A fully discrete stabilized finite element method for ... - Semantic Scholar
Applied Mathematics and Computation 215 (2009) 85–99
Contents lists available at ScienceDirect
Applied Mathematics and Computation journal homepage: www.elsevier.com/locate/amc
A fully discrete stabilized finite element method for the time-dependent Navier–Stokes equations q Li Shan *, Yanren Hou Faculty of Science, Xi’an Jiaotong University, Xi’an 710049, PR China
a r t i c l e
i n f o
Keywords: Navier–Stokes equations Stabilized finite element Local Gauss integration Error estimate
a b s t r a c t In this article, we consider a fully discrete stabilized finite element method based on two local Gauss integrations for the two-dimensional time-dependent Navier–Stokes equations. It focuses on the lowest equal-order velocity–pressure pairs. Unlike the other stabilized method, the present approach does not require specification of a stabilization parameter or calculation of higher-order derivatives, and always leads to a symmetric linear system. The Euler semi-implicit scheme is used for the time discretization. It is shown that the proposed fully discrete stabilized finite element method results in the optimal order bounds for the velocity and pressure. Ó 2009 Elsevier Inc. All rights reserved.
1. Introduction Stable mixed finite element methods for incompressible flows require pressure and velocity approximations that satisfy the inf–sup (or LBB ) compatibility condition (see [1] and the reference therein). This condition does not allow the use of simple interpolations like equal-order ones, which are desirable from a computational view point. Several solutions have been proposed to overcome this restriction, such as the SUPG method [2], the Brezzi–Pitkaranta method [3], the Douglas-Wang method [4] and the well known GLS method [5]. All of them belong to the class of residual-based methods or are closely related to residual-based stabilized methods. For an overview of stabilized finite element methods for the Stokes problem, see [6]. On the other hand, there are several non-residual-based stabilized finite element methods, such as Pressure-Gradient-Projection methods [7,8], in which an extra unknown in the form of the projection of the gradient of the pressure onto the velocity finite element space is introduced, generating a stable problem even for equal-order interpolations; and the Polynomial Pressure Projection methods [9,10], which are based on a detailed study of the instabilities associated to the equal-order and linear-constant pairs, and adding terms to the formulation particularly suited to stabilize these instabilities, these terms depend on projection operators which are shown to be locally computable. Some of these technique have also been extended to transient incompressible flow problems [11,12]. In particular, the work by Li et al. [13] proposed a new kind of stabilized finite element method for the linear interpolation in space, which based on the idea of [9]. The stabilized term now containing only two local Gauss integrations (the difference between the consistent and under-integrated Gauss integration). Actually, if we introduce an appropriate projection of pressure, the method in [13] is exactly the one proposed in [10]. However, viewing the problem from computational angle, the former manifests efficiency because of using local Gauss integrations to deal with the stabilized term. This paper continues the analysis of the stabilized mixed finite element method for solving the two-dimensional timedependent Navier–Stokes problem. [13] has dealt mainly with spatially discrete, time continuous approximations. Here q
Supported by NSFC (Grant No. 10871156) and NSFC (Grant No. 10471110). * Corresponding author. E-mail addresses: [email protected] (L. Shan), [email protected] (Y. Hou).
0096-3003/$ - see front matter Ó 2009 Elsevier Inc. All rights reserved. doi:10.1016/j.amc.2009.04.037
86
L. Shan, Y. Hou / Applied Mathematics and Computation 215 (2009) 85–99
we discuss a fully discrete stabilized finite element approximation, in which time is discreted by the Euler semi-implicit scheme with the time step 0 < Dt < 1. The outline of the paper is as follows: In Section 2, we introduce an abstract functional setting for the two-dimensional Navier–Stokes equations along with some notations. The stabilized finite element methods is stated in Section 3 which is only considering the spatial discretization. In order to make the error estimates for the fully discrete solutions, we provide some key technical lemmas and known results in Section 4. The time discretization using the Euler semi-implicit scheme is displayed in Section 5. Based on the preceding analysis, we present the optimal order error estimates for the approximate velocity and pressure in Section 6. 2. Preliminaries 2.1. Nomenclature In what follows, X denotes a bounded domain in R2 , with a Lipschitz continuous boundary C, satisfying the further condition stated in ðH1Þ below. Throughout the paper, we employ the standard notation Hs ðXÞ; jj jjs ; ð; Þs ; s P 0 for the Sobolev spaces of all functions having square integrable derivatives up to order s on X, the standard Sobolev norm, and inner product, respectively. When s=0, we will write L2 ðXÞ instead of H0 ðXÞ, the L2 -inner product and L2 -norm are denoted by ð; Þ and j j, respectively. As usual, H10 ðXÞ will denote the closure of C 1 0 ðXÞ with respect to the norm jj jj1 . To introduce a variational formulation, we set
X ¼ H10 ðXÞ2 ;
Y ¼ L2 ðXÞ2 ;
V ¼ fv 2 X; div v ¼ 0g;
M ¼ L20 ðXÞ ¼
Z q 2 L2 ðXÞ; qdx ¼ 0 ; X
DðAÞ ¼ ðH2 ðXÞÞ2 \ X:
The spaces H10 ðXÞ and X are equipped with usual scalar product and norm
ððu; v ÞÞ ¼ ðru; rv Þ;
1
jjujj ¼ ððu; uÞÞ2 :
As noted, a further assumption on X is needed: (H1) Assume that X is regular in the sense that the unique solution ðv ; qÞ 2 ðX; MÞ of the steady Stokes problem
Dv þ rq ¼ g; v joX ¼ 0
div v ¼ 0 in X;
for a prescribed g 2 Y exists, and satisfies
jjv jj2 þ jjqjj1 6 cjgj: Throughout the paper, we use c to denote a generic positive constant whose value may change from place to place but that remains independent of the mesh parameter h ¼ maxfhK g. We remark that the validity of assumption ðH1Þ is known (see [14] ) if oX is of C 2 , or if X is a two-dimensional convex polygon. We also note that ðH1Þ implies the following inequalities
jv j2 6 c0 jjv jj2
8v 2 X;
jjv jj2 6 c0 jjv jj22 6 cjAv j2
8v 2 DðAÞ;
ð2:1Þ
where we denote the Laplace operator by A ¼ D and c0 is a positive constant depending only on X. In this paper, we formulate the method for the transient Navier–Stokes problem that use pressure and velocity finite element spaces defined with respect to the same partition Kh of X into triangles K or quadrilaterals K with mesh size 0 < h < 1, assumed to be uniformly regular in the usual sense. Our main focus is on lowest equal-order velocity and pressure pairs. For simplicity of notation and keep the paper brief, we will confine our attention to the Q 1 Q 1 (bilinear velocity, bilinear pressure) quadrilateral element and P 1 P1 (linear velocity, linear pressure) triangular element as follows:
X h ¼ fv h 2 C 0 ðXÞ2 \ X : v h jK 2 R1 ðKÞ2 8K 2 Kh g; M h ¼ fqh 2 C 0 ðXÞ \ M : qh jK 2 R1 ðKÞ 8K 2 Kh g; where
R1 ðKÞ ¼
Q 1 ðKÞ if K is quadrilateral; P 1 ðKÞ if K is triangle:
and the following properties are classical (see [15,16]):
L. Shan, Y. Hou / Applied Mathematics and Computation 215 (2009) 85–99 1
1
jjv h jj 6 ch jv h j; jv h j1 6 cjlnhj2 jjv h jj 8v h 2 X h ; jjP h v jj 6 cjjv jj 8v 2 X;
87
ð2:2Þ ð2:3Þ
where Ph : Y ! X h is a L2 -orthogonal projection operator defined by
ðP h v ; v h Þ ¼ ðv ; v h Þ 8v 2 Y;
v h 2 Xh:
2.2. The time-dependent Navier–Stokes problem Our aim is to solve the following time-dependent, incompressible Navier–Stokes equations
ut mDu þ ðu rÞu þ rp ¼ f ; u ¼ 0 8ðx; tÞ 2 oX ð0; T;
div u ¼ 0 8ðx; tÞ 2 X ð0; T; uðx; 0Þ ¼ u0 ðxÞ 8x 2 X;
ð2:4Þ ð2:5Þ
where u ¼ uðx; tÞ ¼ ðu1 ðx; tÞ; u2 ðx; tÞÞ represents the velocity, p ¼ pðx; tÞ the pressure, f ¼ f ðx; tÞ the prescribed body force, u0 ðxÞ the initial velocity, m the viscosity and T > 0 a finite time. As usual, we make the following assumption on the prescribed data for problem (2.4) and (2.5). (H2) The initial velocity u0 ðxÞ 2 DðAÞ and the body force f ðx; tÞ 2 L2 ð0; T; YÞ are assumed to satisfy
jju0 jj2 þ
Z
T
ðjf ðtÞj2 þ jft ðtÞj2 Þdt
12 6 c:
ð2:6Þ
0
In order to give the weak formulation of the problem (2.4) and (2.5), we define the continuous bilinear forms að; Þ and dð; Þ on X X and X M, respectively, by
aðu; v Þ ¼ mððu; v ÞÞ 8u; v 2 X; dðv ; qÞ ¼ ðq; div v Þ 8v 2 X; q 2 M: Furthermore, we introduce the following bilinear operator:
1 Bðu; v Þ ¼ ðu 5Þv þ ðdiv uÞv 2
8u; v 2 X;
and define a trilinear form on X X X by
1 1 1 bðu; v ; wÞ ¼< Bðu; v Þ; w>X0;X ¼ ððu 5Þv ; wÞ þ ððdiv uÞv ; wÞ ¼ ððu 5Þv ; wÞ ððu 5Þw; v Þ 8u; v ; w 2 X: 2 2 2 It is easy to verify that the trilinear form b satisfies the following important properties (see [14,17]):
bðu; v ; wÞ ¼ bðu; w; v Þ;
1 1 1 1 1 1 jbðu; v ; wÞj þ jbðw; v ; uÞj 6 c0 juj2 jjujj2 ðjjv jj jwj2 jjwjj2 þ jv j2 jjv jj2 jjwjjÞ ;
ð2:7Þ ð2:8Þ
for all u; v ; w 2 X and c0 , like the quantity c1 that appears subsequently, is a positive constant depending on X. With the above notations, the weak form of problem (2.4) and (2.5) can now be written as: Find ðu; pÞ 2 L2 ð0; T; XÞ L2 ð0; T; MÞ; t 2 ð0; T such that for all ðv ; qÞ 2 ðX; MÞ:
ðut ; v Þ þ Bððu; pÞ; ðv ; qÞÞ þ bðu; u; v Þ ¼ ðf ; v Þ; uð0Þ ¼ u0 ;
ð2:9Þ ð2:10Þ
where the generalized bilinear form B on ðX; MÞ ðX; MÞ is given by
Bððu; pÞ; ðv ; qÞÞ ¼ aðu; v Þ dðv ; pÞ þ dðu; qÞ: A simple modification to the argument given in [14,18] allows us to obtain the following regularity results. Theorem 2.1. Assume that ðH1Þ and ðH2Þ hold, then for any given T > 0, the problem (2.9) and (2.10) admit a unique solution ðu; pÞ satisfying the following estimates:
jjuðtÞjj2 þ
Z
t
ðjut j2 þ jAuj2 þ jjpjj21 Þdt 6 c; Z t sðtÞðjAuðtÞj2 þ jjpðtÞjj21 þ jut ðtÞj2 Þ þ sðsÞjjut jj2 ds 6 c; 0 Z t 2 2 2 sðtÞjjut ðtÞjj þ sðsÞðjutt j þ jAut j þ jjpt jj21 Þds 6 c;
ð2:11Þ
0
0
for all t 2 ½0; T, where sðtÞ ¼ minf1; tg.
