Stabilized Finite Element Method Based on the Crank–Nicolson Extrapolation Scheme for the Time-Dependent Navier–Stokes Equations

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MATHEMATICS OF COMPUTATION Volume 76, Number 257, January 2007, Pages 115–136 S 0025-5718(06)01886-2 Article electronically published on September 15, 2006

STABILIZED FINITE ELEMENT METHOD BASED ON THE CRANK–NICOLSON EXTRAPOLATION SCHEME FOR THE TIME-DEPENDENT NAVIER–STOKES EQUATIONS

YINNIAN HE AND WEIWEI SUN

Abstract. This paper provides an error analysis for the Crank–Nicolson extrapolation scheme of time discretization applied to the spatially discrete stabilized finite element approximation of the two-dimensional time-dependent Navier–Stokes problem, where the finite element space pair (Xh,Mh)forthe n n approximation (uh,ph)ofthevelocityu and the pressure p is constructed by the low-order finite element: the Q1 − P0 quadrilateral element or the P1 − P0 triangle element with mesh size h. Error estimates of the numerical solution n n ∈ (uh,ph) to the exact solution (u(tn),p(tn)) with tn (0,T] are derived.

1. Introduction In a primitive variable formulation for solving the Stokes equations and the Navier–Stokes equations, the importance of ensuring the compatibility of discrete velocity and pressure by satisfying the so-called inf-sup condition is widely un- derstood. In particular, it is well known that the simplest conforming low-order elements, such as the P1 − P0 (linear velocity, constant pressure) triangular element and the Q1 − P0 (bilinear velocity, constant pressure) quadrilateral element are not stable. During the last two decades there has been a rapid development in practical stabilization techniques for the P1 − P0 element and the Q1 − P0 element for solving the Stokes problem. For this purpose a local “macroelement condition” and some energy methods have been used. The use of such a macroelement condition as a mean of verifying the (Babuˇska–Brezzi) inf-sup condition is a standard technique (see, for example, Girault and Raviart [18]); the basic idea was first introduced by Boland and Nicolaides [8], and independently by Stenberg [43]. The stabilized mixed finite element approximation under consideration is based on the combination of the standard variational formulation of the Stokes problem and the bilinear form including a jump operator in pressure. The discrete velocity uh and the pressure ph are chosen from finite element subspaces Xh and Mh of the Sobolev spaces X and M defined in Section 2, related to conforming low-order elements like the P1 − P0 triangular element, or the Q1 − P0 quadrilateral element,

Received by the editor November 22, 2004 and, in revised form, September 2, 2005. 2000 Mathematics Subject Classification. Primary 35L70; Secondary 65N30, 76D06. Key words and phrases. Navier–Stokes problem, stabilized finite element, Crank–Nicolson extrapolation scheme. The first author was supported in part by the NSF of the People’s Republic of China (10371095). The second author was supported in part by the Research Grants Council of the Hong Kong Special Administrative Region, People’s Republic of China (Project No. City U 102103).

c 2006 American Mathematical Society 115

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which do not possess the properties required by the inf-sup condition. Recently, Kechkar and Silvester [33, 42], Kay and Silvester [32], Norburn and Silvester [36] and Silvester and Wathen [41] pursued some interesting work on both mathematical analysis and numerical tests of locally stabilized mixed finite element methods for the Stokes problem. Their work has also been extended to the Navier–Stokes problem and other related problems, (see, e.g., [26] and [21]), and some numerical analysis and tests of the stabilized finite element method for the stationary Navier–Stokes equations were provided by He et al. in [24]. The viscous incompressible Navier–Stokes equations with zero boundary conditions are one of the fundamental systems modelling fluid motion. The mathematical theory of these equations and their numerical solution are an important field of research. Here our aim is to solve the following time-dependent viscous incompressible Navier–Stokes problem: ut − ν∆u +(u ·∇)u + ∇p = f, divu =0, (x, t) ∈ Ω × (0,T]; (1.1) u(x, 0) = u0(x),x∈ Ω; u(x, t)|Γ =0,t∈ [0,T], in a bounded two-dimensional domain with some appropriate assumptions stated in Section 2, where u = u(x, t)=(u1(x, t),u2(x, t)) represents the velocity vector of a viscous incompressible fluid, p = p(x, t) the pressure, f = f(x, t) the prescribed body force, u0(x) the initial velocity, ν>0 the viscosity and T>0 a finite time. For the usual spatial discretization, i.e., time-continuous approximations, the finite element space pair (Xh,Mh) needs to satisfy some appropriate approximate properties and the compatibility properties required by the inf-sup condition (see Heywood and Rannacher [27]), where 0

u02 +sup{f(t)0 + ft(t)0 + ftt(t)0} < ∞. t∈[0,T ]

Then the spatial discrete solution (uh(t),ph(t)) satisfies the error following estimates [27]: 1/2 u(t) − uh(t)0 + hu(t) − uh(t)1 + hσ (t)p(t) − ph(t)0 (1.2) ≤ κh2, 0 ≤ t ≤ T, where σ(t)=min{1,t} and κ>0 is a general constant depending on the data (ν, Ω,u0,f,T) which may have different values at its different occurrences. For n n fully discrete approximations, the discrete solution (uh,ph) based on the Crank– Nicolson scheme, in which the viscous term and the nonlinear term are discretized implicitly, satisfies the error estimates [28] − m ≤ 2 3/2 − m ≤ (1.3) σ(tm) uh(tm) uh 0 κτ ,σ(tm) ph(tm) ph 0 κτ,

for all tm = mτ ∈ (0,T], where τ ≤ κ0 is the time step size, and κ0 some ﬁxed value depending on the data (ν, Ω,u0,f,T). s 1 3 It is noted that the factors σ (tn)withs = 2 , 1, 2 that appeared in (1.2)–(1.3) are due to the nonsmoothness of the time derivatives of the velocity u and pressure p at t =0.

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For the usual time discretization, i.e., spatial-continuous approximations, Shen [40] proposed a second-order projection scheme, in which the viscous term and the nonlinear term are treated implicitly and the pressure term explicitly. Guermond and Shen [19] studied a velocity-correction projection scheme for the linearized Navier–Stokes equations. The semi-discrete solution (un,pn) satisﬁes the error estimates m − n2 1/2 ≤ 2 − m ≤ 3/2 (τ u(tn) u 0) κτ , u(tm) u 0 κτ , n=n0 m m (1.4) u(tm) − u 1 + p(tm) − p 0 ≤ κτ, ∈ for all tm = mτ (tn0 ,T] under the assumption (A1), where 0

n n while the fully discrete solution (uh,ph) based on the backward Euler semi-implicit scheme satisﬁes the error estimates 1/2 − m ≤ 2 σ (tm) u(tm) uh 0 κ(h + τ), 1/2 − m − m ≤ 1/2 (1.7) σ (tm) u(tm) uh 1 + σ(tm) ph(tm) ph 0 κ(h + τ ), 1/2 (see He [21]) for all tm = mτ ∈ (0,T]andτ| log h| ≤ κ0. This paper continues our analysis of the stabilized mixed ﬁnite element method based on the Q1 − P0 quadrilateral element and the P1 − P0 triangle element [26, 21] for solving the two-dimensional time-dependent Navier–Stokes equations with respect to the data (u0,f) satisfying the assumption (A1). We consider the second order fully discrete scheme based on the Crank–Nicolson extrapolation scheme in

