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Hindawi Publishing CorporationMathematical Problems in EngineeringVolume 2012, Article ID 297269, 14 pagesdoi:10.1155/2012/297269
Research ArticleA Stabilized Low Order Finite-VolumeMethod for the Three-Dimensional StationaryNavier-Stokes Equations
Jian Li,1, 2 Xin Zhao,2 Jianhua Wu,1 and Jianhong Yang2
1 College of Mathematics and Information Science, Shaanxi Normal University, Xi’an 710062, China2 Department of Mathematics, Baoji University of Arts and Sciences, Baoji 721007, China
Correspondence should be addressed to Jian Li, [email protected]
Received 13 July 2011; Accepted 28 October 2011
Academic Editor: Rafael Martinez-Guerra
Copyright q 2012 Jian Li et al. This is an open access article distributed under the CreativeCommons Attribution License, which permits unrestricted use, distribution, and reproduction inany medium, provided the original work is properly cited.
This paper proposes and analyzes a stabilized finite-volume method (FVM) for the three-dimensional stationary Navier-Stokes equations approximated by the lowest order finite elementpairs. The method studies the new stabilized FVM with the relationship between the stabilizedFEM (FEM) and the stabilized FVM under the assumption of the uniqueness condition. The resultshave three prominent features in this paper. Firstly, the error analysis shows that the stabilizedFVM provides an approximate solution with the optimal convergence rate of the same orderas the usual stabilized FEM solution solving the stationary Navier-Stokes equations. Secondly,superconvergence results on the solutions of the stabilized FEM and stabilized FVM are derived ontheH1-norm and the L2-norm for the velocity and pressure. Thirdly, residual technique is appliedto obtain the L2-norm error for the velocity without additional regular assumption on the exactsolution.
1. Introduction
Recently, the development of stable mixed FEMs is a fundamental component in the searchfor the efficient numerical methods for solving the Navier-Stokes equations governing theflow of an incompressible fluid by using a primitive variable formulation. The object of thiswork is to analyze the stabilized finite volume method for solving the three-dimensionalstationary Navier-Stokes equations.
The importance of ensuring the compatibility of the component approximations ofvelocity and pressure by satisfying the so-called inf-sup condition is widely understood. Thenumerous mixed finite elements satisfying the inf-sup condition have been proposed over theyears. However, elements not satisfying the inf-sup condition may also work well. So far, themost convenient choice of the finite element space from an implementational point of view
2 Mathematical Problems in Engineering
would be the elements of the low polynomial order in the velocity and the pressure with anidentical degree distribution for both the velocity and the pressure.
This paper focuses on the stabilized method called local polynomial pressureprojection for the three-dimensional Navier-Stokes equations [1–5]. The proposed methodis characterized by the following features. First, the method does not require approximationof derivatives, specification of mesh-dependent parameters, edge-based data structures, anda nonstandard assembly procedure. Second, this method is completely local at the elementlevel.
On the other hand, FVM has become an active area in numerical analysis. The mostattractive things are that FVM can keep local conservation and have the advantages of FVMand finite difference methods. The FVM is also termed the control volume method, thecovolume method, or the first-order generalized difference method. Nowadays, it is difficultin analyzing FVM to obtain L2-norm error estimates because trail functions and test functionsare derived from different spaces. Many papers were devoted to its error analysis for second-order elliptic and parabolic partial differential problems [6–10]. Error estimates of optimalorder in theH1-norm are the same as those for the linear FEM [9, 11]. Error estimates of opti-mal order in the L2-norm can be obtained as well [8, 9]. Moreover, the FVM for generalizedStokes problems was studied by many people [11–13]. They analyzed this method by usinga relationship between it and the FEM and obtained its error estimates through those knownfor the latter method. Also, it still requires H3 smoothness assumption of the exact solutionto obtain O(h2) error bound in most previous literatures. However, for the Stokes problemsonly the finite elements that satisfy the discrete inf-sup condition have been studied.