ð2:12Þ ð2:13Þ
88
L. Shan, Y. Hou / Applied Mathematics and Computation 215 (2009) 85–99
3. Stabilized finite element method It is known that the lowest equal-order finite element pair dose not satisfy the inf–sup condition. It requires the compensation for pressure, so we define the following local difference between a consistent and an under-integrated mass matrices the stabilized formulation (cf. [13])
Gðph ; qh Þ ¼ pT ðEk E1 Þq ¼ pT Ek q pT E1 q: Here, we set
pT ¼ ½p1 ; p2 ; . . . ; pN T ; Eij ¼ /i ; /j ;
ph ¼
q ¼ ½q1 ; q2 ; . . . ; qN ;
N X
pi /i ;
i¼1
pi ¼ ph ðxi Þ 8ph 2 M h ;
i; j ¼ 1; 2; . . . ; N;
where /i is the basis function of the pressure on the domain X such that its value is one at node xi and zero at other nodes; the symmetric and positive Ek ; k P 2 and E1 are pressure mass matrix computed by using k-order and 1-order Gauss integrations in each direction, respectively; Also, pi and qi ; i ¼ 1; 2; . . . ; N are the value of ph and qh at the node xi , pTi is the transpose of the matrix pi . Next, we introduce a standard L2 -projection ph : M ! R0 defined by
ðp; qh Þ ¼ ðph p; qh Þ 8p 2 M;
q h 2 R0 ;
with the following properties:
jph pj 6 cjpj 8p 2 M; jp ph pj 6 chjjpjj 8p 2 H1 ðXÞ \ M; where R0 ¼ fqh 2 M : qh jK is a constant; 8 K 2 Kh g. Then we can rewrite the bilinear form Gð; Þ by
Gðp; qÞ ¼ ðp ph p; q ph qÞ:
ð3:1Þ
With the above notation, our stabilized discrete formulation for time-dependent Navier–Stokes problem (2.9) and (2.10) is defined as follows: Find ðuh ; ph Þ 2 ðX h ; M h Þ; t 2 ½0; T, such that for all ðv h ; qh Þ 2 ðX h ; M h Þ,
ðuht ; v h Þ þ Bh ððuh ; ph Þ; ðv h ; qh ÞÞ þ bðuh ; uh ; v h Þ ¼ ðf ; v h Þ;
ð3:2Þ
uh ð0Þ ¼ u0h ;
ð3:3Þ
where u0h is an approximation of u0 , and
Bh ððuh ; ph Þ; ðv h ; qh ÞÞ ¼ aðuh ; v h Þ dðv h ; ph Þ þ dðuh ; qh Þ þ Gðph ; qh Þ is the new stabilized bilinear form. The following theorem establishes the weak coercivity of (3.2) for the equal-order finite element pair. Theorem 3.1 [13]. Let ðX h ; M h Þ be defined as above, then there exists a positive constant b, independent of h, such that
jBh ððu; pÞ; ðv ; qÞÞj 6 cðjjujj þ jpjÞðjjv jj þ jqjÞ 8ðu; pÞ; ðv ; qÞ 2 ðX; MÞ; bðjjuh jj þ jph jÞ 6
sup ðv h ;qh Þ2ðX h ;M h Þ
jBh ððuh ; ph Þ; ðv h ; qh ÞÞj jjv h jj þ jqh j
8ðuh ; ph Þ 2 ðX h ; Mh Þ;
jGðp; qÞj 6 cjp ph pjjq ph qj 8p; q 2 M:
ð3:4Þ ð3:5Þ ð3:6Þ
4. Technical preliminaries In this section, we provide some preliminary estimates and recall some known results which will be useful in the error analysis of the finite element solution ðuh ; ph Þ. Since the bilinear form ððuh ; ph ÞÞ is coercive on X h X h , it generates an invertible operator Ah : X h ! X h through the definition:
1 1 ðAh uh ; v h Þ ¼ A2h uh ; A2h v h ¼ ððuh ; v h ÞÞ 8uh ; v h 2 X h : Also, we define the discrete gradient operator rh for qh 2 M h , as follows:
ðv h ; rh qh Þ ¼ dðv h ; qh Þ 8ðv h ; qh Þ 2 ðX h ; M h Þ: Moreover, from (2.1) and (2.2), the inequalities below hold for any
v h 2 Xh
L. Shan, Y. Hou / Applied Mathematics and Computation 215 (2009) 85–99
jv h j2 6 c0 jjv h jj2 ;
1
jjv h jj2 6 c0 jAh v h j2 ;
jAh v h j 6 ch jjv h jj 8
v h 2 Xh :
89
ð4:1Þ
Besides that, we need to recall the following estimates on the trilinear form b provided by He in [19]. Lemma 4.1. The trilinear form b satisfies the following estimates: 1
1
jbðuh ; v h ; wh Þj þ jbðv h ; uh ; wh Þj 6 c1 jjuh jj2 jAh uh j2 jjv h jj jwh j; 1 2
ð4:2Þ
1 2
jbðuh ; v h ; wh Þj þ jbðuh ; wh ; v h Þj 6 c1 juh j jjv h jj jAh wh j jjwh jj ;
ð4:3Þ
1 2
jbðuh ; v h ; wh Þj 6 c1 jlnhj jjuh jj jjv h jj jwh j;
ð4:4Þ
for any uh ; v h ; wh 2 X h . To derive error estimates for the finite element solution ðuh ; ph Þ, we also define projection operator ðRh ; Q h Þ : ðX; YÞ ! ðX h ; M h Þ by
Bh ððRh ðu; pÞ; Q h ðu; pÞÞ; ðv h ; qh ÞÞ ¼ Bððu; pÞ; ðv h ; qh ÞÞ 8ðu; pÞ 2 ðX; MÞ;
ðv h ; qh Þ 2 ðX h ; Mh Þ;
which is well defined and satisfies the following approximation property(see [13]): 2
jRh ðu; pÞ uj þ hjjRh ðu; pÞ ujj þ hjQ h ðu; pÞ pj 6 ch ðjAuj þ jjpjj1 Þ;
ð4:5Þ
1
for all ðu; pÞ 2 ðDðAÞ; H ðXÞ \ MÞ. Theorem 4.1 [13]. Under the assumptions of Theorems 2.1 and 3.1, ðuh ; ph Þ satisfies
Z
t
2
jju uh jj2 ds þ juðtÞ uh ðtÞj2 þ sðtÞjjuðtÞ uh ðtÞjj2 þ sðtÞjpðtÞ ph ðtÞj2 6 ch ;
ð4:6Þ
0
Z
t
4
juðsÞ uh ðsÞj2 ds þ sðtÞjuðtÞ uh ðtÞj2 6 ch ;
ð4:7Þ
0
for all t 2 ½0; T. Since our error analysis for the time discretization depends heavily on the estimates of the spatial-discrete solution, so next we provide the following smoothness estimates for ðuh ; ph Þ. Theorem 4.2. Under the assumptions of Theorem 4.1, the finite element solution ðuh ; ph Þ satisfies
jjuh ðtÞjj2 þ
Z
t
2 2 2 ðjA1 h ðuhtt þ rh pht Þj þ juht j þ jAh uh j Þds 6 c; Z t 1 juht ðtÞj2 þ sðtÞjAh uh ðtÞj2 þ jAh 2 ðuhtt þ rh pht Þj2 þ jjuht jj2 ds 6 c; 0 Z t 2 sðtÞjjuht ðtÞjj þ sðsÞ juhtt þ rh pht j2 þ jAh uht j2 ds 6 c;
ð4:8Þ
0
ð4:9Þ ð4:10Þ
0
for all t 2 ½0; T. Proof. It follows from (2.2), (2.3) and (4.1) that 1
jjuh ðtÞjj 6 jjuh ðtÞ Ph uðtÞjj þ jjPh uðtÞjj 6 ch juh ðtÞ uðtÞj þ cjjuðtÞjj; jðAh uh ; v h Þj 1 jAh uh j ¼ sup 6 ch jju uh jj þ jAuj; jv h j v h 2X h
ð4:11Þ ð4:12Þ
where u is the weak solution of the problem (2.4) and (2.5). Then from Theorems 2.1 and 4.1 yields
jjuht ðtÞjj2 þ Z
Z
t
Z t 2 jAh uh j2 ds 6 c jjuðtÞjj2 þ jAuj2 ds þ ch ðjuðtÞ uh ðtÞj2
0 t
0 2
jjuðtÞ uh ðsÞjj dsÞ 6 c; 0 6 t 6 T; sðtÞjAh uh ðtÞj2 6 csðsÞ h2 jjuðtÞ uh ðtÞjj2 þ jAuðtÞj2 6 c; þ
ð4:13Þ
0
0 6 t 6 T:
ð4:14Þ
Moreover, by recalling [13], we have the following estimate
juht ðtÞj2 þ
Z t jjuht jj2 þ juht j2 ds 6 c;
0 6 t 6 T:
ð4:15Þ
0
Next, differentiating (3.2) with respect to t result in the equation
ðuhtt þ rh pht ; v h Þ þ aðuht ; v h Þ þ bðuht ; uh ; v h Þ þ bðuh ; uht ; v h Þ ¼ ðft ; v h Þ;
ð4:16Þ
90
for all
L. Shan, Y. Hou / Applied Mathematics and Computation 215 (2009) 85–99
v h 2 X h . From (4.1), (4.3), (2.7) and (2.8), we have the following estimates on b as follows: jbðuht ; uh ; v h Þj þ jbðuh ; uht ; v h Þj 6 cjuht j jjuh jj jAh v h j;
ð4:17Þ
jbðuht ; uh ; v h Þj þ jbðuh ; uht ; v h Þj 6 cjjuht jj jjuh jj jjv h jj:
ð4:18Þ
Then, we obtain from (4.16), (4.17) and (4.13) that
2 2 2 ; jA1 h ðuhtt þ rh pht Þj 6 c juht ðtÞj þ jft ðtÞj 12 jAh ðuhtt þ rh pht Þj2 6 c jjuht ðtÞjj2 þ jft ðtÞj2 :
ð4:19Þ ð4:20Þ
Integrating (4.19) and (4.20) with respect to t and using (2.6) and (4.15) gives
Z t 12 2 2 ds 6 c; jA1 h ðuhtt þ rh pht Þj þ jAh ðuhtt þ rh pht Þj
0 6 t 6 T:
0
Combining with (4.13)–(4.15) yields (4.8) and (4.9). Recalling [13] again, we have
sðtÞjjuht ðtÞjj2 þ
Z
t
sðsÞjuhtt j2 6 c; 0 6 t 6 T;
ð4:21Þ
0
Z
t
sðsÞjjeht jj2 ds 6 ch2 ; 0 6 t 6 T;
ð4:22Þ
0
where eh ¼ Rh ðu; pÞ uh . If we set wh ¼ u Rh ðu; pÞ, then from (4.5), we have
jjwht jj 6 chðjAut j þ jjpt jj1 Þ: Hence, we derive from Theorem 2.1 that
Z
t
sðsÞjjwht jj2 ds 6 ch2
0
Z
t
0
sðsÞ jAut j2 þ jjpt jj21 ds 6 ch2 ; 0 6 t 6 T:
ð4:23Þ
Thanks to the triangle inequality, we have
Z
t
sðsÞjjut uht jj2 ds 6
0
Z
t
sðsÞjjeht jj2 ds þ
Z
0
t
sðsÞjjwht jj2 ds 6 ch2 ; 0 6 t 6 T:
ð4:24Þ
0
Together with (4.12), it follows from Theorem 2.1 that
Z
t
sðsÞjAh uht j2 ds 6
0
Z
t
sðsÞðh2 jjut uht jj2 þ jAut j2 Þds 6 c; 0 6 t 6 T:
ð4:25Þ
0
At last, from (4.1), (4.2) and (4.16), one derives
sðtÞjuhtt þ rh pht j2 6 csðtÞ 1 þ jjuh jj2 jAh uht j2 þ cjft j2 : Integrating with respect to t and using (2.6), (4.8) and (4.25) yields
Z
t
sðsÞjuhtt þ rh pht j2 ds 6 c; 0 6 t 6 T:
ð4:26Þ
0
Combining (4.26) with (4.25) and (4.21) yields (4.10). h For convenience, we recall the time-discrete counterpart of classical Gronwall lemma by slightly improving the argument used in [20] and the dual Gronwall lemma used in [19]. Lemma 4.2. Let C and ak ; bk ; ck ; dk ; for integer k P 0, be non-negative numbers such that
an þ Dt
n X k¼0
bk 6 Dt
n1 X
dk ak þ Dt
n1 X
k¼0
ck þ C
8n P 1:
k¼0
Then
an þ Dt
n X k¼0
bk 6 exp Dt
n1 X k¼0
! dk
Dt
n1 X k¼0
! ck þ C
8n P 1:
91
L. Shan, Y. Hou / Applied Mathematics and Computation 215 (2009) 85–99
Lemma 4.3. Given integer m > 0 and let C and ak ; bk ; ck ; dk , for integers 0 6 k 6 m, be non-negative numbers such that m X
an þ Dt
b k 6 Dt
k¼nþ1
m X
m X
dk ak þ Dt
k¼nþ1
ck þ C;
0 6 n 6 m:
k¼nþ1
Then m X
an þ Dt
m X
bk 6 exp Dt
k¼nþ1
! dk
Dt
k¼nþ1
m X
! ck þ C ;
0 6 n 6 m:
k¼nþ1
5. Fully discrete stabilized finite element method In this section, we consider the time discretization of the stabilized finite element approximation and also present some preliminary estimates. By the end of this section, we introduce the corresponding time-discrete duality problem, which is important for estimating the velocity in L2 -norm. We begin by choosing an integer N and define the time step Dt ¼ NT and discrete times tn ¼ nDt; n ¼ 0; 1; 2; . . . ; N. The Euler semi-implicit scheme applied to the spatially discrete problem (3.2) and (3.3), we obtain the fully discrete scheme as follows: Find functions funh gnP0 X h and fpnh gnP1 Mh as solutions of the recursive linear algebraic equations
n dt unh ; v h þ Bh unh ; pnh ; ðv h ; qh Þ þ bðun1 h ; uh ; v h Þ ¼ ðf ðt n Þ; v h Þ;
for all ðv h ; qh Þ 2 ðX h ; M h Þ with the starting value
u0h
ð5:1Þ
¼ u0h , where
1 n dt unh ¼ u uhn1 : Dt h The solutions funh gnP0 and fpnh gnP1 to (5.1) are expected to be the approximations of fuh ðt n ÞgnP0 and fph ðt n ÞgnP1 with
ph ðt n Þ ¼
1 Dt
Z
tn
ph ðtÞdt: tn1
In order to estimate the discretization error enh ; lnh ¼ uh ðt n Þ unh ; ph ðtn Þ pnh ; we integrate and differentiate (3.2) respectively, to get
Z tn 1 1 ðuh ðt n Þ uh ðt n1 Þ; v h Þ þ Bh ððuh ðtÞ; ph ðtÞÞ; ðv h ; qh ÞÞdt Dt Dt tn1 Z tn 1 bðuh ðtÞ; uh ðtÞ; v h Þdt ¼ ðf n ; v h Þ; þ Dt tn1
ð5:2aÞ
ðuhtt þ rh pht ; v h Þ þ aðuht ; v h Þ þ dðuht ; qh Þ þ Gðpht ; qh Þ þ bðuht ; uh ; v h Þ þ bðuh ; uht ; v h Þ ¼ ðft ; v h Þ;
ð5:2bÞ R tn
for all ðv h ; qh Þ 2 ðX h ; M h Þ, where f n ¼ D1t tn1 f ðtÞdt. Subtracting (5.1) from (5.2a) and using (5.2b) and the relation:
/ðt n Þ
1 Dt
Z
tn
/ðtÞdt ¼ tn1
1 Dt
Z
tn
ðt t n1 Þ/t ðtÞdt
ð5:3Þ
t n1
for all / 2 H1 ðtn1 ; t n ; F Þ for some Hilbert space F, one can find that
n1 n dt enh ; v h þ Bh enh ; lnh ; ðv h ; qh Þ þ b en1 h ; uh ðt n Þ; v h þ b uh ; eh ; v h ¼ ðRn ; v h Þ
for all ðv h ; qh Þ 2 ðX h ; M h Þ, where e0h ; l0h ¼ ð0; 0Þ and
ðRn ; v h Þ ¼
1 Dt
Z
tn
Z ðt tn1 Þðuhtt þ rh pht ; v h Þdt þ b
t n1
tn
uht dt; uh ðt n Þ; v h :
ð5:4Þ
ð5:5Þ
t n1