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which we use an implicit scheme for the viscous and pressure terms and a semi- n n implicit scheme for the nonlinear term. The discrete solution (uh,ph) satisﬁes the error estimates − m ≤ 2 3/2 (1.8) u(tm) uh 0 κ(h + τ ), − m ≤ 3/4 1/2 − m ≤ 3/4 (1.9) u(tm) uh 1 (h + τ ),σ (tm) p¯(tm) p¯h 0 κ(h + τ ), ∈ { }N for all tm (0,T], where Iτ = tm 0 is a given set in the interval [0,T]withatime step size τ =max1≤m≤N (tm − tm−1), and

m 1 m m−1 1 p¯ = (p + p ), p¯(t )= (p(t )+p(t − )). h 2 h h m 2 m m 1 The contents of this paper are divided into sections as follows. In Section 2, the abstract functional setting of the Navier–Stokes problem is given with some basic statements. Stabilized ﬁnite element approximations are recalled in Section 3. Some key technical lemmas and known results are provided in Section 4. The fully discrete stabilized ﬁnite element method with the Crank–Nicolson extrapolation scheme and the corresponding fully discrete duality problem are considered in section Section 5. The L2-andH1-error estimates for the discrete velocity and L2-error estimate for the discrete pressure are derived in Section 6.

2. Functional setting of the Navier–Stokes problem For the mathematical setting of problem (1.1), ﬁrst we introduce the Hilbert spaces 1 2 2 2 2 { ∈ 2 } X = H0 (Ω) ,Y= L (Ω) ,M= L0(Ω) = q L (Ω); qdx =0 . Ω The spaces L2(Ω)m,m =1, 2, 4, are endowed with the L2-scalar product and L2- norm denoted by (·, ·)and·0.ThespaceX is equipped with its equivalent scalar product (∇u, ∇v) and norm |u|1 = ∇u0. Next, let the closed subset V of X be given by V = {v ∈ X;divv =0}, and denote by H the closed subset of Y ; i.e.,

H = {v ∈ Y ;divv =0,v· n|∂Ω =0}. We refer the readers to [1, 2, 6, 7, 18, 27, 45] for more detail on these spaces. We also denote the Laplace operator by A = −∆ and denote the Stokes operator by A¯ = −P ∆, where P is the L2-orthogonal projection of Y onto H. As mentioned above, we need a further assumption on Ω: (A2) Assume that Ω is smooth so that the unique solution (v, q) ∈ (X, M)ofthe steady Stokes problem

−∆v + ∇q = g, divv =0 inΩ,v|∂Ω =0, for prescribed g ∈ Y exists and satisﬁes

v2 + q1 ≤ cg0, where c>0 is a generic constant depending on Ω and ν which may stand for a diﬀerent value at its diﬀerent occurrences, and ·i denotes the usual norm of Sobolev spaces Hi(Ω)m; i =0, 1, 2,m=1, 2, 4.

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We remark that the validity of assumption (A2) is known (see [18, 27, 31, 30, 45]) if ∂ΩisofC2, or if Ω is a two-dimensional convex polygon. From assumption (A2), it is known [1, 27, 34] that 2 ≤2 ≤ ¯ 2 ∈ 2 2 ∩ (2.1) Av 0 v 2 c Av 0,v H (Ω) V, 2 2 (2.2) v0 ≤ γ0|v|1,v∈ X, |v|1 ≤ γ0v2 ≤ cAv0,v∈ H (Ω) ∩ X,

where γ0 is positive constant depending only on Ω. We deﬁne the continuous bilinear forms a(·, ·)andd(·, ·)onX × X and X × M, respectively, by a(u, v)=ν(∇u, ∇v),u,v∈ X, d(v, q)=−(v, ∇q)=(q, divv),v∈ X, q ∈ M, and a trilinear form on X × X × X by 1 b(u, v, w)=((u ·∇)v, w)+ ((divu)v, w) 2 1 1 = ((u ·∇)v, w) − ((u ·∇)w, v), ∀u, v, w ∈ X. 2 2 With the above notations, the variational formulation of problem (1.1) reads as follows. Find (u, p) ∈ (X, M),t∈ (0,T] such that for all (v, q) ∈ (X, M),

(2.3) (ut,v)+a(u, v) − d(v, p)+d(u, q)+b(u, u, v)=(f,v),

(2.4) u(0) = u0. A simple modiﬁcation to the regularity argument given in [29, 27] allows us to obtain the following regularity results. Theorem 2.1. Assume that assumptions (A1) and (A2) are valid. Then the problem (2.3)–(2.4) admits a unique solution (u, p) satisfying the regularity results t 2 2 2 2 ≤ (2.5) u(t) 2 + p(t) 1 + ut(t) 0 + ut 1ds κ, 0 t 2 2 2 2 ≤ (2.6) σ(t) ut(t) 1 + σ(s)( utt 0 + ut 2 + pt 1)ds κ, 0 t 2 2 2 2 2 2 ≤ (2.7) σ (t)( ut(t) 2 + pt(t) 1 + utt(t) 0)+ σ (s) utt 1ds κ, 0 t 3 2 3 2 2 2 ≤ (2.8) σ (t) utt(t) 1 + σ (s)( uttt 0 + utt 2 + ptt 1)ds κ, 0 for all t ∈ [0,T].

3. Stabilized finite element approximation

Let h>0 be a real positive parameter. The ﬁnite element subspace (Xh,Mh) of (X, M) is characterized by τh = τh(Ω), a partitioning of Ω¯ into triangles K or quadrilaterals K, assumed to be uniformly regular as h → 0. For further details, the reader can refer to Ciarlet [13] and Girault and Raviart [18]. The mesh parameter h is given by h =max{hK }, and the set of all interelement boundaries will be denoted by Γh. Finite element subspaces of interest in this paper are deﬁned by setting P1(K)ifK is triangular, (3.1) R1(K)= Q1(K)ifK is quadrilateral,

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giving the continuous piecewise (bi)linear velocity subspace 0 2 Xh = {v ∈ C (Ω)¯ ∩ X; vi|K ∈ R1(K),i=1, 2,K∈ τh}, and the piecewise constant pressure subspace

Mh = {q ∈ M; q|K ∈ P0(K),K ∈ τh}. Note that neither of these methods are stable in the standard Babuˇska–Brezzi sense; the P1 − P0 triangle “locks” on regular grids (since there are more discrete incompressibility constraints than velocity degrees of freedom), and the Q1 − P0 quadrilateral is one example of unstable mixed methods, as elucidated by Sani et al. in [38]. In order to define a locally stabilized formulation of the time-dependent Navier– Stokes problem, we introduce a macroelement partitioning Λh as follows. Given any subdivision τh, a macroelement partitioning Λh may be defined such that each macroelement K is a connected set of adjoining elements from τh. Every element K must lie in exactly one macroelement, which implies that macroelements do not overlap. For each K, the set of interelement edges which are strictly in the interior of K will be denoted by ΓK. The length of edge e ∈ ΓK is denoted by he. With these additional definitions a locally stabilized discrete formulation of the problem (2.3)–(2.4) can be stated as follows.