The work of [14, 15] for the two-dimensional stationary Stokes equations is extendedin this paper for the three-dimensional stationary Navier-Stokes equations approximatedby lowest equal-order finite elements. Following the abstract framework of the relationshipbetween the stabilized FEM and stabilized FVM [14, 15], the stabilized FVM is studied,and the optimal error estimate of the stabilized FVM is obtained for the three-dimensionalstationary Navier-Stokes equations relying on the uniqueness condition. As far as known,there still requires much research on FVM results [16] about the velocity in L2-norm andsuperconvergence result between FEM solution and FVM solution of the three-dimensionalNavier-Stokes equations.
The remainder of the paper is organized as follows. In Section 2, an abstractfunctional setting of the three-dimensional Navier-Stokes problem is given with some basicassumptions. In Section 3, the stability of the stabilized FVM is analyzed and provided byBrouwer’s fixed-point theorem. In Section 4, the optimal error estimates of the stabilizedfinite volume approximation for the three-dimensional stationary Navier-Stokes equationsare obtained.
2. FVM Formulation
Let Ω be a bounded domain in R3, assumed to have a Lipschitz-continuous boundary Γ andto satisfy a further condition stated in (A1) below. The three-dimensional stationary Navier-Stokes equations are considered as follows:
−νΔu +∇p + (u · ∇)u = f, inΩ, (2.1a)
divu = 0, inΩ, (2.1b)
u|∂Ω = 0, on ∂Ω, (2.1c)
Mathematical Problems in Engineering 3
where ν > 0 is the viscosity, u = (u1(x), u2(x), u3(x)) represents the velocity vector, p = p(x)the pressure, and f = (f1(x), f2(x), f3(x)) the prescribed body force.
In order to introduce a variational formulation, we set [17]
X =[H1
0(Ω)]3, Y =
[L2(Ω)
]3, M = L2
0(Ω) ={q ∈ L2(Ω) :
∫
Ωqdx = 0
},
D(A) =[H2(Ω)
]3 ∩X.
(2.2)
As mentioned above, a further assumption on Ω is presented.(A1)Assume thatΩ is regular so that the unique solution (v, q) ∈ (X,M) of the steady
Stokes problem
−Δv +∇q = g, divv = 0 inΩ, v|∂Ω = 0, (2.3)
for a prescribed g ∈ Y exists and satisfies
‖v‖2 +∥∥q∥∥1 ≤ c
∥∥g∥∥0, (2.4)
where c > 0 is a general constant depending on Ω. Here and after, ‖ · ‖i and | · |i denote theusual norm and seminorm of the Sobolev space Hi(Ω) or Hi(Ω)3 for i = 0, 1, 2.
We denote by (·, ·) the inner product on L2(Ω) or Y . The space H10(Ω) and X are
equipped with their equivalent scalar product and norm [17]
((u, v)) = (∇u,∇v), ‖∇u‖0 = ((u, u))1/2. (2.5)
It is well known [18] that for each v ∈ X there hold the following inequalities:
‖v‖L4 ≤ 21/2‖v‖1/40 ‖∇v‖3/40 , ‖v‖0 ≤ γ‖v‖1, (2.6)
where γ is a positive constant depending only on Ω.The continuous bilinear form a(·, ·) on X × X and d(·, ·) on X × M, respectively, are
defined by
a(u, v) = ((u, v)), ∀u, v ∈ X, d(v, q)= −(v,∇q
)=(q,divv
), ∀v ∈ X, q ∈ M. (2.7)
Also, the trilinear term is defined by
b(u, v,w) = ((u · ∇)v,w) +12((divu)v,w)
=12((u · ∇)v,w) − 1
2((u · ∇)w,v), ∀u, v,w ∈ X
(2.8)
4 Mathematical Problems in Engineering
and satisfies
b(u, v,w) ≤ c0‖∇u‖0‖∇v‖0‖∇w‖0. (2.9)
Then the mixed variational form of (2.1a)–(2.1c) is to seek (u, p) ∈ (X,M) such that
a(u, v) − d(v, p)+ d(u, q)+ b(u, u, v) =
(f, v), ∀(v, q) ∈ X ×M. (2.10)
The existence and uniqueness results are classical and can be found in [18–20].We introduce the finite-dimensional subspace (Xh,Mh) ⊂ (X,M), which is
characterized by τh with mesh scale h, a partitioning of Ω into tetrahedron or hexahedron,assumed to be regular in the usual sense(see [20–22]).
