1 Introduction

Let Ω in R² be a bounded domain with a Lipschitz-continuous boundary ∂Ω. We consider the following two-dimensional nonstationary incompressible Navier-Stokes equations:

where u = (u₁,u₂) denotes the fluid velocity vector field,p is the pressure filed,f = (f₁,f₂) is the given vector field,and ν > 0 is the constant inverse Reynolds number.

It is well known that the nonstationary incompressible Navier-Stokes equations are one of the main equations studied in mathematical physics and fluid mechanics fields. Numerous works have been devoted to numerical solutions of the above equations using finite element methods (FEMs). For example,we quote Ref. [1] for the conforming FEM,Ref. [2] for the fully discrete penalty FEM,Ref. [3] for the variational multiscale method,and Refs. [4]-[6] for the different stabilized FEMs. Recently,the nonconforming elements,which are usually easier to be constructed to satisfy the inf-sup condition than the conforming ones,received more and more attention for (1)−(4). In Ref. [7],the finite difference streamline diffusion method about the nonconforming P₁^nc /P₀ element pair of Ref. [8] was discussed with regard to a large Reynolds number. Shi and Wang^[9] extended the above element pair to anisotropic meshes,and the error estimates for the velocity in the L²-norm and the broken H¹-norm,as well as for the pressure in the L²-norm were obtained.

On the other hand,there have appeared some excellent studies on the superconvergent analysis of mixed nonconforming FEMs for the stationary Stokes and Navier-Stokes equations. For example,the L² projection method was discussed for the Stokes problem^[10-11]. The basic idea of L² projection method is to construct a new finite element (FE) approximation in a finite-dimensional space corresponding to a coarse mesh,and the difference in the mesh sizes can be used to achieve a better convergent rate after the post-processing procedure. Later,this idea was extend to solve the nonlinear Navier-Stokes equations^[12]. In Refs. ^[13] and ^[14],the integral identities techniques and interpolation postprocessing operators about the nonconforming CQ₁^rot /P₀ element pair were applied to the Stokes equations and Stokes equations with damping,and obtained the superconvergent results,respectively.

In this paper,we will use CQ₁^rot /P₀ element pair to solve the nonstationary problem (1)−(4) through different approaches from the application of the nonconforming P₁^nc /P₀ element pair in Ref. [9],and we obtain the superclose and superconvergent results for the velocity u in the broken H¹-norm and the pressure p in the L²-norm with reasonable regularity of the exact solution (u,p).

The rest of the paper is organized as follows. In Section 2,we briefly introduce the variational formulation for (1)-(4) and some preliminaries. In Section 3,we state the construction of the nonconforming mixed FE scheme about CQ₁^rot /P₀ element pair. In Section 4,we present the corresponding error estimates. In Section 5,the proper post-processing technique is applied. In Section 6,a numerical experiment is carried out to confirm our theoretical results.

We will use the notations for the Sobolev spaces W^m,q(Ω) with the norm and the semi-norm |·|_m,q,and W^m,q(K) with the norm and the semi-norm |·|_m,q,K,where m and q are nonnegative integer numbers (see Ref. [15]). Especially,for q = 2,q will be omitted in the above norms and semi-norms.

2 Variational formulation and preliminaries

To introduce a variational formulation,set

where

It is well known that for u ∈ H₀¹ (Ω) there hold

Here and later,C denotes a positive constant independent of the mesh parameter h and may be different at each occurrence.

Besides,for all u,v,w ∈ V ,the trilinear form b(· ; · ,·) satisfies the following properties^[16]:

where

In order to guarantee the properties of the existence and uniqueness of the solution (u,p) about problems (1)-(4),the following further assumptions and lemma are needed:

(i) Assume that Ω is regular in the sense that the unique solution (u,p) ∈ (V ,M) of the stationary Stokes problem,i.e.,−Δu+ ∇p = g and ∇u = 0 in Ω,and u|∂Ω = 0 for a prescribed g ∈ Y exists and satisfies .

(ii) The initial velocity u₀ ∈ D(A) and the body force f(x,t) ∈ L²(0,T ;Y ) are assumed to satisfy

Lemma 1^[17] Assume that (i) and (ii) hold. Then,for a given T > 0 there exists a unique solution (u,p) satisfying

where ρ(t) =min{1,t}.