Next, we shall provide some estimations on Rn which are important for estimating the bound on the error ðenh ; lnh Þ. Lemma 5.1. Under the assumptions of Theorem 4.1, Rn satisfies the following bounds:
Dt Dt Dt
m X n¼1 m X n¼1 m X n¼1
2 2 jA1 h R n j 6 c Dt ; 1
jAh 2 Rn j2 6 cDt 2 ;
1 6 m 6 N;
ð5:6Þ
1 6 m 6 N;
ð5:7Þ
si ðtn ÞjRn j2 6 cDtiþ1 ; 1 6 m 6 N; i ¼ 0; 1:
ð5:8Þ
92
L. Shan, Y. Hou / Applied Mathematics and Computation 215 (2009) 85–99
Proof. First, employing the Hölder inequality, it follows from (4.17) and (5.5) that
jA1 h Rn j 6
1 Dt
Z
6 c Dt
tn
ðt tn1 ÞjA1 h ðuhtt þ rh pht Þjdt þ cjjuh ðt n Þjj
t n1 1 2
Z
tn
t n1
Z
tn
juht jdt tn1
12 1 2 2 2 jAh ðuhtt þ rh pht Þj þ jjuh ðtn Þjj juht j dt :
Then applying Theorem 4.2, we obtain 2 2 DtjA1 h R n j 6 c Dt
Z
tn tn1
2 2 jA1 dt: h ðuhtt þ rh pht Þj þ juht j
ð5:9Þ
Summing (5.9) from n ¼ 1 to n ¼ m and applying Theorem 4.2, we derive (5.6). Also, employing Hölder inequality, it follows from(4.18) and (5.5) that
Z tn Z tn 1 1 ðt tn1 ÞjAh 2 ðuhtt þ rh pht Þjdt þ c jjuh ðt n Þjj jjuht jjdt Dt tn1 t n1 Z tn 12 1 6 c Dt jAh 2 ðuhtt þ rh pht Þj2 þ jjuh ðt n Þjj2 jjuht jj2 dt :
1
jAh 2 Rn j 6
ð5:10Þ
t n1
Hence, we deduce from Theorem 4.2 that 1
jAh 2 Rn j2 Dt 6 cDt 2
Z
tn
t n1
1 jAh 2 ðuhtt þ rh pht Þj2 þ jjuht jj2 dt;
ð5:11Þ
Then summing (5.11) from n ¼ 1 to n ¼ m and applying Theorem 4.2, we derives (5.7). Similarly, from (4.17) and (5.5) and applying Hölder inequality, we have
Z tn Z tn 1 ðt t n1 Þjuhtt þ rh pht jdt þ cjAh uh ðt n Þj jjuht jjdt Dt tn1 tn1 Z tn 12 12 Z tn 6c ðt t n1 Þjuhtt þ rh pht j2 dt þ c DtjAh uh ðt n Þj2 jjuht jj2 dt :
jRn j 6
t n1
ð5:12Þ
t n1
Notice the fact
sðtn Þ 6 sðtÞ þ Dt; Dt 6 sðtn Þ; t tn1 6 sðtÞ; t 2 ½tn1 ; tn :
ð5:13Þ
Then applying Theorem 4.2 gives
Dt
m X
jRn j2 6 cDt
n¼1
m Z X n¼1
tn tn1
sðtÞjuhtt þ rh pht j2 dt þ cDt
m X
sðtn ÞjAh uh ðtn Þj2
Z
tn
jjuht jj2 dt 6 cDt
ð5:14aÞ
t n1
n¼1
and
Dt
m X
sðtn ÞjRn j2 6 cDt
n¼1
m X
sðtn Þ
Z
m Z X n¼1
ðt t n1 Þjuhtt þ rh pht j2 dt þ cDt 2
t n1
n¼1
6 c Dt 2
tn
tn
sðtÞjuhtt þ rh pht j2 dt þ cDt2
t n1
m Z X n¼1
m X
sðtn ÞjAh uh ðtn Þj2
n¼1 tn
Z
tn
jjuht jj2 dt
t n1
jjuht jj2 dt 6 cDt 2 :
ð5:14bÞ
t n1
Then (5.14a) and (5.14b) implies (5.8). h To analyze the error ðuh ðt n Þ unh ; ph ðt n Þ pnh Þ, we have to introduce the following time-discrete duality problem corre sponding to (5.4): For any given 1 6 m 6 N and zn 2 X h ; n ¼ 1; 2; . . . m, find Uhn1 ; Whn1 2 ðX h ; M h Þ such that for all ðv h ; qh Þ 2 ðX h ; M h Þ,
v h ; dt Unh
n1 b v h ; uh ðt n Þ; Un1 b uh ðt n1 Þ; v h ; Un1 ¼ ðv h ; zn Þ; Bh ðv h ; qh Þ; Un1 h ; Wh h h
with Um h ¼ 0. In order to analyze the regularity of solutions to problem (5.15), we introduce the following notations
jjPh Bðuh ðtn Þ; Þjj ¼ sup
jPh Bðuh ðtn Þ; v h Þj ; jjv h jj
jjP h Bð; uh ðtn ÞÞjj ¼ sup
jPh Bðv h ; uh ðt n ÞÞj : jjv h jj
v h 2X h v h 2X h
ð5:15Þ
93
L. Shan, Y. Hou / Applied Mathematics and Computation 215 (2009) 85–99
By using an exact similar argument to the one used in [19], we have the result below. Lemma 5.2. Assume that the assumptions of Theorem 4.1 are valid and that Dt satisfies
2
m
c21 jlnhj max jjuh ðtÞjj2 Dt 6 1:
ð5:16Þ
t2½0;T
Then
Dt
N X
jjP h Bðuh ðt n Þ; Þjj2 þ jjPh Bð; uh ðt n ÞÞjj2 6 c:
ð5:17Þ
n¼1
For the existence, uniqueness and smoothness of the solutions to problem (5.15), we provide the following result. Lemma 5.3. Assumethat the assumptions of Theorem 4.1 are valid and that Dt satisfies (5.16), then problem (5.15) admits a unique solution Uhn1 ; Whn1 2 ðX h ; M h Þ for a given Unh . Furthermore, the solution sequence fUnh ; Wnh gm1 n¼0 satisfies the following bound:
sup jjUrh jj2 þ Dt
06r6m
m1 X
jdt Unh j2 6 cDt
n¼1
m X
jzn j2 :
ð5:18Þ
n¼1
Proof. We start by defining the following bilinear form:
Q ððv h ; qh Þ; ðUh ; Wh ÞÞ ¼
1 ðv h ; Uh Þ þ Bh ððv h ; qh Þ; ðUh ; Wh ÞÞ þ bðuh ðt n1 Þ; v h ; Uh Þ þ bðv h ; uh ðt n Þ; Uh Þ: Dt
Consequently, we can rewrite (5.15) as follows:
1 n n1 n ¼ : Q ðv h ; qh Þ; Un1 ; W v ; U z h h h Dt h Following the discussions in [8], we first establish the existence of a constant a1 such that, for all ðUh ; Wh Þ 2 ðX h ; M h Þ
sup ðv h ;qh Þ2ðX h ;Mh Þ
Q ððv h ; qh Þ; ðUh ; Wh ÞÞ P a1 ðjjUh jj þ jWh jÞ: jjv h jj þ jqh j
To this end, we borrow the results from [9] that: For all Wh 2 M h M, there exists a positive constant a1 and g 2 X h such that
dðg; Wh Þ ¼ jWh j2 ;
jjgjj 6 a1 jWh j:
ð5:19Þ
If we choose ðv h ; qh Þ ¼ ðUh þ dg; Wh Þ, where
d¼
1 c 2c2 ðm þ 0 þ max jju ðtÞjj2 Þ1 : 2 Dt m t2½0;T h 2a21
ð5:20Þ
Then it follows from (4.4), (4.18), (5.16), (5.19) and (5.20) that
1 d jUh j2 jUh jjgj þ mjjUh jj2 dmjjgjj jjUh jj þ djWh j2 þ jðI ph ÞWh j2 Dt Dt 1 c1 jlnhj2 jjuh ðt n Þjj jjUh jjjUh j cdðjjuh ðtn Þjj þ jjuh ðtn1 ÞjjÞjjUh jj jjgjj m Dt 2 P c1 jlnhjjjuh ðt n Þjj2 jjUh jj2 þ jðI ph ÞWh j2 2 2 dc a2 c2 da21 þ d 1 mda21 0 1 ðjjuh ðt n Þjj2 þ jjuh ðt n1 Þjj2 Þ jWh j2 2 Dt m m d 2 2 P jjUh jj þ jðI ph ÞWh j þ jWh j2 P a2 ðjjUh jj þ jWh jÞ2 P a3 ðjjUh jj þ jWh jÞðjjv h jj þ jqh jÞ: 2 4
Q ððv h ; qh Þ; ðUh ; Wh ÞÞ P
Here we have used the following fact
jjv h jj þ jqh j 6 jjUh jj þ djjgjj þ jWh j 6 a4 ðjjUh jj þ jWh jÞ: Hence problem (5.15) admits a unique solution Un1 ; Whn1 2 ðX h ; M h Þ. h Next, we will consider the smoothness of Unh ; Wnh , from (5.15) we obtain
a v h ; Uhn1 þ d v h ; Whn1 d dt Unh ; qh G dt Wnh ; qh b v h ; uh ðt n Þ; Un1 h b uh ðtn1 Þ; v h ; Un1 ¼ ðv h ; zn Þ: h
v h ; dt Unh
ð5:21Þ
94
L. Shan, Y. Hou / Applied Mathematics and Computation 215 (2009) 85–99
Setting ðv h ; qh Þ ¼ ðdt Unh ; Whn1 ÞDt in (5.21), then (2.7) implies
m n1 2 2 n1 n 2 þ G Wn1 G Wnh ; Wnh b dt Unh ; uh ðt n Þ; Uhn1 Dt Dtjdt Unh j2 þ jjUh jj þ jjUnh Un1 h ; Wh h jj jjUh jj 2 b uh ðtn1 Þ; dt Unh ; Unh Dt 6 dt Unh ; zn Dt:
ð5:22Þ
Due to (2.8), (4.3) and (4.4), we have
1 2 jb dt Unh ; uh ðtn Þ; Unh Un1 j 6 jdt Unh j2 þ c21 jlnhjjjuh ðtn Þjj2 jjUnh Un1 h h jj ; 4 jb uh ðtn1 Þ; dt Unh ; Unh j þ jb dt Unh ; uh ðtn Þ; Unh j 1 6 jdt Unh j2 þ c jjPh Bðuh ðtn1 Þ; Þjj2 þ jjPh Bð; uh ðt n ÞÞjj2 jjUnh jj2 ; 8 1 j dt Unh ; zn j 6 jdt Unh j2 þ cjzn j2 : 8 Combining (5.22) with the above estimates and using (5.16) yields
2 n1 n1 mjjUnh jj2 G Wnh ; Wnh jdt Unh j2 Dt þ mjjUn1 h jj þ G Wh ; Wh 6 cjzn j2 Dt þ c jjPh Bðuh ðtn1 Þ; Þjj2 þ jjPh Bð; uh ðt n ÞÞjj2 jjUnh jj2 Dt:
ð5:23Þ
Setting r P 0 and summing (5.23) from n ¼ r þ 1 to n ¼ m 1, we obtain
mjjUrh jj2 þ Dt
m1 X
m1 X jdt Unh j2 6 a Um1 ; Uhm1 þ G Wm1 ; Wm1 jzn j2 þ cDt þ c Dt h h h
n¼rþ1
n¼rþ1
m1 X
jjPh Bðuh ðt n1 ÞÞ; jj2 þ jjPh Bð; uh ðtn ÞÞjj2 jjUnh jj2 :
ð5:24Þ
n¼rþ1
m1 Since Um ; Wm1 that h ¼ 0, we obtain from (5.15) and (2.7) with n ¼ m, ðv h ; qh Þ ¼ Uh h
1 m1 2 ; Whm1 þ b Um1 ; uh ðtm Þ; Uhm1 ¼ Um1 ; zm : jUh j þ a Uhm1 ; Uhm1 þ G Wm1 h h h Dt
ð5:25Þ
Due to (4.4) and (5.16), we have
1 jb Uhm1 ; uh ðt m Þ; Um1 jjjUhm1 j j 6 c1 jlnhj2 jjuh ðt m Þjj jjUm1 h h 1 m jj2 ; jUm1 j2 þ jjUm1 2 Dt h 2 h 1 1 jUm1 j2 þ jzm j2 Dt: jðUhm1 Þ; zm Þj 6 2Dt h 2 6
Combining these estimates with (5.25) yields
a Um1 ; Um1 þ G Whm1 ; Wm1 6 cjzm j2 Dt; h h h together with (5.24) gives
mjjUrh jj2 þ Dt
m1 X
jdt Unh j 6 cDt
n¼rþ1
m X
jzn j2 þ cDt
n¼rþ1
m1 X jjP h Bðuh ðt n1 Þ; Þjj2 þ jjPh Bð; uh ðtn ÞÞjj2 ÞjjUnh jj2 : n¼rþ1
Then applying Lemmas 4.3 and 5.2 yields
jjUrh jj2 þ Dt
m1 X n¼rþ1
jdt Unh j2 6 cDt
m X
jzn j2 :
n¼rþ1
Finally, take the supremum for 0 6 r 6 m and complete the proof. h
6. Error analysis In this section, we first provide some estimations for enh ; lnh in various norms, then combining with the results given in Sections 2 and 4, we establish the main error estimations for the fully discrete solutions.
L. Shan, Y. Hou / Applied Mathematics and Computation 215 (2009) 85–99
95
Lemma 6.1. Assume that the assumptions of Theorem 4.1 are valid and Dt satisfied (5.16). Then the error enh ; lnh ; 1 6 n 6 N satisfies the following bound 2 jem h j þ Dt
m X jjenh jj2 þ G lnh ; lnh 6 cDt 2 ;
1 6 m 6 N:
ð6:1Þ
n¼1
Proof. Setting ðv ; qÞ ¼ ðenh ; lnh Þ in (5.4) and using (2.7), we obtain
m n 2 1 n2 1 2 n1 2 n n þ mjjenh jj2 þ Gðlnh ; lnh Þ þ bðen1 jeh j þ jenh en1 jje jj þ m1 jAh 2 Rn j2 : h j jeh j h ; uh ðt n Þ; eh Þ ¼ Rn ; eh 6 2Dt 4 h
ð6:2Þ
For the trilinear term b, it follows from (4.1) and (4.2) that
m n 2 2 n jb en1 jje jj þ cjjP h Bð; uh ðt n ÞÞjj2 jen1 h ; uh ðt n Þ; eh j 6 h j : 4 h Combining (6.2) with the above estimate yields
jenh j2 jehn1 j2 þ
1
2 2 2 mjjenh jj2 þ G lnh ; lnh Dt 6 cjjPh Bð; uh ðtn ÞÞjj2 jen1 h j Dt þ cjAh Rn j Dt:
ð6:3Þ
Summing (6.3) from n ¼ 1 to n ¼ m and applying Lemma 4.2, 5.1 and 5.2 yields (6.1). h Lemma 6.2. Under the assumptions of Lemma 6.1, we have 2 jjem h jj þ Dt
m X
jdt enh j2 6 cDt;
6 m 6 N:
ð6:4Þ
n¼1
Proof. We derive from (5.4) that
n1 n1 n dt enh ; v h þ a enh ; v h d v h ; lnh þ d dt enh ; qh þ G dt ln1 h ; qh þ b eh ; uh ðt n Þ; v h þ b uh ; eh ; v h ¼ ðRn ; v h Þ
for all ðv h ; qh Þ 2 ðX h ; M h Þ. Setting ðv h ; qh Þ ¼
n h
ð6:5Þ
l Dt in (6.5) and using (2.7), we have m n 2 1 n1 n Dtjdt enh j2 þ jjeh jj jjehn1 jj2 þ G lnh ; lnh G ln1 þ b en1 h ; lh h ; uh ðt n Þ; dt eh Dt 2 2 n n þ b uh ðtn1 Þ; ehn1 ; dt enh Dt b en1 h ; eh ; dt eh Dt 6 Rn ; dt enh Dt: dt enh ;
ð6:6Þ
Owing to (4.4) and Cauchy inequality, we have
n n1 n jb en1 h ; uh ðt n Þ; dt eh j þ jbðuh ðt n1 Þ; eh ; dt eh Þj 1 6 jdt enh j2 þ c jjPh Bðuh ðt n1 Þ; Þjj2 þ jjPh Bð; uh ðtn ÞÞjj2 jjehn1 jj2 ; 6 1 1 n n jdt enh j2 þ c21 jlnhjjjehn1 jj2 jjenh jj2 ; jbðehn1 ; enh ; dt enh Þj 6 c1 jlnhj2 jjen1 h jj jjeh jjjdt eh j 6 6 1 j Rn ; dt enh j 6 jdt enh j2 þ cjRn j2 : 6 Combining (6.6) with the above estimates and using Lemma 6.1 yields 2 1 n n n1 n1 1 n 2 jjenh jj2 jjen1 h jj þ m ðGðlh ; lh Þ Gðlh ; lh ÞÞ þ m jdt eh j Dt 1 2 2 2 n1 2 n 2 6 c jjP h Bðuh ðt n1 Þ; Þjj2 þ jjPh Bð; uh ðtn ÞÞjj2 jjen1 h jj Dt þ c 1 jlnhjjjeh jj jjeh jj Dt þ cDtjRn j :
m
ð6:7Þ
Summing (6.7) from n ¼ 1 to n ¼ m and applying Lemmas 5.1, 5.2 and 6.1, we obtain 2 1 jjem h jj þ m Dt
m X
jdt enh j2 6 cDt
1
n¼1 m X
m
n¼1
þ c21 jlnhjDt
m X
2 jjPh Bðuh ðt n1 Þ; Þjj2 þ jjP h Bð; uh ðt n ÞÞjj2 jjen1 h jj
n¼1 2 n 2 jjen1 h jj jjeh jj þ Dt
m X
jRn j2 6 cDt:
n¼1
Lemma 6.3. Under the assumptions of Lemma 6.1, we have
sðtm Þjemh j2 þ Dt
m X jenh j2 þ sðtn Þjjenh jj2 þ sðt n ÞG lnh ; lnh 6 cDt 2 ; n¼1
1 6 m 6 N:
ð6:8Þ
96
L. Shan, Y. Hou / Applied Mathematics and Computation 215 (2009) 85–99
n1 in (5.4), we obtain Proof. Setting ðv ; qÞ ¼ ðenh ; lnh Þ; zn ¼ enh in (5.15) and ðv h ; qh Þ ¼ Un1 h ; Wh