Here, the space (Xh,Mh) satisfies the following approximation properties. For eachv ∈ D(A), p ∈ H1(Ω), there exist approximations Ihv ∈ Xh and Jhq ∈ Mh such that
‖u − Ihu‖0 + h(‖∇(u − Ihu)‖0 +
∥∥p − Jhp∥∥0
) ≤ ch2(‖u‖2 +∥∥p∥∥1
), (2.11)
together with the inverse inequality
‖∇vh‖0 ≤ c1h−1‖vh‖0, ‖uh‖L∞ ≤ c2h
−1/2‖∇uh‖0. (2.12)
The stable and accurate finite element approximational solution of (2.10) requires that(Xh,Mh) satisfies the discrete inf-sup condition
supvh∈Xh
d(vh, qh
)
‖∇vh‖0≥ β∥∥qh∥∥0, (2.13)
where β is positive constant independent of h.The main purpose of this paper is to study a stabilized FVM for the stationary 3D
Navier-Stokes equations. We follow [23, 24] to obtain the dual partition Kh. We first choosean arbitrary point Q in the interior of each tetrahedron K and then connect Q with thebarycentersQijk of its 2D facesΔPiPjPk by straight lines (see Figure 1). On each faceΔPiPjPk,we connect by straight linesQijk with the middle points of the segments PiPj , PjPk, and PkPi,respectively. Then the contribution of K to the control volume K of a vertex P of K is thevolume surrounding P by these straight lines, for example, the contribution from one simplexto the control volume K with the interfaces γ12 and γ13.
Then, the dual finite element space can be constructed for the FVM as follows:
Xh ={v ∈[L2(Ω)
]3: v|K ∈ P0
(K)∀K ∈ Kh; v|∂K = 0
}. (2.14)
Mathematical Problems in Engineering 5
γ12
γ13
P3
D
P1
P1
P2
Q
Q123
Q13
Q12
Figure 1: Control volumes in three-dimensional case.
Obviously, the dimensions of Xh and Xh are the same. Furthermore, there exists an invertiblelinear mapping Γh : Xh → Xh such that for
vh(x) =N∑j=1
vh
(Pj
)φj(x), x ∈ Ω, vh ∈ Xh, (2.15)
with
Γhvh(x) =N∑j=1
vh
(Pj
)χj(x), (2.16)
where {φj} indicates the basis for the finite element space Xh, and {χj} denotes the basisfor the finite volume space Xh that are the characteristic functions associated with the dualpartition Kh:
χj(x) =
⎧⎨⎩1 ifx ∈ Kj ∈ Kh,
0 otherwise.(2.17)
The above idea of connecting the trial and test spaces in the Petrov-Galerkin methodthrough the mapping Γh was first introduced in [25, 26] in the context of elliptic problems.Furthermore, the mapping Γh satisfies the following properties [26].
6 Mathematical Problems in Engineering
Lemma 2.1. Let K ∈ Kh. If vh ∈ Xh and 1 ≤ r ≤ ∞, then
∫
K
(vh − Γhvh)dx = 0, (2.18)
‖Γhvh‖0 ≤ c3‖vh‖0, ‖vh − Γhvh‖Lr(K) ≤ c4hK‖∇vh‖Lr(K),(2.19)
where hK is the diameter of the element K.