Deﬁnition 3.1. Locally Stabilized Formulation. Find (uh,ph) ∈ (Xh,Mh) such that for all t ∈ (0,T]and(vh,qh) ∈ (Xh,Mh),

(3.2) (uht,vh)+Bh((uh,ph); (vh,qh)) + b(uh,uh,vh)=(f,vh),

(3.3) uh(0) = u0h,

where u0h ∈ Xh is an approximation of u0 and

Bh((uh,ph); (vh,qh)) = B((uh,ph); (vh,qh)) + βCh(ph,qh),

for all (uh,ph), (vh,qh) ∈ (Xh,Mh), here B((u, p); (v, q)) = a(u, v) − d(v, p)+d(u, q) ∀(u, p), (v, q) ∈ (X, M), Ch(p, q)= he [p]e[q]eds, K∈ ∈ Λh e ΓK e 1 for all p, q in the algebraic sum H (Ω)+Mh,[·]e is the jump operator across e ∈ ΓK and β>0 is the local stabilization parameter. A general framework for analyzing the locally stabilized formulation (3.2)–(3.3) can be developed using the notion of an equivalent class of macroelements. As in Stenberg [43], each equivalence class, denoted by EKˆ , contains macroelements which are topologically equivalent to a reference macroelement Kˆ. To illustrate the idea, two practical examples of locally stabilized mixed approximations are given below.

Example 3.1. The ﬁrst example is the standard Q1 − P0 approximation pair. A locally stabilized formulation (3.2)–(3.3) can be constructed in this case, if τh is such that the elements K can be grouped into 2 × 2 macroelements

K={K1,K2,K3,K4}, with the reference macroelement

Kˆ = {Kˆ1, Kˆ2, Kˆ3, Kˆ4}.

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An obvious way of constructing such a partitioning in practice is to form the grid τh by uniformly reﬁning a coarse grid Λh, such as by joining the midedge points.

Example 3.2. The triangular P1 − P0 approximation pair can be stabilized similarly if the partitioning τh is constructed such that the elements can be grouped into disjoint macroelements, all consisting of four elements. The following properties are classical (see [7, 13, 47, 48]): −1 (3.4) |vh|1 ≤ ch vh0,vh ∈ Xh. The following stability results of this mixed method for the macroelement partitions deﬁned above were formally established by Kay and Silvester [32] and Kechkar and Silvester [33].

Theorem 3.2. Given a stabilization parameter β ≥ β0 > 0, suppose that every macroelement K∈Λh belongs to one of the equivalence classes EKˆ , and that the following macroelement connectivity condition is valid: for any two neighboring K K ∈ macroelements 1 and 2 with K ∩K ds =0there exists v Xh such that 1 2

(3.5) supp v ⊂K1 ∪K2 and v · nds =0 . K1∩K2 Then, |C |≤ 2 2∇ 2 1/2 2 2∇ 2 1/2 (3.6) h(p, q) c ( ( p 0 + h p 0)dx) ( ( q 0 + h q 0)dx) , K K K∈τh 1 for all p, q ∈ H (Ω) + Mh,and B | | ≤ h((uh,ph); (vh,qh)) (3.7) α( uh 1 + ph 0) sup | | , (vh,qh)∈(Xh,Mh) vh + qh

for all (uh,ph) ∈ (Xh,Mh),and 1 (3.8) Ch(p, qh)=0, Ch(ph,q)=0, Ch(p, q)=0, ∀p, q ∈ H (Ω),ph,qh ∈ Mh,

where α>0 is a constant independent of h and β,andβ0 is some ﬁxed positive constant and n is the unit outward normal vector.

4. Technical preliminaries This section considers preliminary estimates which will be very useful in error estimates of the finite element solution (uh,ph). With the statements in Section 3, a discrete analogue Ah = −∆h of the Laplacian operator A = −∆ is defined through the condition that (−∆huh,vh)=((uh,vh)) for all uh,vh ∈ Xh. The operator Ah : Xh→Xh is invertible, with its inverse denoted −1 −1 Ah .SinceAh is self-adjoint and positive definite, we may define “discrete” Sobolev norms on Xh,ofanyorderr ∈ R, by setting | | r/2 ∀ ∈ vh r = Ah vh 0, vh Xh. These norms will be assumed to have various properties similar to their continuous counterparts, an assumption that implicitly imposes conditions on the structure of the spaces Xh and Mh. In particular, it holds that

|vh|0 = vh0, |vh|1 = ∇vh0, |vh|2 = Ahvh0, ∀vh ∈ Xh.

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By the way, we derive from (2.2) that

(4.1) vh0 ≤ γ0∇vh0, ∇vh0 ≤ γ0Ahvh0, ∀vh ∈ Vh,

where γ0 > 0 is a constant depending only on Ω. Moreover, we deﬁne the discrete gradient operator ∇h for qh ∈ Mh as

(vh, ∇hqh)=−d(vh,qh), ∀vh ∈ Xh. Now, by using a slightly modiﬁed argument on the estimates of the trilinear form b provided in [21, 22, 23, 27, 28, 29], we can obtain the following results on b. Lemma 4.1. The trilinear form b satisﬁes the estimates

(4.2) b(uh,vh,wh)=−b(uh,vh,wh),

(4.3) |b(uh,vh,wh)| + |b(vh,uh,wh)| + |b(wh,uh,vh)| c ≤ 0 u 1/2|u |1/2|v | w 1/2|w |1/2 2 h 0 h 1 h 1 h 0 h 1 c + 0 |u | v 1/2|v |1/2w 1/2|w |1/2, 2 h 1 h 0 h 1 h 0 h 1 (4.4) |b(uh,vh,wh)| + |b(vh,uh,wh)| + |b(wh,uh,vh)| c ≤ 0 A v 1/2v 1/2u 1/2u 1/2w 2 h h 0 h 1 h 0 h 1 h 0 c + 0 A v 1/2v 1/2u w , 2 h h 0 h 0 h 1 h 0

for all uh,vh,wh ∈ Xh,wherec0 > 0 is a constant depending only on Ω.

In order to derive the error estimates of the ﬁnite element solution (uh,ph), we also deﬁne the Galerkin projection (Rh,Qh):(X, Y )→(Xh,Mh) by requiring

Bh((Rh(u, p),Qh(u, p)); (vh,qh)) = B((u, p); (vh,qh)),

(4.5) ∀(u, p) ∈ (X, M), (vh,qh) ∈ (Xh,Mh).

Note that, due to Theorem 3.2, (Rh,Qh) is well defined and satisfies the approximate properties [24] 2 (4.6) Rh(u, p) − u0 + h(|Rh(u, p) − u|1 + Qh(u, p) − p0) ≤ ch (u2 + p1), for all (u, p) ∈ (H2(Ω)2 ∩ X, H1(Ω) ∩ M). 2 Moreover, we need to introduce the following L -orthogonal projection Ph : 2 2 L (Ω) →Xh defined by 2 2 (Phv, vh)=(v, vh),v∈ L (Ω) ,vh ∈ Xh. Using some slight modifications of the literature [26, 27], we can obtain the following error estimates. Theorem 4.2. Assume the assumptions of Theorems 2.1 and 3.2 are valid and set (u0h,p0h)=(Rh(u0,p0),Qh(u0,p0)).Then(uh,ph) satisfies (4.7) u(t) − u (t) + h(u(t) − u (t) + σ1/2(t)p(t) − p (t) ) ≤ κh2, h 0 h 1 h 0 t | − |2 1/2 ≤ (4.8) ( σ(s) ut uht 1ds) κh, 0 1 for all t ∈ (0,T],wherep0 = limt→0 p(t) ∈ H (Ω) ∩ M.

