Figure 4, 
Figure 5, 
Figure 6 and 
Figure 7), respectively. It can be seen that the numerical results are also in good agreement with the theoretical analysis.
Example 2. Let 
 and divide 
 into 
 uniform rectangles. Moreover, the function 
 and the initial and boundary conditions are chosen corresponding to the exact solution [
14]:
        
 We set the final time 
 in the computation. In order to confirm the error estimates in Theorems 1 and 2, we present the numerical errors in 
Table 7 and 
Table 8 at 
. Obviously, it can be seen that numerical results are in agreement with the theoretical analysis.