Multiplying (2.1a) by Γhvh ∈ Xh and integrating over the dual elements K ∈ Kh, (2.1b)by qh ∈ Mh and over the primal elements K ∈ Kh, and applying Green’s formula, we definethe following bilinear forms for the FVM:
A(uh,Γhvh) = −N∑j=1
vh
(Pj
) ·∫
∂Kj
∂uh
∂nds, uh, vh ∈ Xh,
D(Γhvh, ph
)=
N∑j=1
vh
(Pj
) ·∫
∂Kj
phn ds, ph ∈ Mh,
b(uh, vh,Γhwh) = ((uh · ∇)vh,Γhwh) +12(divuhvh,Γhwh),
(f,Γhvh
)=
N∑j=1
vh
(Pj
) ·∫
Kj
f dx, vh ∈ Xh,
(2.20)
where n is the unit normal outward to ∂Kj and these terms are well posed.As noted above, this paper forces on a class of unstable velocity-pressure pairs
consisting of the lowest equal-order finite elements
Xh ={v ∈ X : v|K ∈ [R1(K)]3, ∀K ∈ τh
},
Mh ={q ∈ M : q|K ∈ Ri(K), i = 0, 1, ∀K ∈ τh
},
(2.21)
where Ri(K), i = 0, 1 represent piecewise constant range and continuous range on set K, Ri,i = 0,1 are spaces of polynomials, the maximum degree of which is bounded uniformly withrespect to K ∈ τh and h. The corresponding stabilized FEM is formulated as follows [3]:
a(uh, vh) − d(vh, ph
)+ d(uh, qh
)+G(ph, qh
)+ b(uh, uh, vh) =
(f, vh
), ∀(vh, qh
) ∈ (Xh,Mh).(2.22)
Also, the corresponding stabilized FVM is defined for the solution (uh, ph) ∈ (Xh,Mh) asfollows:
Ch
((uh, ph
);(vh, qh
))+ b(uh, uh,Γhvh) =
(f,Γhvh
) ∀(vh, qh) ∈ (Xh,Mh), (2.23)
Mathematical Problems in Engineering 7
where
Ch
((uh, ph
);(vh, qh
))= A(uh,Γhvh) +D
(Γhvh, ph
)+ d(uh, qh
)+G(ph, qh
). (2.24)
Obviously, the bilinear form G(·, ·) can be defined by the following symmetry form: [1]
G(p, q)=(p −Πhp, q −Πhq
). (2.25)
Note that
Πh =
⎧⎨⎩L2(Ω) −→ R0 if i = 1,
L2(Ω) −→ R1 if i = 0.(2.26)
Here, the operator Πh satisfies the following properties: [1, 4]
(p, qh
)=(Πhp, qh
) ∀p ∈ M, qh ∈ Ri, (2.27)∥∥Πhp
∥∥0 ≤ c5
∥∥p∥∥0 ∀p ∈ M, (2.28)∥∥p −Πhp
∥∥Lp ≤ c6h
∥∥p∥∥H1,p ∀p ∈ H1(Ω) ∩M. (2.29)
In particular, the L2-projection operator Πh can be extended to the vector case.This section concentrates on the study of a relationship between the FEM and FVM for
the Stokes equations.
Lemma 2.2. It holds that [11–13]
A(uh,Γhvh) = a(uh, vh) ∀uh, vh ∈ Xh, (2.30)
with the following properties:
A(uh,Γhvh) = A(vh,Γhuh),
|A(uh,Γhvh)| ≤ c7‖∇uh‖0‖∇vh‖0,
|A(vh,Γhvh)| ≥ ν‖∇vh‖20.
(2.31)
Moreover, the bilinear form D(·, ·) satisfies [14]
D(Γhvh, qh
)= −d(vh, qh
) ∀(vh, qh) ∈ (Xh,Mh). (2.32)
Based on detailed results on existence, uniqueness, and regularity of the solution forthe FVM (2.23), the following result establishes its continuity and weak coercivity.
8 Mathematical Problems in Engineering
Theorem 2.3. It holds that [14]
∣∣Ch
((uh, ph
),(vh, qh
))∣∣ ≤ c(‖uh‖1 +
∥∥ph∥∥0
)(‖vh‖1 +∥∥qh∥∥0
)
∀(uh, ph),(vh, qh
) ∈ (Xh,Mh).(2.33)
Moreover,
sup(vh,qh)∈(Xh,Mh)
∣∣Ch
((uh, ph
),(vh, qh
))∣∣‖vh‖1 +
∥∥qh∥∥0
≥ β(‖uh‖1 +
∥∥ph∥∥0
) ∀(uh, ph) ∈ (Xh,Mh),
(2.34)
where β is independent of h.