n n1 eh ; dt Unh Bh enh ; lnh ; Un1 b enh ; uh ðt n Þ; Un1 b uh ðt n1 Þ; enh ; Un1 ¼ jenh j2 ; h ; Wh h h n1 n1 þ Bh enh ; lnh ; Un1 þ b uhn1 ; enh ; Un1 þ b en1 ¼ Rn ; Un1 : dt enh ; Un1 h ; uh ðt n Þ; Uh h h ; Wh h h
ð6:9Þ ð6:10Þ
Adding (6.9) to (6.10), we obtain
jenh j2 ¼
1 n n n1 n1 n1 n b dt enh ; uh ðt n Þ; Uhn1 Dt b en1 Rn ; Uhn1 : eh ; Uh eh ; Uh h ; eh ; Uh Dt
ð6:11Þ
From (4.1) and (4.2), we have
jbðdt enh ; uh ðtn Þ; Uhn1 Þj 6 cjdt enh j jjPh Bð; uh ðt n ÞÞjj jjUn1 h jj; n1 n1 n jbðehn1 ; enh ; Un1 h Þj 6 cjjeh jj jjeh jj jjUh jj; 1
n1 2 jðRn ; Un1 h Þj 6 jAh Rn j jjUh jj:
Combining (6.11) with the above estimates yields
þ cjjehn1 jj jjenh jj jjUn1 jenh j2 Dt 6 enh ; Unh ehn1 ; Un1 h h jjDt
þ cDt2 jdt enh jkjPh Bð; uh ðtn ÞÞjj jjUn1 h jj
1
þ jAh 2 Rn j jjUn1 h jjDt;
ð6:12Þ
with e0h ¼ Um h ¼ 0. Summing (6.12) from n ¼ 1 to n ¼ m and applying Lemma 5.3, we obtain
Dt
m X
jenh j2 6 cðDt
m X
n¼1
2 jjen1 h jj Þ Dt
n¼1
m X
! jjenh jj2
þ Dt Dt
n¼1
m X
1
jAh 2 Rn j2
! þ cDt 2 Dt
n¼1
m X
! jdt enh j2
n¼1
Dt
m X
! jjPh Bð; uh ðtn ÞÞjj2 :
n¼1
Then applying Lemmas 5.1, 5.2, 6.1 and 6.2, we get
Dt
m X
jenh j2 6 cDt3 ;
1 6 m 6 N:
ð6:13Þ
n¼1
Now, multiplying(6.3) by sðt n Þ and noting the fact
sðtn Þ 6 Dt þ sðtn1 Þ; Dt 6 sðtn1 Þ; 2 6 n 6 N; ehn1 ¼ 0; for n ¼ 1;
ð6:14Þ
we obtain
2 n 2 n n sðtn Þjenh j2 sðtn1 Þjen1 h j þ sðt n Þ mjjeh jj þ Gðlh ; lh Þ Dt
12 2 2 6 cDtjjP h Bð; uh ðt n ÞÞjj2 sðt n1 Þjehn1 j2 þ c jen1 Dt: h j þ jAh Rn j
ð6:15Þ
Summing (6.15) from n ¼ 1 to n ¼ m and applying Lemmas 4.2 and 5.2, we deduce
sðtm Þjemh j2 þ Dt
m X
sðtn Þ mjjenh jj2 þ Gðlnh ; lnh Þ 6 cDt
n¼1
m X 12 2 2 : jen1 h j þ jAh Rn j n¼1
After applying Lemma 5.1 and (6.13), we have
sðtm Þjemh j2 þ Dt
m X
sðtn Þ mjjenh jj2 þ Gðlnh ; lnh Þ 6 cDt2 :
ð6:16Þ
n¼1
This inequality and (6.13) yields (6.8). h Lemma 6.4. Under the assumptions of Lemma 6.1, the error enh ; lnh satisfies the following bound
sðtm Þjjemh jj2 þ Dt
m X n¼1
sðtn Þjdt enh j2 6 cDt2 :
ð6:17Þ
L. Shan, Y. Hou / Applied Mathematics and Computation 215 (2009) 85–99
97
Proof. Multiplying (6.7) by sðtn Þ, noting the fact (6.14), and using Lemmas 5.2 and 6.2, we have
2 1 n1 n1 sðtn Þ jjenh jj2 þ m1 Gðlnh ; lnh Þ sðtn1 Þ jjen1 h jj þ m G lh ; lh
þ m1 sðtn Þjdt enh j2 Dt
1 2 2 2 n1 2 n 2 6 csðtn1 Þ jjPh Bð; uh ðt n ÞÞjj2 þ jjPh Bðuh ðtn1 Þ; Þjj2 jjen1 h jj Dt þ sðt n ÞjRn j Dt þ c 1 Dtjlnhjsðt n Þjjeh jj jjeh jj m 2 2 2 1 n1 n1 n1 2 2 þ Dt jjen1 h jj þ m Gðlh ; lh Þ þ c jjP h Bð; uh ðt n ÞÞjj þ jjP h Bðuh ðt n1 Þ; Þjj jjeh jj Dt :
ð6:18Þ
Summing (6.18) from n ¼ 1 to n ¼ m and applying Lemmas 4.2, 5.1, 5.2, 6.1, 6.2, and 6.3, we obtain
sðtm Þjjemh jj2 þ Dt
m X
sðtn Þjdt enh j2 6 cDt2 :
n¼1
Lemma 6.5. Under the assumptions of Lemma 6.1, the error
Dt
m X
lnh ¼ ph ðtn Þ pnh satisfies the following bound:
sðtn Þjph ðtn Þ pnh j2 6 cDt2 ; 1 6 m 6 N:
ð6:19Þ
n¼1
Proof. It follows from Theorem 3.1 and (4.18) and (5.4) that
jlnh j 6 cjdt enh j þ c jjehn1 jj þ jjuh ðtn1 Þjj jjenh jj þ cjjuh ðt n Þjj jjen1 h jj þ cjRn j: Hence, together with the fact (6.14) and applying Theorem 4.2 and Lemma 6.2, we have
sðtn Þjlnh j2 Dt 6 csðtn Þjdt enh j2 Dt þ csðtn Þjjenh jj2 Dt þ csðtn1 Þjjehn1 jj2 Dt þ cDt2 jjehn1 jj2 þ csðtn ÞjRn j2 Dt:
ð6:20Þ
Summing (6.20) from n ¼ 1 to n ¼ m and applying Lemmas 5.1, 6.2, 6.3 and 6.4, we obtain
Dt
m X
sðtn Þjlnh j2 6 cDt2 ; 1 6 m 6 N:
n¼1
Theorem 6.1. Under the assumptions of Lemma 6.1, the error uðtn Þ unh ; pðt n Þ pnh satisfies the following bound:
Dt
N X
4
2 2 juðt n Þ unh j2 þ sðtm Þjuðt m Þ um h j 6 cðh þ Dt Þ;
ð6:21Þ
n¼1
Dt
N X
2
2 2 jjuðt n Þ unh jj2 þ sðt m Þjjuðt m Þ um h jj 6 cðh þ Dt Þ;
ð6:22Þ
sðtn Þjpðtn Þ pnh j2 6 cðh2 þ Dt2 Þ
ð6:23Þ
n¼1
Dt
N X n¼1
for all t m 2 ð0; T. Proof. Integrating by parts, we obtain
Dtjuðt n Þ uh ðtn Þj2 6 2
Z
Dtjjuðtn Þ uh ðtn Þjj2 6 2
tn
ju uh j2 dt þ
t n1 Z tn
Z
tn
ðt tn1 Þ2 jut uht j2 dt;
ð6:24Þ
tn1
jju uh jj2 dt þ
t n1
Z
tn
ðt tn1 Þ2 jjut uht jj2 dt:
ð6:25Þ
t n1
Recalling [13], we have
Z
T
sðtÞjut uht j2 dt 6 ch2 :
ð6:26Þ
0
Summing (6.24) and (6.25) from n ¼ 1 to n ¼ N, respectively, using (4.24) and (6.26) and noting the fact (5.13), we obtain
Dt
N X
juðt n Þ uh ðt n Þj2 6 2
Z
N X n¼1
2
ju uh j2 dt þ ch Dt;
ð6:27Þ
0
n¼1
Dt
T
jjuðt n Þ uh ðtn Þjj2 6 2
Z
T
2
jju uh jj2 dt þ ch Dt:
0
Then we can easily obtain (6.21) and (6.22) by employing Theorem 4.1, Lemmas 6.3 and 6.4.
ð6:28Þ
98
L. Shan, Y. Hou / Applied Mathematics and Computation 215 (2009) 85–99
Furthermore, we derive from (2.6), (3.5) and Theorem 4.2 that
Z
t
sðtÞjpht j2 ds 6 c
0
Z t
sðsÞjuhtt j2 þ jjuht jj4 þ jft j2 ds 6 c:
ð6:29Þ
0
Hence, applying the relation (5.3) and Theorem 2.1 yields
sðt1 Þjpðt1 Þ ph ðt1 Þj ¼
Z
tn
t n1
ðt tn1 Þpht ðtÞdt 6 Dtð
Z 0
Dt
1
sðsÞjpht j2 dsÞ2 6 cDt:
ð6:30Þ
In other words, we obtain Dtsðt n Þjpðtn Þ ph ðtn Þj2 6 cDt2 when n ¼ 1. For the case of n P 2, from the triangle inequality and integration by parts, we have
Dt
N X
sðtn Þjpðtn Þ ph ðtn Þj2 6 Dt
n¼2
N X
sðtn Þjpðtn Þ pðtn Þj2 þ Dt
n¼2
N X
sðtn Þjpðtn Þ ph ðtn Þj2
n¼2
! Z tn 2 Z tn N X sðtn Þ 2 ¼ ðpðt n Þ pðtÞÞdt þ ð ðpðtÞ ph ðtÞÞdtÞ Dt t n1 t n1 n¼2 Z tn 2 Z tn 2 ! N X sðtn Þ ¼ ðt t n1 Þpt dt þ ðpðtÞ ph ðtÞÞdt Dt t n1 t n1 n¼2 Z tn Z tn Z tn N N X X sðtn Þ 6 Dt sðtn Þ ðt t n1 Þjpt j2 dt þ dt jp ph j2 dt Dt t n1 tn1 t n1 n¼2 n¼2 Z T Z T 6 Dt 2 sðtÞjpt j2 dt þ 2 sðtÞjp ph j2 dt: t1
ð6:31Þ
t1
On the other hand, we subtract (3.2) from (2.9) with ðv ; qÞ ¼ ðv h ; qh Þ and using the properties of the Galerkin projection ðRh ; Q h Þ, we obtain
ðut uht ; v h Þ þ Bh ðeh ; lh Þ; ðv h ; qh Þ þ bðu uh ; u; v h Þ þ bðuh ; u uh ; v h Þ ¼ 0;
ð6:32Þ
where eh ¼ Rh ðu; pÞ uh ; lh ¼ Q h ðu; pÞ ph , then from (2.1), (2.8) and (3.5), we obtain
Z
T
0
sðtÞjlh j2 dt 6 c
Z
T
sðtÞjut uht j2 þ sðtÞjju uh jj2 jjujj2 þ jjuh jj2 dt:
0
Thus, it follows from (2.11), (4.6), (4.8) and (6.26) that
Z
T
0
sðtÞjlh j2 dt 6 ch2 :
ð6:33Þ
Combining with (2.11) and (4.5) yields
Z
T
0
sðtÞjp ph j2 dt 6 ch2
Z 0
T
Z jAuj2 þ jjpjj21 dt þ
0
T
sðtÞjlh j2 dt 6 ch2 :
ð6:34Þ
Now, substituting (6.34) into (6.31) and using (2.13) and (6.30) and Theorem 2.1, we get
Dt
N X
sðtn Þjpðtn Þ ph ðtn Þj2 6 cðh2 þ Dt2 Þ:
ð6:35Þ
n¼1
Finally, combining (6.35) with (6.19), we have
Dt
N X
sðtn Þjpðtn Þ pnh j2 6 Dt
n¼1
N X
sðtn Þ jpðtn Þ ph ðtn Þj2 þ jph ðtn Þ pnh j2 6 cðh2 þ Dt2 Þ:
n¼1
7. Conclusions In this paper, we have established a fully discrete stabilized finite element method for the two-dimensional time-dependent Navier–Stokes problem. The spatial discretization is based on the linear equal-order interpolations for velocity and pressure; the time discretization is based on the Euler semi-implicit scheme. We have derived the error estimates for the fully discrete stabilized finite element solution unh ; pnh . It has been shown that our method has the optimal convergence rate for velocity and pressure. Acknowledgement We would like to thank the anonymous referees for their valuable comments and suggestions that helped to improve this paper.
L. Shan, Y. Hou / Applied Mathematics and Computation 215 (2009) 85–99
99
References [1] V. Girault, P.A. Raviart, Finite Element Method for the Navier–Stokes Equations: Theory and Algorithms, Springer-Verlag, Berlin, Heidelberg, 1986. [2] T.J.R. Hughes, L.P. Franca, A new finite element formulation for computational fluids dynamics: V. Circumventing the Bavuska–Brezzi condition: a stable Ptrov–Galerkin formulation of the Stokes problem accommodating equal-order interpolations, Comput. Methods Appl. Mech. Eng. 62 (1987) 85– 99. [3] R. Codina, J. Blasco, Stabilized finite element method for the transient Navier–Stokes equations based on a pressure gradient projection, Comput. Methods Appl. Mech. Eng. 182 (2000) 277–300. [4] J. Douglas, J. Wang, An absolutely stabilized finite element method for the Stokes problem, Math. Comput. 52 (1989) 495–508. [5] L. Franca, F. Frey, Stabilized finite element methods:II. The incompressible Navier–Stokes equations, Comput. Methods Appl. Mech. Eng. 99 (1992) 209– 233. [6] T. Barth, P. Bochev, M. Gunzburger, J. Shahid, A taxonomy of consistently stabilized finite element methods for the Stokes problem, SIAM J. Sci. Comput. 25 (2004) 1585–1607. [7] R. Codina, J. Blasco, A finite element formulation for the Stokes problem allowing equal-pressure interpolation, Comput. Methods Appl. Mech. Eng. 143 (1997) 373–391. [8] R. Codina, J. Blasco, Analysis of pressure-stabilized finite element approximation of the stationary Navier–Stokes equations, Numer. Math. 87 (2000) 59–81. [9] P.B. Bochev, C.R. Dohrmann, M.D. Gunzburger, Stabilization of low-order mixed finite elements for the Stokes equations, SIAM J. Numer. Anal. 44 (2006) 82–101. [10] C.R. Dohrmann, P.B. Bochev, A stabilized finite element method for the Stokes problem based on polynomial pressure projections, Int. J. Numer. Methods Fluids 46 (2004) 183–201. [11] G.R. Barrenechca, J. Blasco, Pressure stabilization of finite element approximations of time-dependent incompressible flow problems, Comput. Methods Appl. Mech. Eng. 197 (2007) 219–231. [12] L. Blasco, R. Codina, Space and time estimates for a first order pressure stabilized finite element method for the incompressible Navier–Stokes equations, Appl. Numer. Math. 38 (2001) 475–497. [13] J. Li, Y. He, Z. Chen, A new stabilized finite element method for the transient Navier–Stokes equations, Comput. Methods 197 (2007) 22–35. [14] J.G. Heywood, R. Rannacher, Finite element approximation of the non-stationary Navier–Stokes problem I: regularity of solutions and second-order error estimates for spatial discretization, SIAM J. Numer. Anal. 19 (1982) 275–311. [15] P.G. Ciarlet, The Finite Element Method for Elliptic Problems, North-Holland, Amsterdam, 1978. [16] J.H. Bramble, J. Xu, Some estimates for a weighted L2 projection, Math. Comput. 56 (1991) 463–576. [17] R. Temam, Navier-Stokes Equations, Theory and Numerical Analysis, North-Holland, Amsterdam, 1984. [18] J.H. Hill, E. Süli, Approximation of the global attractors for the incompressible Navier–Stokes equations, IMA J. Numer. Anal. 20 (2000) 633–667. [19] Y. He, A fully discrete stabilized finite-element method for the time-dependent Navier–Stokes problem, IMA J. Numer. Anal. 23 (2003) 665–691. [20] J.G. Heywood, R. Rannacher, Finite-Element approximation of the nonstationary Navier–Stokes problem Part IV: error analysis for second-order time discretization, SIAM J. Numer. Anal. 27 (1990) 353–384.