3 Construction of nonconforming mixed FE scheme

Let T_h be a rectangular partition of the convex polygon domain Ω. For a given element K ∈ T_h with a center point (xK,yK),its four vertices are denoted by a₁(x₁,y₁),a₂(x₂,y₂),a₃(x₃,y₃),a₄(x₄,y₄) and four edges by (i = 1,2,3,4 (modulus 4)). h_x,K and h_y,K are the lengths of edges along the x-axis and y-axis,respectively. . Let = [−1,1]² be the reference element in the (ξ,η) plane with nodes ₁(−1,−1),₂(1,−1),₃(1,1),₄(−1,1) and edges (i = 1,2,3,4 (modulus 4)).

Define the affine mapping F_K : by

The Q₁^rot element space R_h is defined by^[18-20]

where ,[v_h] denotes the jump value of v_h across the boundary F, and [v_h] = v_h if F ⊂ ∂Ω.

Then,the CQ₁^rot element space CR_h is defined by^{[13, 21]}

For the velocity,we choose Vh = CRh ×CRh as the approximation space. Furthermore,we define

which is a norm over V_h.

Let

be a piecewise constant FE space,and M_h ⊂ Q_h is used to approximate the pressure,which consists of piecewise constants with respect to T_h. The local basis functions for M_h on a 2 × 2 patch of T are indicated in Fig. 1,i.e.,

and the subdivision T_2h is obtained by dividing each element of T_h into four congruent rectangles.

The associated interpolation operators are

Then,for all u ∈ H₂(Ω) ∩H₀¹(Ω) or q ∈ H¹(Ω) ∩ L₀² (Ω),there hold

Lemma 2^[14] Assume that u ∈ H^m+1(Ω) ∩H₀¹ 0 (Ω),we have

Lemma 3^[13-14] For v_h ∈ V_h,we have

Lemma 4 For v_h ∈ V_h,we have

Proof The proof of first result of (16) is similar to Lemma 4.3 in Ref. [22]. Thus,we only give the proof of the second one. Let Then,there holds

Since is a constant,we have

Hence,for a given element K,there holds

A direct computation implies

which leads to

Similarly,

The second desired result in (16) follows from (19) and (20). The proof is completed.

4 Convergence analysis

The nonconforming FE approximation of (5)-(7) is as follows: to find (u_h,p_h) ∈ V_h ×M_h such that

where

The above trilinear form b_h(·; ·,·) satisfies the following properties^[22]:

where

With the similar argument to Ref. [23],we have

which leads to

Therefore,

Using the Sobolev inequalities^[24-25] and combining (25) with k = 1,we can derive

Furthermore,we have

It has been shown in Refs. ^[13] and ^[21] that the pair (V_h,M_h) satisfies the discrete inf-sup condition,i.e.,

where β is a positive constant independent of h.

Now,we introduce a subspace Z_h of V_h as

In order to carry out the error estimates,we introduce the notations ρ = u − Π_hu and θ = Π_hu − u_h,where θ ∈ Z_h.

Lemma 5 Under the assumptions (i) and (ii) and ,there hold

Here and later,

Proof Firstly,for any v_h ∈ Z_h,taking v_h = 2u_h in (21),there holds

Integrating the above inequality from 0 to t and noting

we obtain the result (30) by Lemma 1.

Secondly,combining (1) with (21),we find that

where

Choosing v_h = θ and rearranging terms in (34),we get

Using (26) and Lemmas 3 and 4,we have

and

Combining these inequalities with (34),and noting ,we find that

Then,by integrating (37) from 0 to t and noting that

it follows from Lemma 1 and (30) that

To estimate the term ,we set v_h = u_ht in (21) to obtain

The terms b_h(u_h;u,u_ht) and (f,u_ht) can be bounded as follows:

Obviously,we have

Subsequently,combining the above estimates with (40),we get

Now,integrating (42) from 0 to t and using (39),we have

which implies

Combining (43)-(44) with (39) gives the desired results (31) and (32).