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Since our error analysis for the time discretization depends heavily on some regularity estimates of the semi-discrete solution (uh,ph), we will provide the following regularity results.

Theorem 4.3. Under the assumptions of Theorems 4.2, the finite element solution (uh,ph) satisfies the regularities t −1 ∇ 2 2 2 (4.9) ( Ah (uhtt + Ph hpht) 0 + uht 0 + Ahuh 0)ds 0 2 + ν|uh(t)| + βCh(ph(t)a, ph(t)) ≤ κ, 1 t −1/2 ∇ 2 | |2 C (4.10) ( Ah (uhtt + Ph hpht) 0 + ν uht 1 + β h(pht,pht))ds 0 2 2 2 +(uht(t) + ∇hph(t) + Ahuh(t) ) ≤ κ, 0 0 0 t 2 ∇ 2 2 ≤ (4.11) σ(s)( uhtt 0 + Ph hpht 0 + Ahuht 0)ds κ, 0 t −1 ∇ 2 (4.12) σ(s) Ah (uhttt + Ph hphtt) 0ds 0 2 + σ(t)(ν|uht(t)| + βCh(pht(t),pht(t))) ≤ κ, 1 t 2 −1/2 ∇ 2 | |2 C (4.13) σ (s)( Ah (uhttt + Ph hphtt) 0 + ν uhtt 1 + β h(phtt,phtt))ds 0 2 2 + σ (t)uhtt(t) ≤ κ, 0 t 3 2 ∇ 2 2 ≤ (4.14) σ (s)( uhttt 0 + Ph hphtt 0 + Ahuhtt 0)ds κ, 0 for all t ∈ [0,T]. Proof. The proof is by a fairly standard energy argument. Hereafter, we will make frequent use of (4.1)–(4.4) without explicit mention. First, differentiating (3.2) with respect to t results in the equations

(4.15) (uhtt + Ph∇hpht,vh)+a(uht,vh)+d(uht,qh)+βCh(pht,qh)

+ bt(uh,uh,vh)=(ft,vh),

(4.16) (uhttt + Ph∇hphtt,vh)+a(uhtt,vh)+d(uhtt,qh)+βCh(phtt,qh)

+ btt(uh,uh,vh)=(ftt,vh).

If we take qh = 0 in (4.15)–(4.16), a simple calculation yields t t −i/2 ∇ 2 ≤ 1−i/2 2 | |2 2 (4.17) Ah (uhtt + Ph hpht) 0ds c Ah uht 0(1 + uh 1)+ ft 0)ds, 0 0 t t 3−i −i/2 ∇ 2 ≤ 3−i 1−i/2 2| |2 (4.18) σ (s) Ah (uhttt + Ph hphtt) 0ds c σ (s) Ah uht 0 uht 1ds 0 0 t 3−i 1−i/2 2 | |2 2 + c (σ (s) Ah uhtt 0(1 + uh 1)+ ftt 0)ds, 0 for i =2, 1, 0. Next, using (3.4), we obtain | | (Ahuh,vh) ≤ −1| − | (4.19) Ahuh 0 =sup (ch u uh 1 + Au 0), vh∈Xh vh 0

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for all uh ∈ Xh and u ∈ X. Hence, we derive from Theorems 2.1 and 4.2 that 2 ≤ −2| − |2 2 ≤ ∀ ≤ ≤ (4.20) Ahuh(t) 0 c(h u(t) uh(t) 1 + Au(t) 0) κ, 0 t T. Now, by recalling [26, 21], we have t | |2 C 2 ≤ ∀ ≤ ≤ (4.21) ν uh(t) 1 + β h(ph(t),ph(t)) + uht 0ds κ, 0 t T. 0

Taking (vh,qh)=(uht,pht) in (4.15) and using (2.3) and (4.1)–(4.3), we obtain 1 d (4.22) u 2 + ν|u |2 + βC (p ,p )+b(u ,u ,u )=(f ,u ), 2 dt ht 0 ht 1 h ht ht ht h ht t ht ν (4.23) |b(u ,u ,u )|≤cu |u | A u ≤ |u |2 + cA u 2u 2, ht h ht ht 0 ht 1 h h 0 4 ht 1 h h 0 ht 0 ν (4.24) (f ,u ) ≤ |u |2 + cf 2. t ht 4 ht 1 t 0

From the deﬁnition of (uh(0),ph(0)) = (Rh(u0,p0),Qh(u0,p0)), it holds that

(4.25) (uht(0),vh) − ν(∆u0,vh)+(vh, ∇p0)+b(uh(0),uh(0),vh)=(f(0),vh),

which with (4.1) and (4.2) yield

(4.26) uht(0)0 ≤ cu02 + p01 + cAhuh(0)0|uh(0)|1 + f(0)0.

Now we integrate (4.22) and use (4.23)–(4.24), (4.26) and (4.20)–(4.21) to obtain t 2 | |2 C ≤ ≤ ≤ (4.27) uht(t) 0 + (ν uht 1 + β h(pht,pht))ds κ, 0 t T. 0 Combining (4.17) with (4.20)–(4.21) and (4.27) and using (3.2) yields (4.9)–(4.10). Furthermore, we derive from (4.19), (4.8), (4.2) and (2.6) that t t 2 ≤ −2| − |2 2 ≤ (4.28) σ(s) Ahuht 0ds c σ(s)(h ut uht 1 + Aut 0)ds κ, 0 0 for all 0 ≤ t ≤ T , which (4.9) and (4.17) with i =0yield t ∇ 2 ≤ ∀ ≤ ≤ (4.29) σ(s) uhtt + hpht 0ds κ, 0 t T. 0 Moreover, we derive from (4.15) that

(uhtt,vh)+a(uht,vh) − d(vh,pht)+d(uhtt,qh)+βCh(phtt,qh)

(4.30) + b(uht,uh,vh)+b(uh,uht,vh)=(ft,vh).

By taking (vh,qh)=σ(t)(uhtt,pht) in (4.30), we obtain 1 d σ(t)u 2 + (σ(t)ν|u |2 + σ(t)βC (p ,p )) htt 0 2 dt ht 1 h ht ht + σ(t)b(uht,uh,uhtt)+σ(t)b(uh,uht,uhtt) 1 d (4.31) = σ(t)(ν|u |2 + βC (p ,p )) + σ(t)(f ,u ). 2 dt ht 1 h ht ht t htt

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Using (4.1)–(4.3), we have

σ(t)|b(uht,uh,uhtt)| + σ(t)|b(uh,uht,uhtt)| ≤ | | 2 cσ(t) uht 1 Ahuh 0 uhtt 0 1 ≤ σ(t)u 2 + cσ(t)|u |2A u 2, 4 htt 0 ht 1 h h 0 1 σ(t)|(f ,u )|≤ σ(t)u 2 + cf 2. t htt 4 htt 0 t 0 Combining these inequalities with (4.31) yields d σ(t)u 2 + (σ(t)ν|u |2 + σ(t)βC (p ,p )) htt 0 dt ht 1 h ht ht ≤ | |2 C | |2 2 2 (4.32) ν uht 1 + β h(pht,pht)+cσ(t) uht 1 Ahuh 0 + ft 0.