Contents lists available at ScienceDirect
Applied Mathematics and Computation journal homepage: www.elsevier.com/locate/amc
A fully discrete stabilized finite element method for the time-dependent Navier–Stokes equations q Li Shan *, Yanren Hou Faculty of Science, Xi’an Jiaotong University, Xi’an 710049, PR China
a r t i c l e
i n f o
Keywords: Navier–Stokes equations Stabilized finite element Local Gauss integration Error estimate
a b s t r a c t In this article, we consider a fully discrete stabilized finite element method based on two local Gauss integrations for the two-dimensional time-dependent Navier–Stokes equations. It focuses on the lowest equal-order velocity–pressure pairs. Unlike the other stabilized method, the present approach does not require specification of a stabilization parameter or calculation of higher-order derivatives, and always leads to a symmetric linear system. The Euler semi-implicit scheme is used for the time discretization. It is shown that the proposed fully discrete stabilized finite element method results in the optimal order bounds for the velocity and pressure. Ó 2009 Elsevier Inc. All rights reserved.
1. Introduction Stable mixed finite element methods for incompressible flows require pressure and velocity approximations that satisfy the inf–sup (or LBB ) compatibility condition (see [1] and the reference therein). This condition does not allow the use of simple interpolations like equal-order ones, which are desirable from a computational view point. Several solutions have been proposed to overcome this restriction, such as the SUPG method [2], the Brezzi–Pitkaranta method [3], the Douglas-Wang method [4] and the well known GLS method [5]. All of them belong to the class of residual-based methods or are closely related to residual-based stabilized methods. For an overview of stabilized finite element methods for the Stokes problem, see [6]. On the other hand, there are several non-residual-based stabilized finite element methods, such as Pressure-Gradient-Projection methods [7,8], in which an extra unknown in the form of the projection of the gradient of the pressure onto the velocity finite element space is introduced, generating a stable problem even for equal-order interpolations; and the Polynomial Pressure Projection methods [9,10], which are based on a detailed study of the instabilities associated to the equal-order and linear-constant pairs, and adding terms to the formulation particularly suited to stabilize these instabilities, these terms depend on projection operators which are shown to be locally computable. Some of these technique have also been extended to transient incompressible flow problems [11,12]. In particular, the work by Li et al. [13] proposed a new kind of stabilized finite element method for the linear interpolation in space, which based on the idea of [9]. The stabilized term now containing only two local Gauss integrations (the difference between the consistent and under-integrated Gauss integration). Actually, if we introduce an appropriate projection of pressure, the method in [13] is exactly the one proposed in [10]. However, viewing the problem from computational angle, the former manifests efficiency because of using local Gauss integrations to deal with the stabilized term. This paper continues the analysis of the stabilized mixed finite element method for solving the two-dimensional timedependent Navier–Stokes problem. [13] has dealt mainly with spatially discrete, time continuous approximations. Here q
Supported by NSFC (Grant No. 10871156) and NSFC (Grant No. 10471110). * Corresponding author. E-mail addresses: [email protected] (L. Shan), [email protected] (Y. Hou).
0096-3003/$ - see front matter Ó 2009 Elsevier Inc. All rights reserved. doi:10.1016/j.amc.2009.04.037
86
L. Shan, Y. Hou / Applied Mathematics and Computation 215 (2009) 85–99
we discuss a fully discrete stabilized finite element approximation, in which time is discreted by the Euler semi-implicit scheme with the time step 0 < Dt < 1. The outline of the paper is as follows: In Section 2, we introduce an abstract functional setting for the two-dimensional Navier–Stokes equations along with some notations. The stabilized finite element methods is stated in Section 3 which is only considering the spatial discretization. In order to make the error estimates for the fully discrete solutions, we provide some key technical lemmas and known results in Section 4. The time discretization using the Euler semi-implicit scheme is displayed in Section 5. Based on the preceding analysis, we present the optimal order error estimates for the approximate velocity and pressure in Section 6. 2. Preliminaries 2.1. Nomenclature In what follows, X denotes a bounded domain in R2 , with a Lipschitz continuous boundary C, satisfying the further condition stated in ðH1Þ below. Throughout the paper, we employ the standard notation Hs ðXÞ; jj jjs ; ð; Þs ; s P 0 for the Sobolev spaces of all functions having square integrable derivatives up to order s on X, the standard Sobolev norm, and inner product, respectively. When s=0, we will write L2 ðXÞ instead of H0 ðXÞ, the L2 -inner product and L2 -norm are denoted by ð; Þ and j j, respectively. As usual, H10 ðXÞ will denote the closure of C 1 0 ðXÞ with respect to the norm jj jj1 . To introduce a variational formulation, we set
X ¼ H10 ðXÞ2 ;
Y ¼ L2 ðXÞ2 ;
V ¼ fv 2 X; div v ¼ 0g;
M ¼ L20 ðXÞ ¼
Z q 2 L2 ðXÞ; qdx ¼ 0 ; X
DðAÞ ¼ ðH2 ðXÞÞ2 \ X:
The spaces H10 ðXÞ and X are equipped with usual scalar product and norm
ððu; v ÞÞ ¼ ðru; rv Þ;
1
jjujj ¼ ððu; uÞÞ2 :
As noted, a further assumption on X is needed: (H1) Assume that X is regular in the sense that the unique solution ðv ; qÞ 2 ðX; MÞ of the steady Stokes problem
Dv þ rq ¼ g; v joX ¼ 0
div v ¼ 0 in X;
for a prescribed g 2 Y exists, and satisfies
jjv jj2 þ jjqjj1 6 cjgj: Throughout the paper, we use c to denote a generic positive constant whose value may change from place to place but that remains independent of the mesh parameter h ¼ maxfhK g. We remark that the validity of assumption ðH1Þ is known (see [14] ) if oX is of C 2 , or if X is a two-dimensional convex polygon. We also note that ðH1Þ implies the following inequalities
jv j2 6 c0 jjv jj2
8v 2 X;
jjv jj2 6 c0 jjv jj22 6 cjAv j2
8v 2 DðAÞ;
ð2:1Þ
where we denote the Laplace operator by A ¼ D and c0 is a positive constant depending only on X. In this paper, we formulate the method for the transient Navier–Stokes problem that use pressure and velocity finite element spaces defined with respect to the same partition Kh of X into triangles K or quadrilaterals K with mesh size 0 < h < 1, assumed to be uniformly regular in the usual sense. Our main focus is on lowest equal-order velocity and pressure pairs. For simplicity of notation and keep the paper brief, we will confine our attention to the Q 1 Q 1 (bilinear velocity, bilinear pressure) quadrilateral element and P 1 P1 (linear velocity, linear pressure) triangular element as follows:
X h ¼ fv h 2 C 0 ðXÞ2 \ X : v h jK 2 R1 ðKÞ2 8K 2 Kh g; M h ¼ fqh 2 C 0 ðXÞ \ M : qh jK 2 R1 ðKÞ 8K 2 Kh g; where
R1 ðKÞ ¼
Q 1 ðKÞ if K is quadrilateral; P 1 ðKÞ if K is triangle:
and the following properties are classical (see [15,16]):
L. Shan, Y. Hou / Applied Mathematics and Computation 215 (2009) 85–99 1
1
jjv h jj 6 ch jv h j; jv h j1 6 cjlnhj2 jjv h jj 8v h 2 X h ; jjP h v jj 6 cjjv jj 8v 2 X;
87
ð2:2Þ ð2:3Þ
where Ph : Y ! X h is a L2 -orthogonal projection operator defined by
ðP h v ; v h Þ ¼ ðv ; v h Þ 8v 2 Y;
v h 2 Xh:
2.2. The time-dependent Navier–Stokes problem Our aim is to solve the following time-dependent, incompressible Navier–Stokes equations
ut mDu þ ðu rÞu þ rp ¼ f ; u ¼ 0 8ðx; tÞ 2 oX ð0; T;
div u ¼ 0 8ðx; tÞ 2 X ð0; T; uðx; 0Þ ¼ u0 ðxÞ 8x 2 X;
ð2:4Þ ð2:5Þ
where u ¼ uðx; tÞ ¼ ðu1 ðx; tÞ; u2 ðx; tÞÞ represents the velocity, p ¼ pðx; tÞ the pressure, f ¼ f ðx; tÞ the prescribed body force, u0 ðxÞ the initial velocity, m the viscosity and T > 0 a finite time. As usual, we make the following assumption on the prescribed data for problem (2.4) and (2.5). (H2) The initial velocity u0 ðxÞ 2 DðAÞ and the body force f ðx; tÞ 2 L2 ð0; T; YÞ are assumed to satisfy
jju0 jj2 þ
Z
T
ðjf ðtÞj2 þ jft ðtÞj2 Þdt
12 6 c:
ð2:6Þ
0
In order to give the weak formulation of the problem (2.4) and (2.5), we define the continuous bilinear forms að; Þ and dð; Þ on X X and X M, respectively, by
aðu; v Þ ¼ mððu; v ÞÞ 8u; v 2 X; dðv ; qÞ ¼ ðq; div v Þ 8v 2 X; q 2 M: Furthermore, we introduce the following bilinear operator:
1 Bðu; v Þ ¼ ðu 5Þv þ ðdiv uÞv 2
8u; v 2 X;
and define a trilinear form on X X X by
1 1 1 bðu; v ; wÞ ¼< Bðu; v Þ; w>X0;X ¼ ððu 5Þv ; wÞ þ ððdiv uÞv ; wÞ ¼ ððu 5Þv ; wÞ ððu 5Þw; v Þ 8u; v ; w 2 X: 2 2 2 It is easy to verify that the trilinear form b satisfies the following important properties (see [14,17]):
bðu; v ; wÞ ¼ bðu; w; v Þ;
1 1 1 1 1 1 jbðu; v ; wÞj þ jbðw; v ; uÞj 6 c0 juj2 jjujj2 ðjjv jj jwj2 jjwjj2 þ jv j2 jjv jj2 jjwjjÞ ;
ð2:7Þ ð2:8Þ
for all u; v ; w 2 X and c0 , like the quantity c1 that appears subsequently, is a positive constant depending on X. With the above notations, the weak form of problem (2.4) and (2.5) can now be written as: Find ðu; pÞ 2 L2 ð0; T; XÞ L2 ð0; T; MÞ; t 2 ð0; T such that for all ðv ; qÞ 2 ðX; MÞ:
ðut ; v Þ þ Bððu; pÞ; ðv ; qÞÞ þ bðu; u; v Þ ¼ ðf ; v Þ; uð0Þ ¼ u0 ;
ð2:9Þ ð2:10Þ
where the generalized bilinear form B on ðX; MÞ ðX; MÞ is given by
Bððu; pÞ; ðv ; qÞÞ ¼ aðu; v Þ dðv ; pÞ þ dðu; qÞ: A simple modification to the argument given in [14,18] allows us to obtain the following regularity results. Theorem 2.1. Assume that ðH1Þ and ðH2Þ hold, then for any given T > 0, the problem (2.9) and (2.10) admit a unique solution ðu; pÞ satisfying the following estimates:
jjuðtÞjj2 þ
Z
t
ðjut j2 þ jAuj2 þ jjpjj21 Þdt 6 c; Z t sðtÞðjAuðtÞj2 þ jjpðtÞjj21 þ jut ðtÞj2 Þ þ sðsÞjjut jj2 ds 6 c; 0 Z t 2 2 2 sðtÞjjut ðtÞjj þ sðsÞðjutt j þ jAut j þ jjpt jj21 Þds 6 c;
ð2:11Þ
0
0
for all t 2 ½0; T, where sðtÞ ¼ minf1; tg.