Lemma 6 Under the assumptions of Lemma 5,there hold

Proof By differentiating (21) with respect to the time t and noting v_h ∈ Z_h,it holds

Set v_h = u_ht,we have

which implies

Using (28) and Young’s inequality,we have

which,combining with (49),gives

Integrating (50) and using Lemma 5,we obtain the result (45).

Taking v_h = u_htt in (47),we find

which can be rewritten as

It follows from (41) that

and

Then,combining these estimates with (52),it follows from Lemma 5 that

Similarly,multiplying (53) by t,integrating from 0 to t,and using Lemma 5,we obtain that

i.e.,

he proof is completed.

Theorem 1 Let (u,p) ∈ V ×M and (u_h,p_h) ∈ V_h ×M_h be the solutions of (1)-(4) and (21)-(23),respectively. Assume that u,u_t,u_tt ∈ H³(Ω),p,p_t ∈ H²(Ω),and ν⁻¹ 1 − δ (0 < δ < 1). We have

where σ(t) = .

Proof From (1) and (21),we know

where

Setting v_h = θ in (57),we obtain

Applying (24),Lemmas 2-5,and the results (3.11) and (3.13) in Ref. [13],we have

and

Noting ν⁻¹ − δ again,by the Schwarz inequality and Lemma 1,we get

Integrating (62) from 0 to t and noting θ(0) = 0,we obtain

By the Gronwall lemma,it holds

Then,(54) can be derived directly.

Secondly,we prove (55). Differentiating both sides of (57) with respect to t and setting v_h = θ_t,we have

By (24),(28),and Lemmas 2-5,we can obtain

and

Substituting (66) and (67) into (65) and combining with ν⁻¹ − δ,we get

Rearranging terms in (68) gives

Multiplying both sides of (69) by t and integrating it with respect to t,with the result (64),we have

Let σ(t) = max. By the Gronwall lemma,it holds

which can derive the result (55) directly.

Taking v_h = θ_t in (57),we have

Using Lemmas 1 and 5,we can obtain

Similar to (67),it holds

Substituting (73)-(74) into (72),we have

Multiplying both sides of (75) by t,integrating it with respect to t,and using (71),we have

Furthermore,we can derive

The desired result (56) follows from (77),and the proof is completed.

Theorem 2 Under the assumptions of Theorem 1,we have

Proof Subtracting (21) from (1),for all v_h ∈ V_h,we obtain the error equation,i.e.,

By (24),(25) and Lemma 5,we have

and

Combining (79)-(81) and Theorem 1,we have

Using the discrete inf-sup condition,we deduce

which implies the desired result. The proof is completed.

5 Superconvergent results

Now,we start to introduce the post-processing operators (Π_2h,I_2h)^[13] such that

where Q₁(T) and Q₂(T) are the spaces of the bilinear and quadratic functions on T ∈ T_2h,respectively. We have the following superconvergent results.

Theorem 3 Under the assumptions of Theorem 1,we have

Proof Noticing that

by (83) and interpolation error estimates,we have

Consequently,it follows from (83) and the result (56) of Theorem 1 that

which implies the result (85).

Similarly,we can derive the result (86). The proof is completed.

6 Numerical experiment

In this section,we present a numerical example to confirm our theoretical analysis.

To get enough accuracy,we use the linearly extrapolated Crank-Nicolson time-stepping scheme (see Ref. [26]) to perform the numerical experiment.

Consider the problem (1)−(4) on Ω = [0, 1]²,the exact solutions are

Now,we divide the domain into m × n uniform rectangles and obtain the errors of the velocity u and the pressure p in the broken H¹-norm and the L²-norm at different time with m×n = 8×12,16 × 24,32 × 48,64 × 96,respectively.

The numerical results at different time and values of ν are listed in Tables 1−6. For convenience,we just plot the exact solutions u,p and the FEM solutions u_h,p_h at t = 1 with ν = 1 (see Figs. 2-4).

It can be seen from Tables 1−6 that |u − u_h|_h and are convergent at the rate of O(h),,|Π_hu − u_h|_h,|u − Π_2hu|_h,,and are convergent at the rate of O(h²),which coincides with the present theoretical analysis.