3. Stability
In this section, we analyze the results of FVM for the three-dimensional stationary Navier-Stokes equations. Firstly, we are now in a position to show the well-posedness of system(2.23)
h0 =4c2c3c4γh1/2
∥∥f∥∥0ν2
. (3.1)
Theorem 3.1 (stability). For each h > 0 such that
0 < h0 ≤ 12, (3.2)
system (2.23) admits a solution (uh, ph) ∈ (Xh,Mh). Moreover, if the viscosity ν > 0, the body forcef ∈ Y , and the mesh size h > 0 satisfy
0 < h0 ≤ 14, 1 − 2c0c5γν−2
∥∥f∥∥0 > 0, (3.3)
then the solution (uh, ph) ∈ (Xh,Mh) is unique. Furthermore, it satisfies
‖∇uh‖0 ≤2c3γν
∥∥f∥∥0,∥∥ph∥∥0 ≤ β−1c3γ
∥∥f∥∥0(4c3c8ν−2γ
∥∥f∥∥0 + 1). (3.4)
Proof. For fixed f ∈ Y , we introduce the set
BM ={(
vh, qh) ∈ (Xh,Mh) : ‖∇uh‖0≤
2c3γν
∥∥f∥∥0,∥∥ph∥∥0≤β−1c3γ
∥∥f∥∥0(4c3c8ν−2γ
∥∥f∥∥0+1)}
.
(3.5)
Mathematical Problems in Engineering 9
Then we define the mapping Th : (Xh,Mh) → (Xh,Mh) by [19]
A(Phwh,Γhvh) +D(Γhvh, ph
)+ d(Thwh, qh
)+G(ph, qh
)+ b(wh, Thwh,Γhvh) =
(f,Γhvh
)
∀(vh, qh) ∈ (Xh,Mh),
(3.6)
where Th(wh, ph) :≡ (T1wh, T2ph) = (uh, ph). We will prove that Th maps BM into BM.First, taking (vh, qh) = (uh, ph) ∈ (Xh,Mh) in (3.6), and using (2.12) and (2.19), we see
that for for all (vh, qh) ∈ (Xh,Mh)
ν‖∇uh‖20 ≤ A(uh,Γhuh) +G(ph, ph
)
≤ ∥∥f∥∥0‖Γhuh‖0 + 2c2c4h1/2‖∇wh‖20‖∇uh‖20
≤ c3γ∥∥f∥∥0‖∇uh‖0 +
4c2c4c3γh1/2∥∥f∥∥0
ν‖∇uh‖20
(3.7)
since
b(wh, uh,Γhuh) = b(wh, uh,Γhuh − uh)
≤(‖wh‖L∞‖∇uh‖0 +
√32
‖uh‖L∞‖∇wh‖0)‖Γhuh − uh‖0
≤ 2c2c4h1/2‖∇wh‖0‖∇uh‖20.
(3.8)
Thus, we have
ν
(1 − 4c2c3c4γh1/2
∥∥f∥∥0ν2
)≤ c3γ
∥∥f∥∥0‖∇uh‖0, (3.9)
which implies
‖∇uh‖0 ≤2c3γν
∥∥f∥∥0. (3.10)
Then, using the definition of b(·; ·, ·) and Ch(·, ·), (2.19), setting c8 = 2max{2c2c4h1/2, c0}, andthe same approach as above gives that
Ch
((uh, ph
),(vh, qh
))
‖∇vh‖0 +∥∥qh∥∥0
≥ β∥∥ph∥∥0,
|b(wh;uh,Γhvh)| = |b(wh;uh,Γhvh − vh) + b(wh;uh, vh)|≤ c8‖∇wh‖0‖∇uh‖0‖∇vh‖0,
∣∣(f,Γhvh
)∣∣ ≤ ∥∥f∥∥0‖Γhvh‖0≤ c3γ
∥∥f∥∥0‖∇vh‖0,
(3.11)
10 Mathematical Problems in Engineering
which, together with (3.10), gives
∥∥ph∥∥0 ≤ β−1c3γ
∥∥f∥∥0(4c3c8ν−2γ
∥∥f∥∥0 + 1). (3.12)
Since the mapping Th is well defined, it follows from Brouwer’s fixed-point theorem thatthere exists a solution to system (2.23).