ð2:12Þ ð2:13Þ
88
L. Shan, Y. Hou / Applied Mathematics and Computation 215 (2009) 85–99
3. Stabilized finite element method It is known that the lowest equal-order finite element pair dose not satisfy the inf–sup condition. It requires the compensation for pressure, so we define the following local difference between a consistent and an under-integrated mass matrices the stabilized formulation (cf. [13])
Gðph ; qh Þ ¼ pT ðEk E1 Þq ¼ pT Ek q pT E1 q: Here, we set
pT ¼ ½p1 ; p2 ; . . . ; pN T ; Eij ¼ /i ; /j ;
ph ¼
q ¼ ½q1 ; q2 ; . . . ; qN ;
N X
pi /i ;
i¼1
pi ¼ ph ðxi Þ 8ph 2 M h ;
i; j ¼ 1; 2; . . . ; N;
where /i is the basis function of the pressure on the domain X such that its value is one at node xi and zero at other nodes; the symmetric and positive Ek ; k P 2 and E1 are pressure mass matrix computed by using k-order and 1-order Gauss integrations in each direction, respectively; Also, pi and qi ; i ¼ 1; 2; . . . ; N are the value of ph and qh at the node xi , pTi is the transpose of the matrix pi . Next, we introduce a standard L2 -projection ph : M ! R0 defined by
ðp; qh Þ ¼ ðph p; qh Þ 8p 2 M;
q h 2 R0 ;
with the following properties:
jph pj 6 cjpj 8p 2 M; jp ph pj 6 chjjpjj 8p 2 H1 ðXÞ \ M; where R0 ¼ fqh 2 M : qh jK is a constant; 8 K 2 Kh g. Then we can rewrite the bilinear form Gð; Þ by
Gðp; qÞ ¼ ðp ph p; q ph qÞ:
ð3:1Þ
With the above notation, our stabilized discrete formulation for time-dependent Navier–Stokes problem (2.9) and (2.10) is defined as follows: Find ðuh ; ph Þ 2 ðX h ; M h Þ; t 2 ½0; T, such that for all ðv h ; qh Þ 2 ðX h ; M h Þ,
ðuht ; v h Þ þ Bh ððuh ; ph Þ; ðv h ; qh ÞÞ þ bðuh ; uh ; v h Þ ¼ ðf ; v h Þ;
ð3:2Þ
uh ð0Þ ¼ u0h ;
ð3:3Þ
where u0h is an approximation of u0 , and
Bh ððuh ; ph Þ; ðv h ; qh ÞÞ ¼ aðuh ; v h Þ dðv h ; ph Þ þ dðuh ; qh Þ þ Gðph ; qh Þ is the new stabilized bilinear form. The following theorem establishes the weak coercivity of (3.2) for the equal-order finite element pair. Theorem 3.1 [13]. Let ðX h ; M h Þ be defined as above, then there exists a positive constant b, independent of h, such that
jBh ððu; pÞ; ðv ; qÞÞj 6 cðjjujj þ jpjÞðjjv jj þ jqjÞ 8ðu; pÞ; ðv ; qÞ 2 ðX; MÞ; bðjjuh jj þ jph jÞ 6
sup ðv h ;qh Þ2ðX h ;M h Þ
jBh ððuh ; ph Þ; ðv h ; qh ÞÞj jjv h jj þ jqh j
8ðuh ; ph Þ 2 ðX h ; Mh Þ;
jGðp; qÞj 6 cjp ph pjjq ph qj 8p; q 2 M:
ð3:4Þ ð3:5Þ ð3:6Þ
4. Technical preliminaries In this section, we provide some preliminary estimates and recall some known results which will be useful in the error analysis of the finite element solution ðuh ; ph Þ. Since the bilinear form ððuh ; ph ÞÞ is coercive on X h X h , it generates an invertible operator Ah : X h ! X h through the definition:
1 1 ðAh uh ; v h Þ ¼ A2h uh ; A2h v h ¼ ððuh ; v h ÞÞ 8uh ; v h 2 X h : Also, we define the discrete gradient operator rh for qh 2 M h , as follows:
ðv h ; rh qh Þ ¼ dðv h ; qh Þ 8ðv h ; qh Þ 2 ðX h ; M h Þ: Moreover, from (2.1) and (2.2), the inequalities below hold for any
v h 2 Xh
L. Shan, Y. Hou / Applied Mathematics and Computation 215 (2009) 85–99
jv h j2 6 c0 jjv h jj2 ;
1
jjv h jj2 6 c0 jAh v h j2 ;
jAh v h j 6 ch jjv h jj 8
v h 2 Xh :
89
ð4:1Þ
Besides that, we need to recall the following estimates on the trilinear form b provided by He in [19]. Lemma 4.1. The trilinear form b satisfies the following estimates: 1
1
jbðuh ; v h ; wh Þj þ jbðv h ; uh ; wh Þj 6 c1 jjuh jj2 jAh uh j2 jjv h jj jwh j; 1 2
ð4:2Þ
1 2
jbðuh ; v h ; wh Þj þ jbðuh ; wh ; v h Þj 6 c1 juh j jjv h jj jAh wh j jjwh jj ;
ð4:3Þ
1 2
jbðuh ; v h ; wh Þj 6 c1 jlnhj jjuh jj jjv h jj jwh j;
ð4:4Þ
for any uh ; v h ; wh 2 X h . To derive error estimates for the finite element solution ðuh ; ph Þ, we also define projection operator ðRh ; Q h Þ : ðX; YÞ ! ðX h ; M h Þ by
Bh ððRh ðu; pÞ; Q h ðu; pÞÞ; ðv h ; qh ÞÞ ¼ Bððu; pÞ; ðv h ; qh ÞÞ 8ðu; pÞ 2 ðX; MÞ;
ðv h ; qh Þ 2 ðX h ; Mh Þ;
which is well defined and satisfies the following approximation property(see [13]): 2
jRh ðu; pÞ uj þ hjjRh ðu; pÞ ujj þ hjQ h ðu; pÞ pj 6 ch ðjAuj þ jjpjj1 Þ;
ð4:5Þ
1
for all ðu; pÞ 2 ðDðAÞ; H ðXÞ \ MÞ. Theorem 4.1 [13]. Under the assumptions of Theorems 2.1 and 3.1, ðuh ; ph Þ satisfies
Z
t
2
jju uh jj2 ds þ juðtÞ uh ðtÞj2 þ sðtÞjjuðtÞ uh ðtÞjj2 þ sðtÞjpðtÞ ph ðtÞj2 6 ch ;
ð4:6Þ
0
Z
t
4
juðsÞ uh ðsÞj2 ds þ sðtÞjuðtÞ uh ðtÞj2 6 ch ;
ð4:7Þ
0
for all t 2 ½0; T. Since our error analysis for the time discretization depends heavily on the estimates of the spatial-discrete solution, so next we provide the following smoothness estimates for ðuh ; ph Þ. Theorem 4.2. Under the assumptions of Theorem 4.1, the finite element solution ðuh ; ph Þ satisfies
jjuh ðtÞjj2 þ
Z
t
2 2 2 ðjA1 h ðuhtt þ rh pht Þj þ juht j þ jAh uh j Þds 6 c; Z t 1 juht ðtÞj2 þ sðtÞjAh uh ðtÞj2 þ jAh 2 ðuhtt þ rh pht Þj2 þ jjuht jj2 ds 6 c; 0 Z t 2 sðtÞjjuht ðtÞjj þ sðsÞ juhtt þ rh pht j2 þ jAh uht j2 ds 6 c;
ð4:8Þ
0
ð4:9Þ ð4:10Þ
0
for all t 2 ½0; T. Proof. It follows from (2.2), (2.3) and (4.1) that 1
jjuh ðtÞjj 6 jjuh ðtÞ Ph uðtÞjj þ jjPh uðtÞjj 6 ch juh ðtÞ uðtÞj þ cjjuðtÞjj; jðAh uh ; v h Þj 1 jAh uh j ¼ sup 6 ch jju uh jj þ jAuj; jv h j v h 2X h
ð4:11Þ ð4:12Þ
where u is the weak solution of the problem (2.4) and (2.5). Then from Theorems 2.1 and 4.1 yields
jjuht ðtÞjj2 þ Z
Z
t
Z t 2 jAh uh j2 ds 6 c jjuðtÞjj2 þ jAuj2 ds þ ch ðjuðtÞ uh ðtÞj2
0 t
0 2
jjuðtÞ uh ðsÞjj dsÞ 6 c; 0 6 t 6 T; sðtÞjAh uh ðtÞj2 6 csðsÞ h2 jjuðtÞ uh ðtÞjj2 þ jAuðtÞj2 6 c; þ
ð4:13Þ
0
0 6 t 6 T:
ð4:14Þ
Moreover, by recalling [13], we have the following estimate
juht ðtÞj2 þ
Z t jjuht jj2 þ juht j2 ds 6 c;
0 6 t 6 T:
ð4:15Þ
0
Next, differentiating (3.2) with respect to t result in the equation
ðuhtt þ rh pht ; v h Þ þ aðuht ; v h Þ þ bðuht ; uh ; v h Þ þ bðuh ; uht ; v h Þ ¼ ðft ; v h Þ;
ð4:16Þ
90
for all
L. Shan, Y. Hou / Applied Mathematics and Computation 215 (2009) 85–99
v h 2 X h . From (4.1), (4.3), (2.7) and (2.8), we have the following estimates on b as follows: jbðuht ; uh ; v h Þj þ jbðuh ; uht ; v h Þj 6 cjuht j jjuh jj jAh v h j;
ð4:17Þ
jbðuht ; uh ; v h Þj þ jbðuh ; uht ; v h Þj 6 cjjuht jj jjuh jj jjv h jj:
ð4:18Þ
Then, we obtain from (4.16), (4.17) and (4.13) that
2 2 2 ; jA1 h ðuhtt þ rh pht Þj 6 c juht ðtÞj þ jft ðtÞj 12 jAh ðuhtt þ rh pht Þj2 6 c jjuht ðtÞjj2 þ jft ðtÞj2 :
ð4:19Þ ð4:20Þ
Integrating (4.19) and (4.20) with respect to t and using (2.6) and (4.15) gives
Z t 12 2 2 ds 6 c; jA1 h ðuhtt þ rh pht Þj þ jAh ðuhtt þ rh pht Þj
0 6 t 6 T:
0
Combining with (4.13)–(4.15) yields (4.8) and (4.9). Recalling [13] again, we have
sðtÞjjuht ðtÞjj2 þ
Z
t
sðsÞjuhtt j2 6 c; 0 6 t 6 T;
ð4:21Þ
0
Z
t
sðsÞjjeht jj2 ds 6 ch2 ; 0 6 t 6 T;
ð4:22Þ
0
where eh ¼ Rh ðu; pÞ uh . If we set wh ¼ u Rh ðu; pÞ, then from (4.5), we have
jjwht jj 6 chðjAut j þ jjpt jj1 Þ: Hence, we derive from Theorem 2.1 that
Z
t
sðsÞjjwht jj2 ds 6 ch2
0
Z
t
0
sðsÞ jAut j2 þ jjpt jj21 ds 6 ch2 ; 0 6 t 6 T:
ð4:23Þ
Thanks to the triangle inequality, we have
Z
t
sðsÞjjut uht jj2 ds 6
0
Z
t
sðsÞjjeht jj2 ds þ
Z
0
t
sðsÞjjwht jj2 ds 6 ch2 ; 0 6 t 6 T:
ð4:24Þ
0
Together with (4.12), it follows from Theorem 2.1 that
Z
t
sðsÞjAh uht j2 ds 6
0
Z
t
sðsÞðh2 jjut uht jj2 þ jAut j2 Þds 6 c; 0 6 t 6 T:
ð4:25Þ
0
At last, from (4.1), (4.2) and (4.16), one derives
sðtÞjuhtt þ rh pht j2 6 csðtÞ 1 þ jjuh jj2 jAh uht j2 þ cjft j2 : Integrating with respect to t and using (2.6), (4.8) and (4.25) yields
Z
t
sðsÞjuhtt þ rh pht j2 ds 6 c; 0 6 t 6 T:
ð4:26Þ
0
Combining (4.26) with (4.25) and (4.21) yields (4.10). h For convenience, we recall the time-discrete counterpart of classical Gronwall lemma by slightly improving the argument used in [20] and the dual Gronwall lemma used in [19]. Lemma 4.2. Let C and ak ; bk ; ck ; dk ; for integer k P 0, be non-negative numbers such that
an þ Dt
n X k¼0
bk 6 Dt
n1 X
dk ak þ Dt
n1 X
k¼0
ck þ C
8n P 1:
k¼0
Then
an þ Dt
n X k¼0
bk 6 exp Dt
n1 X k¼0
! dk
Dt
n1 X k¼0
! ck þ C
8n P 1:
91
L. Shan, Y. Hou / Applied Mathematics and Computation 215 (2009) 85–99
Lemma 4.3. Given integer m > 0 and let C and ak ; bk ; ck ; dk , for integers 0 6 k 6 m, be non-negative numbers such that m X
an þ Dt
b k 6 Dt
k¼nþ1
m X
m X
dk ak þ Dt
k¼nþ1
ck þ C;
0 6 n 6 m:
k¼nþ1
Then m X
an þ Dt
m X
bk 6 exp Dt
k¼nþ1
! dk
Dt
k¼nþ1
m X
! ck þ C ;
0 6 n 6 m:
k¼nþ1
5. Fully discrete stabilized finite element method In this section, we consider the time discretization of the stabilized finite element approximation and also present some preliminary estimates. By the end of this section, we introduce the corresponding time-discrete duality problem, which is important for estimating the velocity in L2 -norm. We begin by choosing an integer N and define the time step Dt ¼ NT and discrete times tn ¼ nDt; n ¼ 0; 1; 2; . . . ; N. The Euler semi-implicit scheme applied to the spatially discrete problem (3.2) and (3.3), we obtain the fully discrete scheme as follows: Find functions funh gnP0 X h and fpnh gnP1 Mh as solutions of the recursive linear algebraic equations
n dt unh ; v h þ Bh unh ; pnh ; ðv h ; qh Þ þ bðun1 h ; uh ; v h Þ ¼ ðf ðt n Þ; v h Þ;
for all ðv h ; qh Þ 2 ðX h ; M h Þ with the starting value
u0h
ð5:1Þ
¼ u0h , where
1 n dt unh ¼ u uhn1 : Dt h The solutions funh gnP0 and fpnh gnP1 to (5.1) are expected to be the approximations of fuh ðt n ÞgnP0 and fph ðt n ÞgnP1 with
ph ðt n Þ ¼
1 Dt
Z
tn
ph ðtÞdt: tn1
In order to estimate the discretization error enh ; lnh ¼ uh ðt n Þ unh ; ph ðtn Þ pnh ; we integrate and differentiate (3.2) respectively, to get
Z tn 1 1 ðuh ðt n Þ uh ðt n1 Þ; v h Þ þ Bh ððuh ðtÞ; ph ðtÞÞ; ðv h ; qh ÞÞdt Dt Dt tn1 Z tn 1 bðuh ðtÞ; uh ðtÞ; v h Þdt ¼ ðf n ; v h Þ; þ Dt tn1
ð5:2aÞ
ðuhtt þ rh pht ; v h Þ þ aðuht ; v h Þ þ dðuht ; qh Þ þ Gðpht ; qh Þ þ bðuht ; uh ; v h Þ þ bðuh ; uht ; v h Þ ¼ ðft ; v h Þ;
ð5:2bÞ R tn
for all ðv h ; qh Þ 2 ðX h ; M h Þ, where f n ¼ D1t tn1 f ðtÞdt. Subtracting (5.1) from (5.2a) and using (5.2b) and the relation:
/ðt n Þ
1 Dt
Z
tn
/ðtÞdt ¼ tn1
1 Dt
Z
tn
ðt t n1 Þ/t ðtÞdt
ð5:3Þ
t n1
for all / 2 H1 ðtn1 ; t n ; F Þ for some Hilbert space F, one can find that
n1 n dt enh ; v h þ Bh enh ; lnh ; ðv h ; qh Þ þ b en1 h ; uh ðt n Þ; v h þ b uh ; eh ; v h ¼ ðRn ; v h Þ
for all ðv h ; qh Þ 2 ðX h ; M h Þ, where e0h ; l0h ¼ ð0; 0Þ and
ðRn ; v h Þ ¼
1 Dt
Z
tn
Z ðt tn1 Þðuhtt þ rh pht ; v h Þdt þ b
t n1
tn
uht dt; uh ðt n Þ; v h :
ð5:4Þ
ð5:5Þ
t n1