In view of (4.10), there exists a sequence n→0 such that | |2 C → σ( n)ν uht( n) 1 + σ( n)β h(pht( n),pht( n)) 0.

Therefore, integrating (4.32) from n to t and letting n→0, one ﬁnds t 2 | |2 C σ(s) uhtt 0ds + σ(t)ν uht 1 + σ(t)β h(pht(t),pht(t)) 0 t ≤ | |2 C (4.33) (ν uht 1 + β h(pht,pht))ds 0 t t | |2 2 2 + c σ(s) uht 1 Ahuh 0ds + c ft 0ds. 0 0 Applying the Gronwall lemma to (4.33) and using (4.10), we obtain t 2 | |2 C ≤ ≤ ≤ (4.34) σ(s) uhtt 0ds + σ(t)ν uht(t) 1 + σ(t)β h(pht(t),pht(t)) k, 0 t T. 0 Combining (4.34) with (4.28)–(4.29) and using (4.18) with i=2 yields (4.11)–(4.12). Similarly, we can prove (4.13)–(4.14) by using (3.2), (3.4), (4.1)–(4.4) and (4.9)– (4.12).

Also, we will introduce some discrete versions of the Gronwall lemma by a slightly improved argument used in [16, 39].

Lemma 4.4. Let C, τ and an,bn,dn, for integers n ≥ n1, be nonnegative numbers such that m m−1 (4.35) am + τ bn ≤ τ andn + C, ∀m ≥ n1.

n=n1 n=n1 Then,

m m−1 (4.36) am + τ bn ≤ exp(τ dn)C, ∀m ≥ n1.

n=n1 n=n1

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5. Fully discrete stabilized finite element method In this section we consider the time discretization of the stabilized finite element T approximation. Let tn = nτ,whereτ = N and N is an integer. The Crank– Nicolson extrapolation scheme applied to the stabilized finite element approxima- { n}N ⊂ { n}N ⊂ tion is to determine the series uh n=1 Xh, ph n=1 Mh as the solution of the recursive linear equation n B n n n n ¯ (5.1) (dtuh,vh)+ h((¯uh, p¯h); (vh,qh)) + b(φ(uh), u¯h,vh)=(f(tn),vh), ∈ 0 0 for all (vh,qh) (Xh,Mh) for the initial value (uh,ph)=(u0h,p0h). Here and after, −1 3/2 1/2 we define a small time step size τ0 = T τ on time interval (0,τ ] and a large T 1/2 time step size τ = on [τ ,T] for some fixed integer n0,where n0 T − τ 1/2 2T τ n = τ 1/2,N= n + = − τ −1/2, 0 0 0 τ τ 1 d un = (un − un−1),k = τ as 1 ≤ n ≤ n ,k= τ as n +1≤ n ≤ N, t h k h h 0 0 0 3 1 φ(u0 )=φ(u1 )=u0 ,φ(un)= un−1 − un−2, 2 ≤ n ≤ N, h h h h 2 h 2 h n 1 n n−1 1 u¯ = (u + u ), u¯ (t )= (u (t )+u (t − )). h 2 h h h n 2 h n h n 1 From the definition of φ, we find that the Crank–Nicolson extrapolation scheme is an implicit scheme for the viscous and pressure terms and a semi-implicit scheme for the nonlinear term. n n − n − n In order to analyze the discretization errors (eh, µ¯ )=(uh(tn) uh, p¯h(tn) p¯h) with (e0 ,µ0 )=(0, 0), we deduce from (3.2) that h h 1 1 tn (uh(tn) − uh(tn−1),vh)+ Bh((uh(t),ph(t)); (vh,qh))dt k k t − (5.2) n 1 1 tn 1 tn + b(uh(t),uh(t),vh)dt = (f(t),vh)dt k tn−1 k tn−1

for all (vh,qh) ∈ (Xh,Mh). Subtracting (5.1) from (5.2) and using the following formulas

d(¯uh(tn),qh)+βCh(¯ph(tn),qh)=0, 1 tn (d(u (t),q )+βC (p (t),q ))dt =0, k h h h h h tn−1 tn tn ¯ 1 1 φ(tn) − φ(t)dt = (tn − t)(t − tn−1)φtt(t)dt, k tn−1 2k tn−1 2 for all φ ∈ H (tn−1,tn; F )forsomeHilbertspaceF , then using (3.2), results in n n n n n n (dte ,vh)+Bh((¯e , µ˜ ); (vh,qh)) + b(φ(e ), u¯h(tn),vh)+β(φ(u ), e¯ ,vh) h h h h h h (5.3) β tn =(E˜n,vh) − (tn − t)(t − tn−1)Ch(phtt,qh)dt, 2k tn−1

for all (vh,qh) ∈ (Xh,Mh), and n B n n n n n (dteh,vh)+ h((¯eh, µ¯h); (vh,qh)) + b(φ(eh), u¯h(tn),vh)+b(φ(uh), e¯h,vh) (5.4) =(En,vh),

License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use STABILIZED FEM BASED ON THE C-N EXTRAPOLATION SCHEME 127 n 1 tn n for all (vh,qh) ∈ (Xh,Mh), whereµ ˜ = ph(t)dt − p¯ and h k tn−1 h 1 t1 (5.5) (E˜ ,v )=− (t − t)(t − t )(u + P ∇ p ,v )dt 1 h 2k 1 0 httt h h htt h t0 1 t1 − b( u dt, u (t ),v ), 2 ht h 1 h t0 1 tn (5.6) (E˜ ,v )=− (t − t)(t − t − )(u + P ∇ p ,v )dt n h 2k n n 1 httt h h htt h tn−1 1 tn tn−1 − b( (t − t)u dt − (t − t − )u dt, u¯ (t ),v ) 2 n htt n 2 htt h n h tn−1 tn−2 1 tn tn − b( uhtdt, uhtdt, vh), 4 tn−1 tn−1

for all 2 ≤ n ≤ N,and 1 t1 (5.7) (E ,v )=− (t − t)(t − t )(u ,v )dt 1 h 2k 1 0 httt h t0 1 t1 − b( u dt, u (t ),v ), 2 ht h 1 h t0 1 tn (5.8) (E ,v )=− (t − t)(t − t − )(u ,v )dt n h 2k n n 1 httt h tn−1 1 tn tn−1 − b( (t − t)u dt − (t − t − )u dt, u¯ (t ),v ) 2 n htt n 2 htt h n h tn−1 tn−2 1 tn tn − b( uhtdt, uhtdt, vh), 4 tn−1 tn−1

for all 2 ≤ n ≤ N. n n In order to provide the bound of the error (eh, µ¯h), we need to provide the bound of E˜n and En.