To prove uniqueness, assume that (u1, p1) and (u2, p2) are two solutions to (2.23). Thenwe see that
Ch
((u1 − u2, p1 − p2
),(vh, qh
))+ b(u1 − u2;u1,Γhvh) + b(u2;u1 − u2,Γhvh) = 0. (3.13)
Letting (vh, qh) = (u1 − u2, p1 − p2) = (e, η), we obtain
Ch
((e, η),(e, η)) ≥ ν‖∇e‖20,
|b(e;u1,Γhe) + b(u2; e,Γhe)| = |b(e;u1,Γhe − e) + b(e;u1, e) + b(u2; e,Γhe − e)|
≤ 2c2c4h1/2(‖∇u1‖0 + ‖∇u2‖0)‖∇e‖20 + c0‖∇u1‖0‖∇e‖20
≤ 2(νh0 + c0c5γν
−1∥∥f∥∥0)‖∇e‖20,
(3.14)
which together with (3.3) and (3.13), gives
0 ≤ ν(1 − 2c0c5γν−2
∥∥f∥∥0)‖∇e‖20 ≤ 0, (3.15)
which shows that e = 0 by (3.15); that is, u1 = u2. Next, applying (3.3) to (3.13) and (2.34)yields that p1 = p2. Therefore, it follows that (2.23) has a unique solution.
4. Optimal Error Estimates
Theorem 4.1 (optimal error and superconvergent results). Assume that h > 0 satisfies (3.2) andf ∈ Y and ν > 0 satisfy (3.2). Let (u, p) ∈ (X,M) and (uh, ph) ∈ (Xh,Mh) be the solution of (2.10)and (2.23), respectively. Then it holds
‖u − uh‖1 +∥∥p − ph
∥∥0 ≤ κh
(‖u‖2 +∥∥p∥∥1 +
∥∥f∥∥0). (4.1)
Also, if f ∈ [H1(Ω)]3, there holds for the solution (uh, ph) of (2.22) that
‖uh − uh‖1 +∥∥ph − ph
∥∥0 ≤ κh3/2(‖u‖2 +
∥∥p∥∥1 +∥∥f∥∥1
). (4.2)
Proof. Subtracting (2.10) from (2.23) gives that
Ch
((e, η);(vh, qh
))+ b(e, uh,Γhvh) + b(uh, e, vh) + b(uh, uh, vh − Γhvh) = 0, (4.3)
Mathematical Problems in Engineering 11
with (e, η) = (uh − uh, ph − ph). By (vh, qh) = (e, η), it follows that
ν‖∇e‖20 +G(η, η)+ b(e, uh, e) + b(uh, uh,Γhe − e) =
(f, vh − Γhvh
). (4.4)
Using Theorem 3.1, (2.12), (2.23), and (2.25) gives
b(e, uh, e) ≤ c0‖∇u‖0‖∇e‖20≤ 2c2c4h1/2‖∇uh‖0‖∇e‖20
≤ 4c2c4c5γh1/2∥∥f∥∥0
ν2‖∇e‖20
= νh0‖∇e‖20,
b(uh, uh,Γhe − e) ≤∣∣∣∣(((uh −Πhuh) · ∇)uh +
12divuh(uh −Πhuh), e − Γhe
)∣∣∣∣
≤ ‖∇uh‖∞‖uh −Πhuh‖0‖e − Γhe‖0≤ ch3/2‖∇uh‖0‖∇e‖20.
(4.5)
Similarly, by Lemma 2.1 and (2.25), we have
∣∣(f, e − Γhe)∣∣ = ∣∣(f −Πhf, e − Γhe
)∣∣
≤ Chi∥∥f∥∥i‖e − Γhe‖0
≤ ch2(i+1)∥∥f∥∥2i +ν04‖e‖21, i = 0, 1.
(4.6)
Combining the above inequalities with (4.3) gives
‖e‖1 ≤ c(h3/2 + hi+1
)∥∥f∥∥i, i = 0, 1. (4.7)
In the same argument, it follows from (2.34) that
∥∥η∥∥0 ≤ c(h3/2 + hi+1
)∥∥f∥∥i, i = 0, 1. (4.8)
Noting that [3]
‖u − uh‖1 +∥∥p − ph
∥∥0 ≤ ch
(‖u‖2 +∥∥p∥∥1 +
∥∥f∥∥0), (4.9)
(4.6)–(4.8), and using a triangle inequality completes the proof of Theorem 4.1.