Next, we shall provide some estimations on Rn which are important for estimating the bound on the error ðenh ; lnh Þ. Lemma 5.1. Under the assumptions of Theorem 4.1, Rn satisfies the following bounds:
Dt Dt Dt
m X n¼1 m X n¼1 m X n¼1
2 2 jA1 h R n j 6 c Dt ; 1
jAh 2 Rn j2 6 cDt 2 ;
1 6 m 6 N;
ð5:6Þ
1 6 m 6 N;
ð5:7Þ
si ðtn ÞjRn j2 6 cDtiþ1 ; 1 6 m 6 N; i ¼ 0; 1:
ð5:8Þ
92
L. Shan, Y. Hou / Applied Mathematics and Computation 215 (2009) 85–99
Proof. First, employing the Hölder inequality, it follows from (4.17) and (5.5) that
jA1 h Rn j 6
1 Dt
Z
6 c Dt
tn
ðt tn1 ÞjA1 h ðuhtt þ rh pht Þjdt þ cjjuh ðt n Þjj
t n1 1 2
Z
tn
t n1
Z
tn
juht jdt tn1
12 1 2 2 2 jAh ðuhtt þ rh pht Þj þ jjuh ðtn Þjj juht j dt :
Then applying Theorem 4.2, we obtain 2 2 DtjA1 h R n j 6 c Dt
Z
tn tn1
2 2 jA1 dt: h ðuhtt þ rh pht Þj þ juht j
ð5:9Þ
Summing (5.9) from n ¼ 1 to n ¼ m and applying Theorem 4.2, we derive (5.6). Also, employing Hölder inequality, it follows from(4.18) and (5.5) that
Z tn Z tn 1 1 ðt tn1 ÞjAh 2 ðuhtt þ rh pht Þjdt þ c jjuh ðt n Þjj jjuht jjdt Dt tn1 t n1 Z tn 12 1 6 c Dt jAh 2 ðuhtt þ rh pht Þj2 þ jjuh ðt n Þjj2 jjuht jj2 dt :
1
jAh 2 Rn j 6
ð5:10Þ
t n1
Hence, we deduce from Theorem 4.2 that 1
jAh 2 Rn j2 Dt 6 cDt 2
Z
tn
t n1
1 jAh 2 ðuhtt þ rh pht Þj2 þ jjuht jj2 dt;
ð5:11Þ
Then summing (5.11) from n ¼ 1 to n ¼ m and applying Theorem 4.2, we derives (5.7). Similarly, from (4.17) and (5.5) and applying Hölder inequality, we have
Z tn Z tn 1 ðt t n1 Þjuhtt þ rh pht jdt þ cjAh uh ðt n Þj jjuht jjdt Dt tn1 tn1 Z tn 12 12 Z tn 6c ðt t n1 Þjuhtt þ rh pht j2 dt þ c DtjAh uh ðt n Þj2 jjuht jj2 dt :
jRn j 6
t n1
ð5:12Þ
t n1
Notice the fact
sðtn Þ 6 sðtÞ þ Dt; Dt 6 sðtn Þ; t tn1 6 sðtÞ; t 2 ½tn1 ; tn :
ð5:13Þ
Then applying Theorem 4.2 gives
Dt
m X
jRn j2 6 cDt
n¼1
m Z X n¼1
tn tn1
sðtÞjuhtt þ rh pht j2 dt þ cDt
m X
sðtn ÞjAh uh ðtn Þj2
Z
tn
jjuht jj2 dt 6 cDt
ð5:14aÞ
t n1
n¼1
and
Dt
m X
sðtn ÞjRn j2 6 cDt
n¼1
m X
sðtn Þ
Z
m Z X n¼1
ðt t n1 Þjuhtt þ rh pht j2 dt þ cDt 2
t n1
n¼1
6 c Dt 2
tn
tn
sðtÞjuhtt þ rh pht j2 dt þ cDt2
t n1
m Z X n¼1
m X
sðtn ÞjAh uh ðtn Þj2
n¼1 tn
Z
tn
jjuht jj2 dt
t n1
jjuht jj2 dt 6 cDt 2 :
ð5:14bÞ
t n1
Then (5.14a) and (5.14b) implies (5.8). h To analyze the error ðuh ðt n Þ unh ; ph ðt n Þ pnh Þ, we have to introduce the following time-discrete duality problem corre sponding to (5.4): For any given 1 6 m 6 N and zn 2 X h ; n ¼ 1; 2; . . . m, find Uhn1 ; Whn1 2 ðX h ; M h Þ such that for all ðv h ; qh Þ 2 ðX h ; M h Þ,
v h ; dt Unh
n1 b v h ; uh ðt n Þ; Un1 b uh ðt n1 Þ; v h ; Un1 ¼ ðv h ; zn Þ; Bh ðv h ; qh Þ; Un1 h ; Wh h h
with Um h ¼ 0. In order to analyze the regularity of solutions to problem (5.15), we introduce the following notations
jjPh Bðuh ðtn Þ; Þjj ¼ sup
jPh Bðuh ðtn Þ; v h Þj ; jjv h jj
jjP h Bð; uh ðtn ÞÞjj ¼ sup
jPh Bðv h ; uh ðt n ÞÞj : jjv h jj
v h 2X h v h 2X h
ð5:15Þ
93
L. Shan, Y. Hou / Applied Mathematics and Computation 215 (2009) 85–99
By using an exact similar argument to the one used in [19], we have the result below. Lemma 5.2. Assume that the assumptions of Theorem 4.1 are valid and that Dt satisfies
2
m
c21 jlnhj max jjuh ðtÞjj2 Dt 6 1:
ð5:16Þ
t2½0;T
Then
Dt
N X
jjP h Bðuh ðt n Þ; Þjj2 þ jjPh Bð; uh ðt n ÞÞjj2 6 c:
ð5:17Þ
n¼1
For the existence, uniqueness and smoothness of the solutions to problem (5.15), we provide the following result. Lemma 5.3. Assumethat the assumptions of Theorem 4.1 are valid and that Dt satisfies (5.16), then problem (5.15) admits a unique solution Uhn1 ; Whn1 2 ðX h ; M h Þ for a given Unh . Furthermore, the solution sequence fUnh ; Wnh gm1 n¼0 satisfies the following bound:
sup jjUrh jj2 þ Dt
06r6m
m1 X
jdt Unh j2 6 cDt
n¼1
m X
jzn j2 :
ð5:18Þ
n¼1
Proof. We start by defining the following bilinear form:
Q ððv h ; qh Þ; ðUh ; Wh ÞÞ ¼
1 ðv h ; Uh Þ þ Bh ððv h ; qh Þ; ðUh ; Wh ÞÞ þ bðuh ðt n1 Þ; v h ; Uh Þ þ bðv h ; uh ðt n Þ; Uh Þ: Dt
Consequently, we can rewrite (5.15) as follows:
1 n n1 n ¼ : Q ðv h ; qh Þ; Un1 ; W v ; U z h h h Dt h Following the discussions in [8], we first establish the existence of a constant a1 such that, for all ðUh ; Wh Þ 2 ðX h ; M h Þ
sup ðv h ;qh Þ2ðX h ;Mh Þ
Q ððv h ; qh Þ; ðUh ; Wh ÞÞ P a1 ðjjUh jj þ jWh jÞ: jjv h jj þ jqh j
To this end, we borrow the results from [9] that: For all Wh 2 M h M, there exists a positive constant a1 and g 2 X h such that
dðg; Wh Þ ¼ jWh j2 ;
jjgjj 6 a1 jWh j:
ð5:19Þ
If we choose ðv h ; qh Þ ¼ ðUh þ dg; Wh Þ, where
d¼
1 c 2c2 ðm þ 0 þ max jju ðtÞjj2 Þ1 : 2 Dt m t2½0;T h 2a21
ð5:20Þ
Then it follows from (4.4), (4.18), (5.16), (5.19) and (5.20) that
1 d jUh j2 jUh jjgj þ mjjUh jj2 dmjjgjj jjUh jj þ djWh j2 þ jðI ph ÞWh j2 Dt Dt 1 c1 jlnhj2 jjuh ðt n Þjj jjUh jjjUh j cdðjjuh ðtn Þjj þ jjuh ðtn1 ÞjjÞjjUh jj jjgjj m Dt 2 P c1 jlnhjjjuh ðt n Þjj2 jjUh jj2 þ jðI ph ÞWh j2 2 2 dc a2 c2 da21 þ d 1 mda21 0 1 ðjjuh ðt n Þjj2 þ jjuh ðt n1 Þjj2 Þ jWh j2 2 Dt m m d 2 2 P jjUh jj þ jðI ph ÞWh j þ jWh j2 P a2 ðjjUh jj þ jWh jÞ2 P a3 ðjjUh jj þ jWh jÞðjjv h jj þ jqh jÞ: 2 4
Q ððv h ; qh Þ; ðUh ; Wh ÞÞ P
Here we have used the following fact
jjv h jj þ jqh j 6 jjUh jj þ djjgjj þ jWh j 6 a4 ðjjUh jj þ jWh jÞ: Hence problem (5.15) admits a unique solution Un1 ; Whn1 2 ðX h ; M h Þ. h Next, we will consider the smoothness of Unh ; Wnh , from (5.15) we obtain
a v h ; Uhn1 þ d v h ; Whn1 d dt Unh ; qh G dt Wnh ; qh b v h ; uh ðt n Þ; Un1 h b uh ðtn1 Þ; v h ; Un1 ¼ ðv h ; zn Þ: h
v h ; dt Unh
ð5:21Þ
94
L. Shan, Y. Hou / Applied Mathematics and Computation 215 (2009) 85–99
Setting ðv h ; qh Þ ¼ ðdt Unh ; Whn1 ÞDt in (5.21), then (2.7) implies
m n1 2 2 n1 n 2 þ G Wn1 G Wnh ; Wnh b dt Unh ; uh ðt n Þ; Uhn1 Dt Dtjdt Unh j2 þ jjUh jj þ jjUnh Un1 h ; Wh h jj jjUh jj 2 b uh ðtn1 Þ; dt Unh ; Unh Dt 6 dt Unh ; zn Dt:
ð5:22Þ
Due to (2.8), (4.3) and (4.4), we have
1 2 jb dt Unh ; uh ðtn Þ; Unh Un1 j 6 jdt Unh j2 þ c21 jlnhjjjuh ðtn Þjj2 jjUnh Un1 h h jj ; 4 jb uh ðtn1 Þ; dt Unh ; Unh j þ jb dt Unh ; uh ðtn Þ; Unh j 1 6 jdt Unh j2 þ c jjPh Bðuh ðtn1 Þ; Þjj2 þ jjPh Bð; uh ðt n ÞÞjj2 jjUnh jj2 ; 8 1 j dt Unh ; zn j 6 jdt Unh j2 þ cjzn j2 : 8 Combining (5.22) with the above estimates and using (5.16) yields
2 n1 n1 mjjUnh jj2 G Wnh ; Wnh jdt Unh j2 Dt þ mjjUn1 h jj þ G Wh ; Wh 6 cjzn j2 Dt þ c jjPh Bðuh ðtn1 Þ; Þjj2 þ jjPh Bð; uh ðt n ÞÞjj2 jjUnh jj2 Dt:
ð5:23Þ
Setting r P 0 and summing (5.23) from n ¼ r þ 1 to n ¼ m 1, we obtain
mjjUrh jj2 þ Dt
m1 X
m1 X jdt Unh j2 6 a Um1 ; Uhm1 þ G Wm1 ; Wm1 jzn j2 þ cDt þ c Dt h h h
n¼rþ1
n¼rþ1
m1 X
jjPh Bðuh ðt n1 ÞÞ; jj2 þ jjPh Bð; uh ðtn ÞÞjj2 jjUnh jj2 :
ð5:24Þ
n¼rþ1
m1 Since Um ; Wm1 that h ¼ 0, we obtain from (5.15) and (2.7) with n ¼ m, ðv h ; qh Þ ¼ Uh h
1 m1 2 ; Whm1 þ b Um1 ; uh ðtm Þ; Uhm1 ¼ Um1 ; zm : jUh j þ a Uhm1 ; Uhm1 þ G Wm1 h h h Dt
ð5:25Þ
Due to (4.4) and (5.16), we have
1 jb Uhm1 ; uh ðt m Þ; Um1 jjjUhm1 j j 6 c1 jlnhj2 jjuh ðt m Þjj jjUm1 h h 1 m jj2 ; jUm1 j2 þ jjUm1 2 Dt h 2 h 1 1 jUm1 j2 þ jzm j2 Dt: jðUhm1 Þ; zm Þj 6 2Dt h 2 6
Combining these estimates with (5.25) yields
a Um1 ; Um1 þ G Whm1 ; Wm1 6 cjzm j2 Dt; h h h together with (5.24) gives
mjjUrh jj2 þ Dt
m1 X
jdt Unh j 6 cDt
n¼rþ1
m X
jzn j2 þ cDt
n¼rþ1
m1 X jjP h Bðuh ðt n1 Þ; Þjj2 þ jjPh Bð; uh ðtn ÞÞjj2 ÞjjUnh jj2 : n¼rþ1
Then applying Lemmas 4.3 and 5.2 yields
jjUrh jj2 þ Dt
m1 X n¼rþ1
jdt Unh j2 6 cDt
m X
jzn j2 :
n¼rþ1
Finally, take the supremum for 0 6 r 6 m and complete the proof. h
6. Error analysis In this section, we first provide some estimations for enh ; lnh in various norms, then combining with the results given in Sections 2 and 4, we establish the main error estimations for the fully discrete solutions.
L. Shan, Y. Hou / Applied Mathematics and Computation 215 (2009) 85–99
95
Lemma 6.1. Assume that the assumptions of Theorem 4.1 are valid and Dt satisfied (5.16). Then the error enh ; lnh ; 1 6 n 6 N satisfies the following bound 2 jem h j þ Dt
m X jjenh jj2 þ G lnh ; lnh 6 cDt 2 ;
1 6 m 6 N:
ð6:1Þ
n¼1
Proof. Setting ðv ; qÞ ¼ ðenh ; lnh Þ in (5.4) and using (2.7), we obtain
m n 2 1 n2 1 2 n1 2 n n þ mjjenh jj2 þ Gðlnh ; lnh Þ þ bðen1 jeh j þ jenh en1 jje jj þ m1 jAh 2 Rn j2 : h j jeh j h ; uh ðt n Þ; eh Þ ¼ Rn ; eh 6 2Dt 4 h
ð6:2Þ
For the trilinear term b, it follows from (4.1) and (4.2) that
m n 2 2 n jb en1 jje jj þ cjjP h Bð; uh ðt n ÞÞjj2 jen1 h ; uh ðt n Þ; eh j 6 h j : 4 h Combining (6.2) with the above estimate yields
jenh j2 jehn1 j2 þ
1
2 2 2 mjjenh jj2 þ G lnh ; lnh Dt 6 cjjPh Bð; uh ðtn ÞÞjj2 jen1 h j Dt þ cjAh Rn j Dt:
ð6:3Þ
Summing (6.3) from n ¼ 1 to n ¼ m and applying Lemma 4.2, 5.1 and 5.2 yields (6.1). h Lemma 6.2. Under the assumptions of Lemma 6.1, we have 2 jjem h jj þ Dt
m X
jdt enh j2 6 cDt;
6 m 6 N:
ð6:4Þ
n¼1
Proof. We derive from (5.4) that
n1 n1 n dt enh ; v h þ a enh ; v h d v h ; lnh þ d dt enh ; qh þ G dt ln1 h ; qh þ b eh ; uh ðt n Þ; v h þ b uh ; eh ; v h ¼ ðRn ; v h Þ
for all ðv h ; qh Þ 2 ðX h ; M h Þ. Setting ðv h ; qh Þ ¼
n h
ð6:5Þ
l Dt in (6.5) and using (2.7), we have m n 2 1 n1 n Dtjdt enh j2 þ jjeh jj jjehn1 jj2 þ G lnh ; lnh G ln1 þ b en1 h ; lh h ; uh ðt n Þ; dt eh Dt 2 2 n n þ b uh ðtn1 Þ; ehn1 ; dt enh Dt b en1 h ; eh ; dt eh Dt 6 Rn ; dt enh Dt: dt enh ;
ð6:6Þ
Owing to (4.4) and Cauchy inequality, we have
n n1 n jb en1 h ; uh ðt n Þ; dt eh j þ jbðuh ðt n1 Þ; eh ; dt eh Þj 1 6 jdt enh j2 þ c jjPh Bðuh ðt n1 Þ; Þjj2 þ jjPh Bð; uh ðtn ÞÞjj2 jjehn1 jj2 ; 6 1 1 n n jdt enh j2 þ c21 jlnhjjjehn1 jj2 jjenh jj2 ; jbðehn1 ; enh ; dt enh Þj 6 c1 jlnhj2 jjen1 h jj jjeh jjjdt eh j 6 6 1 j Rn ; dt enh j 6 jdt enh j2 þ cjRn j2 : 6 Combining (6.6) with the above estimates and using Lemma 6.1 yields 2 1 n n n1 n1 1 n 2 jjenh jj2 jjen1 h jj þ m ðGðlh ; lh Þ Gðlh ; lh ÞÞ þ m jdt eh j Dt 1 2 2 2 n1 2 n 2 6 c jjP h Bðuh ðt n1 Þ; Þjj2 þ jjPh Bð; uh ðtn ÞÞjj2 jjen1 h jj Dt þ c 1 jlnhjjjeh jj jjeh jj Dt þ cDtjRn j :
m
ð6:7Þ
Summing (6.7) from n ¼ 1 to n ¼ m and applying Lemmas 5.1, 5.2 and 6.1, we obtain 2 1 jjem h jj þ m Dt
m X
jdt enh j2 6 cDt
1
n¼1 m X
m
n¼1
þ c21 jlnhjDt
m X
2 jjPh Bðuh ðt n1 Þ; Þjj2 þ jjP h Bð; uh ðt n ÞÞjj2 jjen1 h jj
n¼1 2 n 2 jjen1 h jj jjeh jj þ Dt
m X
jRn j2 6 cDt:
n¼1
Lemma 6.3. Under the assumptions of Lemma 6.1, we have
sðtm Þjemh j2 þ Dt
m X jenh j2 þ sðtn Þjjenh jj2 þ sðt n ÞG lnh ; lnh 6 cDt 2 ; n¼1
1 6 m 6 N:
ð6:8Þ
96
L. Shan, Y. Hou / Applied Mathematics and Computation 215 (2009) 85–99
n1 in (5.4), we obtain Proof. Setting ðv ; qÞ ¼ ðenh ; lnh Þ; zn ¼ enh in (5.15) and ðv h ; qh Þ ¼ Un1 h ; Wh