Lemma 5.1. Under the assumptions of Theorem 4.2, it holds that

n0 i −1/2 ˜ 2 ≤ i+2 (5.9) τ0 σ (tn) Ah PhEn 0 κτ0 ,i=0, 1, n=1 N −1/2 ˜ 2 ≤ 3 (5.10) τ Ah PhEn 0 κτ , n=n0+1 n0 i 2 ≤ i+1 (5.11) τ0 σ (tn) PhEn 0 κτ0 ,i=0, 1, n=1 N i 2 ≤ 5/2+i/2 (5.12) τ σ (tn) PhEn 0 κτ ,i=0, 1. n=n0+1

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Proof. In this proof for brevity we will make frequent use of (4.1)–(4.4) and the facts

σ(tn) ≤ σ(tn−1)+τ0, 1 ≤ n ≤ n0, (5.13) σ(tn−1) ≤ 2σ(tn−2),n0 +1≤ n ≤ N, without explicit mention in this section. First, we derive from (5.6) and the Schwarz inequality that | ˜ | −1/2 ˜ (En,vh) Ah PhEn 0 =sup ∈ 1/2 vh Xh A v h h 0 tn 1 −1/2 ≤ (t − t)(t − t − )A (u + P ∇ p ) dt k n n 1 h httt h h htt 0 tn−1 tn tn−1 + κ( (tn − t)|uhtt|1dt + (t − tn−2)|uhtt|1dt) tn−1 tn−2 tn (5.14) 2 + κ( |uht|1dt) tn−1 tn −1/2 2 2 ≤ κk ( (tn − t) k tn−1 −1/2 2 2 1/2 × (A (uhttt + Ph∇hphtt) + |uhtt| )dt) h 0 1 tn−1 tn −1/2 − 2 2| |2 1/2 | |2 + κk ( (t tn−2) k uhtt 1dt) + κk uht 1dt, tn−2 tn−1

with k = τ0 for 1 ≤ n ≤ n0 and k = τ for n0 +1≤ n ≤ N. Similarly, we deduce from (5.5) that −1/2 ˜ ≤ Ah PhE1 0 c0γ0 Ahuh(t1) 0 sup uht(t) 0τ0 t ≤t≤t 0 1 (5.15) t1 1/2 2 −1/2 ∇ 2 1/2 + κτ0 ( σ (t) Ah (uhttt + Ph hphtt) 0dt) . t0 Thus, by using (5.13)–(5.14) and Theorem 4.3, we obtain i −1/2 2 (5.16) σ (tn)A E˜n τ0 h 0 tn ≤ 2+i 2 −1/2 ∇ 2 | |2 κτ0 (σ (t) Ah (uhttt + Ph hphtt) 0 + uht 1)dt tn−1 tn 2+i 2 | |2 ≤ ≤ + κτ0 σ (t) uhtt 1dt, 2 n n0,i=0, 1, tn−2 −1/2 2 (5.17) A E˜n τ h 0 tn ≤ 4 −2 2 −1/2 ∇ 2 κτ σ (tn−1) σ (t)( Ah (uhttt + Ph hphtt) 0 tn−1 tn | |2 4 −2 2 | |2 + uht 1)dt + κτ σ (tn−1) σ (t) uhtt 1dt, tn−2

for all n0 +1≤ n ≤ N. Summing (5.16) from n =2ton = n0 and (5.17) from n = n0 +1 to n = N, respectively, using (5.15) and Theorem 4.3 and noting −2 −2 −1 − ≤ ≤ ≤ σ (tn 1) tn0 = τ for all n0 +1 n N, we have obtained (5.9) and (5.10).

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Furthermore, we derive from (5.7)–(5.8) that

|(E1,vh)| (5.18) PhE10 =sup ∈ v vh Xh h 0 t1 ≤ 1/2 | |2 1/2 c0γ0τ0 Ahuh(t1) 0( uht(t) 1dt) t0 +2 sup uht(t)0, t0≤t≤t1

|(En,vh)| (5.19) PhEn0 =sup ∈ vh0 vh Xh tn −1/2 2 2 1/2 ≤ k ( (tn − t) (t − tn−1) uhttt0dt) Ahu¯h(tn)0 tn−1 tn −1/2 − 2 2| |2 1/2 + ck ( (tn t) k uhtt 1dt) tn−1 tn−1 −1/2 − 2 2| |2 1/2 + ck ( (t tn−2) k uhtt 1dt) Ahu¯h(tn) 0 tn−2 tn tn 1/2 2 1/2 | |2 1/2 + cκ ( k Ahuht 0dt) ( uht 1dt) , tn−1 tn−1 for all 2 ≤ n ≤ N. Hence, we derive from (5.19), (5.13), and Theorem 4.3 that

(5.20) tn i 2 ≤ i+1 3 2 | |2 σ (tn) PhEn 0τ0 κτ0 (σ (t) uhttt 0 + uht 1)dt tn−1 tn i+1 2 | |2 ≤ ≤ + κτ0 σ (t) uhtt 1dt, 2 n n0, tn−2 (5.21) tn i 2 ≤ 4 −3+i 3 2 | |2 σ (tn) PhEn 0τ κτ σ (tn−1) (σ (t) uhttt 0 + uht 1)dt tn−1 tn 4 −3+i 3 | |2 ≤ ≤ + κτ σ (tn−1) σ (t) uhtt 1dt, n0 +1 n N. tn−2

Summing (5.20) from n =2ton = n0 and (5.21) from n = n0 +1 to n = N, −1 −1 −1/2 − ≤ ≤ respectively, using Theorem 4.3 and (5.18) and noting σ (tn 1) tn0 τ for all n0 +1≤ n ≤ N, we arrive at (5.11) and (5.12). 6. Error analysis n n In this section we will analyze the error (eh, µ¯h). With the aid of Lemma 5.1, we shall obtain the following lower-order error estimates. Lemma 6.1. Under the assumptions of Theorem 4.2, it holds that m m2 | n|2 C n n ≤ 2 ≤ ≤ (6.1) eh 0 + τ0 (ν e¯h 1 + β h(˜µh, µ˜h)) κτ0 , 1 m n0, n=1 m m2 | n|2 C n n ≤ 2 3 ≤ ≤ (6.2) eh 0 + τ (ν e¯h 1 + β h(˜µh, µ˜h)) κτ0 + κτ ,n0 +1 m N. n=1

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n n Proof. First, we take (vh,qh)=2(¯eh, µ˜h)k in (5.3) and use (4.1)–(4.2) and (4.4), obtaining n2 − n−12 | n|2 C n n eh 0 eh 0 +2ν e¯h 1k +2β h(˜µh, µ˜h)k n n n = −2b(φ(e ), u¯h(tn), e¯ )k +2(E˜n, e¯ )k h h h tn − − − C n β (t tn)(t tn−1) h(phtt, µ˜h)dt t − (6.3) n 1 ≤ | n|2 C n n −1 −1/2 ˜ 2 ν e¯h 1k + β h(˜µh, µ˜h)k +4ν Ah PhEn 0k −1 2 2 2 n 2 +4ν c γ Ahu¯h(tn) φ(e ) k 0 0 0 h 0 tn 2 2 + β (t − tn) (t − tn−1) Ch(phtt,phtt)dt, tn−1

for all 1 ≤ n ≤ N with k = τ0 for 1 ≤ n ≤ n0 and k = τ for n0 +1≤ n ≤ N. Summing (6.3) from n =1ton = m for m ≤ n0 and n = n0 +1 to n = m for 0 −2 −2 −1 n +1≤ m ≤ N, respectively, noting e =0,σ (t − ) ≤ τ = τ ,n +1≤ 0 h n 1 n0 0 n ≤ N and using Lemma 5.1 and Theorem 4.3, we obtain m m2 | n|2 C n n (6.4) e 0 + τ0 (ν e¯h 1 + β h(˜µh, µ˜h)) n=1 m−1 ≤ 2 n2 ≤ ≤ κτ0 + κτ0 eh 0, 1 m n0, n=1 m m2 | n|2 C n n (6.5) e 0 + τ (ν e¯h 1 + β h(˜µh, µ˜h)) n=1 m−1 ≤ n0 2 3 n2 ≤ ≤ eh 0 + κτ + κτ eh 0,n0 +1 m N. n=1 Applying Lemma 4.4 to (6.4) and (6.5), respectively, we arrive at (6.1) and (6.2).

Lemma 6.2. Under the assumptions of Theorem 4.2, the following estimates hold: m | m|2 C m m n2 ≤ ≤ ≤ (6.6) ν eh 1 + β h(µh ,µh )+τ0 dteh 0 κτ0, 1 m n0, n=1

m | m|2 C m m n2 ≤ 2 ≤ ≤ (6.7) ν eh 1 + β h(µh ,µh )+τ dteh 0 κτ0 + κτ ,n0 +1 m N, n=n0+1 where µ0 =0,µn =2¯µn − µn−1. Proof. In view of the fact 0 C 0 ∀ ∈ d(eh,qh)+β h(µh,qh)=0, qh Mh, we derive from (5.4) that n n − n n C n (dteh,vh)+a(¯eh,vh) d(vh, µ¯h)+d(dteh,qh)+β h(dtµh,qh) n n n (6.8) + b(φ(uh), e¯h,vh)+b(φ(eh), u¯h(tn),vh)

=(En,vh), ∀(vh,qh) ∈ (Xh,Mh).

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n n By setting (vh,qh)=2(dteh, µ¯h)k in (6.8), we obtain n2 | n|2 −| n−1|2 C n n −C n−1 n−1 2 dteh 0k + ν( eh 1 eh 1)+β( h(µh,µh) h(µh ,µh )) n n n n (6.9) +2b(φ(eh), u¯h(tn),dteh)k +2b(φ(uh(tn)), e¯h,dteh)k − n n n n 2b(φ(eh), e¯h,dteh)k =2(En,dteh)k. Thus, it follows from (4.1)–(4.4), Theorem 4.3 and Lemma 6.1 that | n n | | n n | 2 b(φ(eh), u¯h(tn),dteh) k +2b(φ(uh(tn)), e¯h,dteh) k ≤ | n| | n | n 2c0γ0( Ahφ(uh(tn)) 0 e¯h 1 + Ahu¯h(tn) 0 φ(eh) 1) dte 0k 1 ≤ d en2k + κ(|en−1|2 + |en−2|2 + |e¯n|2)k, 2 t 0 h 1 h 1 h 1 1 2|(E ,d en)|k ≤ d en2k +4P E 2k, n t h 4 t 0 h n 0 | n n n | ≤ n 1/2| n |1/2| n| n1/2| n|1/2 2 b(φ(eh), e¯h,dteh) k c0 φ(eh) 0 φ(eh) 1 e¯h 1 dteh 0 dteh 1 k n 1/2| n |1/2 | n1/2| n|1/2| n| + c0 φ(eh) 0 φ(eh) 1 e¯h 0 e¯h 1 dteh 1k ≤ | n |1/2 | n| | n − n−1|1/2 | n|1/2| n − n−1| κ φ(eh) 1 ( e¯h 1 eh eh 1 + e¯h 1 eh eh 1)k ≤ | n|2 | n−1|2 | n−2|2 κ( e¯ 1 + eh 1 + eh 1)k. Combining these inequalities with (6.9) yields | n|2 −| n−1|2 C n n −C n−1 n−1 n2 (6.10) ν( eh 1 eh 1)+β( h(µh,µh) h(µh ,µh )) + dteh 0τ0 ≤ | n−1|2 | n−2|2 | n|2 2 κ( eh 1 + eh 1)τ0 + κ( e¯h 1 + PhEn 0)τ0, | n|2 −| n−1|2 C n n −C n−1 n−1 n2 (6.11) ν( eh 1 eh 1)+β( h(µh,µh) h(µh ,µh )) + dteh 0τ ≤ | n−1|2 | n−2|2 | n|2 2 κ( eh 1 + eh 1)τ + κ( e¯h 1 + PhEn 0)τ,

Summing (6.10) from n =1ton = m for 1 ≤ m ≤ n0 and summing (6.11) from n = n0 +1 to n = m for n0 +1≤ m ≤ N, respectively, then using Lemmas 5.1 and 6.1 and Theorem 4.3, we obtain m | m|2 C m m n2 (6.12) ν eh 1 + β h(µh ,µh )+τ0 dteh 0 n=1 m−1 ≤ | n|2 ≤ ≤ κτ0 + κτ0 eh 1, 1 m n0, n=1 m | m|2 C m m n2 (6.13) ν eh 1 + β h(µh ,µh )+τ0 dteh 0 n=1 m−1 ≤ | n0 |2 C n0 n0 2 | n|2 ≤ ≤ ν eh 1 +β h(µh ,µh )+κτ +κτ eh 1,n0 +1 n N. n=n0+1 Applying Lemma 4.4 to (6.12) and 6.13), respectively, we obtain (6.6)–(6.7).

Lemma 6.3. Under the assumptions of Theorem 4.2, we have m2 ≤ ≤ ≤ (6.14) σ(tm) dteh 0 κτ0, 1 n n0, m2 ≤ 3/2 ≤ ≤ (6.15) σ(tm) dteh 0 κτ0 + κτ ,n0 +1 n N.

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Proof. From (5.3) we derive

1 (d en − d en−1,v )+B ((d e¯n,d µ˜n); (v ,q )) k t h t h h h t h t h h h n n n−1 n + b(φ(dtuh), e¯h,vh)+b(φ(dtuh ),dte¯h,vh) n n−1 + b(φ(dte ), u¯h(tn),vh)+b(φ(e ),dtu¯h(tn),vh) h h (6.16) β tn =(d E˜ ,v ) − (t − t)(t − t − )C (p ,q )dt t n h 2k2 n n 1 h htt h tn−1 β tn−1 − − − − − C 2 (tn 1 t)(t tn 2) h(phtt,qh)dt, 2k tn−2

∀(vh,qh) ∈ (Xh,Mh),

whereweusethenotationoft−2 = t−1 = t0. n n By setting (vh,qh)=2(dte¯h,dtµ¯h)k in (6.16) and using (4.2), we obtain

n2 − n−12 | n|2 C n n dteh 0 dteh 0 +2(ν dte¯h 1 + β h(dtµ˜h,dtµ˜h))k − n n n n n + 2b(φ(dteh), e¯h,dte¯h)k +2b(φ(dteh), u¯h(tn),dte¯h) n n n−1 n +2b(φ(dtuh(tn)), e¯ ,dte¯ )k +2b(φ(e ),dtu¯h(tn),dte¯ )k h h h h (6.17) tn n β n =2(d E˜ ,d e¯ )k − (t − t)(t − t − )C (p ,d µ˜ )dt t n t h 2k n n 1 h htt t h tn−1 tn−1 − β − − C n (tn−1 t)(t tn−2) h(phtt,dtµ˜h)dt. 2k tn−2