n n1 eh ; dt Unh Bh enh ; lnh ; Un1 b enh ; uh ðt n Þ; Un1 b uh ðt n1 Þ; enh ; Un1 ¼ jenh j2 ; h ; Wh h h n1 n1 þ Bh enh ; lnh ; Un1 þ b uhn1 ; enh ; Un1 þ b en1 ¼ Rn ; Un1 : dt enh ; Un1 h ; uh ðt n Þ; Uh h h ; Wh h h
ð6:9Þ ð6:10Þ
Adding (6.9) to (6.10), we obtain
jenh j2 ¼
1 n n n1 n1 n1 n b dt enh ; uh ðt n Þ; Uhn1 Dt b en1 Rn ; Uhn1 : eh ; Uh eh ; Uh h ; eh ; Uh Dt
ð6:11Þ
From (4.1) and (4.2), we have
jbðdt enh ; uh ðtn Þ; Uhn1 Þj 6 cjdt enh j jjPh Bð; uh ðt n ÞÞjj jjUn1 h jj; n1 n1 n jbðehn1 ; enh ; Un1 h Þj 6 cjjeh jj jjeh jj jjUh jj; 1
n1 2 jðRn ; Un1 h Þj 6 jAh Rn j jjUh jj:
Combining (6.11) with the above estimates yields
þ cjjehn1 jj jjenh jj jjUn1 jenh j2 Dt 6 enh ; Unh ehn1 ; Un1 h h jjDt
þ cDt2 jdt enh jkjPh Bð; uh ðtn ÞÞjj jjUn1 h jj
1
þ jAh 2 Rn j jjUn1 h jjDt;
ð6:12Þ
with e0h ¼ Um h ¼ 0. Summing (6.12) from n ¼ 1 to n ¼ m and applying Lemma 5.3, we obtain
Dt
m X
jenh j2 6 cðDt
m X
n¼1
2 jjen1 h jj Þ Dt
n¼1
m X
! jjenh jj2
þ Dt Dt
n¼1
m X
1
jAh 2 Rn j2
! þ cDt 2 Dt
n¼1
m X
! jdt enh j2
n¼1
Dt
m X
! jjPh Bð; uh ðtn ÞÞjj2 :
n¼1
Then applying Lemmas 5.1, 5.2, 6.1 and 6.2, we get
Dt
m X
jenh j2 6 cDt3 ;
1 6 m 6 N:
ð6:13Þ
n¼1
Now, multiplying(6.3) by sðt n Þ and noting the fact
sðtn Þ 6 Dt þ sðtn1 Þ; Dt 6 sðtn1 Þ; 2 6 n 6 N; ehn1 ¼ 0; for n ¼ 1;
ð6:14Þ
we obtain
2 n 2 n n sðtn Þjenh j2 sðtn1 Þjen1 h j þ sðt n Þ mjjeh jj þ Gðlh ; lh Þ Dt
12 2 2 6 cDtjjP h Bð; uh ðt n ÞÞjj2 sðt n1 Þjehn1 j2 þ c jen1 Dt: h j þ jAh Rn j
ð6:15Þ
Summing (6.15) from n ¼ 1 to n ¼ m and applying Lemmas 4.2 and 5.2, we deduce
sðtm Þjemh j2 þ Dt
m X
sðtn Þ mjjenh jj2 þ Gðlnh ; lnh Þ 6 cDt
n¼1
m X 12 2 2 : jen1 h j þ jAh Rn j n¼1
After applying Lemma 5.1 and (6.13), we have
sðtm Þjemh j2 þ Dt
m X
sðtn Þ mjjenh jj2 þ Gðlnh ; lnh Þ 6 cDt2 :
ð6:16Þ
n¼1
This inequality and (6.13) yields (6.8). h Lemma 6.4. Under the assumptions of Lemma 6.1, the error enh ; lnh satisfies the following bound
sðtm Þjjemh jj2 þ Dt
m X n¼1
sðtn Þjdt enh j2 6 cDt2 :
ð6:17Þ
L. Shan, Y. Hou / Applied Mathematics and Computation 215 (2009) 85–99
97
Proof. Multiplying (6.7) by sðtn Þ, noting the fact (6.14), and using Lemmas 5.2 and 6.2, we have
2 1 n1 n1 sðtn Þ jjenh jj2 þ m1 Gðlnh ; lnh Þ sðtn1 Þ jjen1 h jj þ m G lh ; lh
þ m1 sðtn Þjdt enh j2 Dt
1 2 2 2 n1 2 n 2 6 csðtn1 Þ jjPh Bð; uh ðt n ÞÞjj2 þ jjPh Bðuh ðtn1 Þ; Þjj2 jjen1 h jj Dt þ sðt n ÞjRn j Dt þ c 1 Dtjlnhjsðt n Þjjeh jj jjeh jj m 2 2 2 1 n1 n1 n1 2 2 þ Dt jjen1 h jj þ m Gðlh ; lh Þ þ c jjP h Bð; uh ðt n ÞÞjj þ jjP h Bðuh ðt n1 Þ; Þjj jjeh jj Dt :
ð6:18Þ
Summing (6.18) from n ¼ 1 to n ¼ m and applying Lemmas 4.2, 5.1, 5.2, 6.1, 6.2, and 6.3, we obtain
sðtm Þjjemh jj2 þ Dt
m X
sðtn Þjdt enh j2 6 cDt2 :
n¼1
Lemma 6.5. Under the assumptions of Lemma 6.1, the error
Dt
m X
lnh ¼ ph ðtn Þ pnh satisfies the following bound:
sðtn Þjph ðtn Þ pnh j2 6 cDt2 ; 1 6 m 6 N:
ð6:19Þ
n¼1
Proof. It follows from Theorem 3.1 and (4.18) and (5.4) that
jlnh j 6 cjdt enh j þ c jjehn1 jj þ jjuh ðtn1 Þjj jjenh jj þ cjjuh ðt n Þjj jjen1 h jj þ cjRn j: Hence, together with the fact (6.14) and applying Theorem 4.2 and Lemma 6.2, we have
sðtn Þjlnh j2 Dt 6 csðtn Þjdt enh j2 Dt þ csðtn Þjjenh jj2 Dt þ csðtn1 Þjjehn1 jj2 Dt þ cDt2 jjehn1 jj2 þ csðtn ÞjRn j2 Dt:
ð6:20Þ
Summing (6.20) from n ¼ 1 to n ¼ m and applying Lemmas 5.1, 6.2, 6.3 and 6.4, we obtain
Dt
m X
sðtn Þjlnh j2 6 cDt2 ; 1 6 m 6 N:
n¼1
Theorem 6.1. Under the assumptions of Lemma 6.1, the error uðtn Þ unh ; pðt n Þ pnh satisfies the following bound:
Dt
N X
4
2 2 juðt n Þ unh j2 þ sðtm Þjuðt m Þ um h j 6 cðh þ Dt Þ;
ð6:21Þ
n¼1
Dt
N X
2
2 2 jjuðt n Þ unh jj2 þ sðt m Þjjuðt m Þ um h jj 6 cðh þ Dt Þ;
ð6:22Þ
sðtn Þjpðtn Þ pnh j2 6 cðh2 þ Dt2 Þ
ð6:23Þ
n¼1
Dt
N X n¼1
for all t m 2 ð0; T. Proof. Integrating by parts, we obtain
Dtjuðt n Þ uh ðtn Þj2 6 2
Z
Dtjjuðtn Þ uh ðtn Þjj2 6 2
tn
ju uh j2 dt þ
t n1 Z tn
Z
tn
ðt tn1 Þ2 jut uht j2 dt;
ð6:24Þ
tn1
jju uh jj2 dt þ
t n1
Z
tn
ðt tn1 Þ2 jjut uht jj2 dt:
ð6:25Þ
t n1
Recalling [13], we have
Z
T
sðtÞjut uht j2 dt 6 ch2 :
ð6:26Þ
0
Summing (6.24) and (6.25) from n ¼ 1 to n ¼ N, respectively, using (4.24) and (6.26) and noting the fact (5.13), we obtain
Dt
N X
juðt n Þ uh ðt n Þj2 6 2
Z
N X n¼1
2
ju uh j2 dt þ ch Dt;
ð6:27Þ
0
n¼1
Dt
T
jjuðt n Þ uh ðtn Þjj2 6 2
Z
T
2
jju uh jj2 dt þ ch Dt:
0
Then we can easily obtain (6.21) and (6.22) by employing Theorem 4.1, Lemmas 6.3 and 6.4.
ð6:28Þ
98
L. Shan, Y. Hou / Applied Mathematics and Computation 215 (2009) 85–99
Furthermore, we derive from (2.6), (3.5) and Theorem 4.2 that
Z
t
sðtÞjpht j2 ds 6 c
0
Z t
sðsÞjuhtt j2 þ jjuht jj4 þ jft j2 ds 6 c:
ð6:29Þ
0
Hence, applying the relation (5.3) and Theorem 2.1 yields
sðt1 Þjpðt1 Þ ph ðt1 Þj ¼
Z
tn
t n1
ðt tn1 Þpht ðtÞdt 6 Dtð
Z 0
Dt
1
sðsÞjpht j2 dsÞ2 6 cDt:
ð6:30Þ
In other words, we obtain Dtsðt n Þjpðtn Þ ph ðtn Þj2 6 cDt2 when n ¼ 1. For the case of n P 2, from the triangle inequality and integration by parts, we have
Dt
N X
sðtn Þjpðtn Þ ph ðtn Þj2 6 Dt
n¼2
N X
sðtn Þjpðtn Þ pðtn Þj2 þ Dt
n¼2
N X
sðtn Þjpðtn Þ ph ðtn Þj2
n¼2
! Z tn 2 Z tn N X sðtn Þ 2 ¼ ðpðt n Þ pðtÞÞdt þ ð ðpðtÞ ph ðtÞÞdtÞ Dt t n1 t n1 n¼2 Z tn 2 Z tn 2 ! N X sðtn Þ ¼ ðt t n1 Þpt dt þ ðpðtÞ ph ðtÞÞdt Dt t n1 t n1 n¼2 Z tn Z tn Z tn N N X X sðtn Þ 6 Dt sðtn Þ ðt t n1 Þjpt j2 dt þ dt jp ph j2 dt Dt t n1 tn1 t n1 n¼2 n¼2 Z T Z T 6 Dt 2 sðtÞjpt j2 dt þ 2 sðtÞjp ph j2 dt: t1
ð6:31Þ
t1
On the other hand, we subtract (3.2) from (2.9) with ðv ; qÞ ¼ ðv h ; qh Þ and using the properties of the Galerkin projection ðRh ; Q h Þ, we obtain
ðut uht ; v h Þ þ Bh ðeh ; lh Þ; ðv h ; qh Þ þ bðu uh ; u; v h Þ þ bðuh ; u uh ; v h Þ ¼ 0;
ð6:32Þ
where eh ¼ Rh ðu; pÞ uh ; lh ¼ Q h ðu; pÞ ph , then from (2.1), (2.8) and (3.5), we obtain
Z
T
0
sðtÞjlh j2 dt 6 c
Z
T
sðtÞjut uht j2 þ sðtÞjju uh jj2 jjujj2 þ jjuh jj2 dt:
0
Thus, it follows from (2.11), (4.6), (4.8) and (6.26) that
Z
T
0
sðtÞjlh j2 dt 6 ch2 :
ð6:33Þ
Combining with (2.11) and (4.5) yields
Z
T
0
sðtÞjp ph j2 dt 6 ch2
Z 0
T
Z jAuj2 þ jjpjj21 dt þ
0
T
sðtÞjlh j2 dt 6 ch2 :
ð6:34Þ
Now, substituting (6.34) into (6.31) and using (2.13) and (6.30) and Theorem 2.1, we get
Dt
N X
sðtn Þjpðtn Þ ph ðtn Þj2 6 cðh2 þ Dt2 Þ:
ð6:35Þ
n¼1
Finally, combining (6.35) with (6.19), we have
Dt
N X
sðtn Þjpðtn Þ pnh j2 6 Dt
n¼1
N X
sðtn Þ jpðtn Þ ph ðtn Þj2 þ jph ðtn Þ pnh j2 6 cðh2 þ Dt2 Þ:
n¼1
7. Conclusions In this paper, we have established a fully discrete stabilized finite element method for the two-dimensional time-dependent Navier–Stokes problem. The spatial discretization is based on the linear equal-order interpolations for velocity and pressure; the time discretization is based on the Euler semi-implicit scheme. We have derived the error estimates for the fully discrete stabilized finite element solution unh ; pnh . It has been shown that our method has the optimal convergence rate for velocity and pressure. Acknowledgement We would like to thank the anonymous referees for their valuable comments and suggestions that helped to improve this paper.
L. Shan, Y. Hou / Applied Mathematics and Computation 215 (2009) 85–99
99
References [1] V. Girault, P.A. Raviart, Finite Element Method for the Navier–Stokes Equations: Theory and Algorithms, Springer-Verlag, Berlin, Heidelberg, 1986. [2] T.J.R. Hughes, L.P. Franca, A new finite element formulation for computational fluids dynamics: V. Circumventing the Bavuska–Brezzi condition: a stable Ptrov–Galerkin formulation of the Stokes problem accommodating equal-order interpolations, Comput. Methods Appl. Mech. Eng. 62 (1987) 85– 99. [3] R. Codina, J. Blasco, Stabilized finite element method for the transient Navier–Stokes equations based on a pressure gradient projection, Comput. Methods Appl. Mech. Eng. 182 (2000) 277–300. [4] J. Douglas, J. Wang, An absolutely stabilized finite element method for the Stokes problem, Math. Comput. 52 (1989) 495–508. [5] L. Franca, F. Frey, Stabilized finite element methods:II. The incompressible Navier–Stokes equations, Comput. Methods Appl. Mech. Eng. 99 (1992) 209– 233. [6] T. Barth, P. Bochev, M. Gunzburger, J. Shahid, A taxonomy of consistently stabilized finite element methods for the Stokes problem, SIAM J. Sci. Comput. 25 (2004) 1585–1607. [7] R. Codina, J. Blasco, A finite element formulation for the Stokes problem allowing equal-pressure interpolation, Comput. Methods Appl. Mech. Eng. 143 (1997) 373–391. [8] R. Codina, J. Blasco, Analysis of pressure-stabilized finite element approximation of the stationary Navier–Stokes equations, Numer. Math. 87 (2000) 59–81. [9] P.B. Bochev, C.R. Dohrmann, M.D. Gunzburger, Stabilization of low-order mixed finite elements for the Stokes equations, SIAM J. Numer. Anal. 44 (2006) 82–101. [10] C.R. Dohrmann, P.B. Bochev, A stabilized finite element method for the Stokes problem based on polynomial pressure projections, Int. J. Numer. Methods Fluids 46 (2004) 183–201. [11] G.R. Barrenechca, J. Blasco, Pressure stabilization of finite element approximations of time-dependent incompressible flow problems, Comput. Methods Appl. Mech. Eng. 197 (2007) 219–231. [12] L. Blasco, R. Codina, Space and time estimates for a first order pressure stabilized finite element method for the incompressible Navier–Stokes equations, Appl. Numer. Math. 38 (2001) 475–497. [13] J. Li, Y. He, Z. Chen, A new stabilized finite element method for the transient Navier–Stokes equations, Comput. Methods 197 (2007) 22–35. [14] J.G. Heywood, R. Rannacher, Finite element approximation of the non-stationary Navier–Stokes problem I: regularity of solutions and second-order error estimates for spatial discretization, SIAM J. Numer. Anal. 19 (1982) 275–311. [15] P.G. Ciarlet, The Finite Element Method for Elliptic Problems, North-Holland, Amsterdam, 1978. [16] J.H. Bramble, J. Xu, Some estimates for a weighted L2 projection, Math. Comput. 56 (1991) 463–576. [17] R. Temam, Navier-Stokes Equations, Theory and Numerical Analysis, North-Holland, Amsterdam, 1984. [18] J.H. Hill, E. Süli, Approximation of the global attractors for the incompressible Navier–Stokes equations, IMA J. Numer. Anal. 20 (2000) 633–667. [19] Y. He, A fully discrete stabilized finite-element method for the time-dependent Navier–Stokes problem, IMA J. Numer. Anal. 23 (2003) 665–691. [20] J.G. Heywood, R. Rannacher, Finite-Element approximation of the nonstationary Navier–Stokes problem Part IV: error analysis for second-order time discretization, SIAM J. Numer. Anal. 27 (1990) 353–384.