Then, a simple calculation, using (4.1)–(4.4), Theorem 4.3 and Lemmas 6.1 and 6.2, yields

| n n n | ≤ | n | | n| | n| 2 b(φ(dteh), e¯h,dte¯h) k 2c0γ0 φ(dteh) 1 e¯h 1 dte¯h 1k ν ≤ |d e¯n|2k +4ν−1c2γ2|φ(en − en−1)|2|e¯n|2k−1, 4 t h 1 0 0 h h 1 h 1 n−1 n n−1 n 2|b(φ(e ),dtu¯h(tn),dte¯ )|k ≤ 2c0γ0|φ(e )|1|dtu¯h(tn)|1|dte¯ |1k h h h h tn ≤ ν | n|2 −1 2 2 | |2 | n−1 |2 dte¯h 1k +4ν c0γ0 uht 1dt φ(eh ) 1, 4 tn−2 n n n n 2|b(φ(dtuh(tn)), e¯ ,dte¯ )|k ≤ 2c0γ0|φ(dtuh(tn))|1|e¯ |1|dte¯ |1k h h h h tn−1 ≤ ν | n|2 −1 2 2 | |2 | n|2 dte¯h 1k +5ν c0γ0 uht 1dt e¯h 1, 4 tn−3 | n n | ≤ n | n| 2 b(φ(dteh), u¯h(tn),dte¯h) k 2c0γ0 φ(dteh) 0 Ahu¯h(tn) 0 dte¯h 1k ν ≤ |d e¯n|2k +4ν−1c2γ2φ(d en)2A u¯ (t )2k, 4 t h 1 0 0 t h 0 h h n 0

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ν − 2|(d E ,d e¯n)|k ≤ |d e¯n|2k +8ν−1(A 1/2P E 2 t n t h 4 t h 1 h h n 0 −1/2 2 −1 + A PhEn−1 )k , h 0 tn β n β n n | (t − t)(t − t − )C (p ,d µ˜ )dt|≤ C (d µ˜ ,d µ˜ )k 2k n n 1 h htt t h 2 h t h t h tn−1 tn β 2 2 + (t − t) (t − t − ) C (p ,p )dt, 8k2 n n 1 h htt htt tn−1 tn−1 β n β n n | (t − − t)(t − t − )C (p ,d µ˜ )dt|≤ C (d µ˜ ,d µ˜ )k 2k n 1 n 2 h htt t h 2 h t h t h tn−2 tn−1 β 2 2 − − − − C + 2 (tn 1 t) (t tn 2) h(phtt,phtt)dt, 8k tn−2 for all 2 ≤ n ≤ N. Combining these inequalities with (6.17) and using (5.13) yields n2 − n−12 ≤ n−12 σ(tn) dteh 0 σ(tn−1) dteh 0 dteh 0k −1 −1/2 2 −1/2 2 −1 +8ν (σ(tn)A PhEn +2σ(tn−1)A PhEn−1 )k h 0 h 0 tn −1 2 2| n − n−1 |2| n|2 −1 −1 2 2 | |2 | n−1 |2 +4ν c0γ0 φ(eh eh ) 1 e¯h 1k +4ν c0γ0 uht 1dt φ(eh ) 1 tn−2 tn−1 (6.18) −1 2 2 | |2 | n|2 −1 2 2 n 2 2 +5ν c0γ0 uht 1dt e¯h 1 +4ν c0γ0 φ(dteh) 0 Ahu¯h(tn) 0k tn−3 β tn + k (t − t − )σ(t)C (p ,p )dt 4 n 1 h htt htt tn−1 β tn−1 + k (t − tn−2)σ(t)Ch(phtt,phtt)dt. 4 tn−2

Summing (6.18) from n =1ton = m for 1 ≤ m ≤ n0 and (6.18) from n = n0 +1 to n = m for n0 +1≤ m ≤ N, then using Lemmas 5.1, 6.1, and 6.2, and Theorem −1 −1 −1/2 − ≤ ≤ ≤ 4.3 and noting σ (tn 1) tn0 = τ for all n0 +1 n N, we obtain (6.14) and (6.15).

m − m Lemma 6.4. Under the assumptions of Theorem 4.2,theerrorµ¯ =¯p(tm) p¯h satisﬁes the bound 1/2 − m ≤ 1/2 3/4 ≤ ≤ (6.19) σ (tm) p¯h(tm) p¯h 0 κτ0 + κτ , 1 m N. Proof. First, we derive from (5.4) and Theorem 3.2 that 1/2 m ≤ | m| m | m | | | | | | m| σ (tm) µ¯h 0 c e¯h 1 + c dteh 0 + c φ(eh ) 1 uh(tm) 1 + c φ(uh(tm)) 1 e¯h 1 | m | | m| 1/2 ≤ ≤ + c φ(eh ) 1 e¯h 1 + cσ (tm) Em 0, 1 m N. Then, by using Theorem 4.3 and Lemmas 6.2, 6.3, and 5.1 in the above estimate, we obtain 1/2 m ≤ | m| | m | 1/2 m 1/2 σ (tm) µ¯h 0 κ( e¯h 1 + φ(eh ) 1 + σ (tm) dteh 0)+cσ (tm) Em 0 ≤ 1/2 3/4 ≤ ≤ (6.20) κτ0 + κτ , 1 m N, which is (6.19).

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Theorem 6.5. Under the assumptions of Theorem 4.2,theerror(em, µ¯m)= − m − m (u(tm) uh , p¯(tm) p¯h ) satisﬁes the bound − m ≤ 2 3/2 (6.21) u(tm) uh 0 κ(h + τ ), − m ≤ 3/4 1/2 − m ≤ 3/4 (6.22) u(tm) uh 1 (h + τ ),σ (tm) p¯(tm) p¯h 0 κ(h + τ ),

for all tm ∈ (0,T]. Proof. By combining Lemmas 6.1, 6.2, and 6.4 with Theorem 4.2 and noting that 1 3/2 τ0 = T τ , completes the proof of Theorem 6.5. Remark. We have presented the convergence analysis of a stabilized ﬁnite element method with the Crank–Nicolson extrapolation scheme in time direction for the two-dimensional time-dependent Navier–Stokes problem. We have proved that the scheme is unconditionally stable. However, the error estimate obtained in this paper is not optimal in time direction, which is mainly for lack of discrete Stokes operator on the ﬁnite element space pair (Xh,Mh)bytheQ1 − P0 quadrilateral element or the P1 − P0 triangle element. In this case, the estimate N − n2 ≤ 4 τ u¯h(tn) u¯h −1 κτ , n=1 used in [28] is no longer available. Acknowledgments The authors would like to thank the editor and referees for their valuable com- ments and suggestions which helped to improve the results of this paper. References

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Faculty of Science, Xi’an Jiaotong University, Xi’an 710049, People’s Republic of China E-mail address: [email protected] Department of Mathematics, City University of Hong Kong, Hong Kong, People’s Republic of China E-mail address: [email protected]

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