12 Mathematical Problems in Engineering
As noted above, it is still difficult to achieve an optimal error estimate for the velocityin the L2-norm for the three-dimensional stationary Navier-Stokes equations. Here, thefollowing dual problem is proposed and analyzed:
a(v,Φ) + d(v,Ψ) − d(Φ, q)+ b(u;v,Φ) + b(v;u,Φ) = (u − uh, v). (4.10)
Because of convexity of the domain Ω, this problem has a unique solution that satisfies theregularity property [18]
‖Φ‖2 + ‖Ψ‖1 ≤ C‖u − uh‖0. (4.11)
Below set (Φh,Ψh) = (IhΦ, JhΨ) ∈ (Xh,Mh), which satisfies, by (3.2),
‖Φ −Φh‖0 + h(‖Φ −Φh‖1 + ‖Ψ −Ψh‖0) ≤ Ch2(‖Φ‖2 + ‖Ψ‖1). (4.12)
Theorem 4.2 (optimal L2-error for the velocity). Let (u, p) be the solution of (2.1a)–(2.1c) andlet (uh, ph)be the solution of (4.3). Then, under the assumptions of Theorem 4.1, it holds
‖u − uh‖0 ≤ Ch2(‖u‖2 +∥∥p∥∥1 +
∥∥f∥∥1). (4.13)
Proof. Multiplying (2.1a) and (2.1b) by ΓhΦh ∈ Xh andΨh ∈ Mh and integrating over the dualelements K and the primary elements K, respectively, and adding the resulting equations to(2.23)with (vh, qh) = (Φh,Ψh), we see that
A(e,ΓhΦh) +D(ΓhΦh, η
)+ d(e,Ψh) +G
(η,Ψh
)
+ b(e;u,ΓhΦh) + b(u; e,ΓhΦh) − b(e; e,ΓhΦh) = G(p,Ψh
),
(4.14)
where (e, η) = (u − uh, p − ph). Subtracting (4.14) from (4.10)with (v, q) = (e, η) to obtain
‖e‖20 = a(e,Φ −Φh) + d(e,Ψ −Ψh) − d(Φ −Φh, η
) −G(η,Ψh
)+G(p,Ψh
)+ a(e,Φh)
−A(e,ΓhΦh) − d(Φh, η
) −D(ΓhΦh, η
)+ b(u; e,Φ − ΓhΦh) + b(e;u,Φ − ΓhΦh)
+ b(e; e,ΓhΦh)
= a(e,Φ −Φh) + d(e,Ψ −Ψh) − d(Φ −Φh, η
) −G(η,Ψh
)+G(p,Ψh
)+ b(e; e,ΓhΦh)
+ b(u; e,Φ − ΓhΦh) + b(e;u,Φ − ΓhΦh) +(f − (u · ∇)u,Φh − ΓhΦh
).
(4.15)
Mathematical Problems in Engineering 13
Obviously, we deduce from Theorem 3.1, (2.27)–(2.29), (4.11), the inverse inequality (2.12),and the Cauchy inequality that∣∣a(e,Φ −Φh) + d(e,Ψ −Ψh) − d
(Φ −Φh, η
)∣∣ ≤ c(‖e‖1 +
∥∥η∥∥0)(‖Φ −Φh‖1 + ‖Ψ −Ψh‖0)
≤ ch2(‖u‖2 +∥∥p∥∥1
)(‖Φ‖2 + ‖Ψ‖1)
≤ ch2(‖u‖2 +∥∥p∥∥1
)‖e‖0,∣∣G(η,Ψh
) −G(p,Ψh
)∣∣ ≤ ch(∥∥p −Πp
∥∥0 +∥∥η∥∥0
)‖Ψ‖1≤ ch2(‖u‖2 +
∥∥p∥∥1)‖e‖0,
∣∣(f − (u · ∇)u,Φh − ΓhΦh
)∣∣ = ∣∣([f −Πhf]
−[(u · ∇)u −Πh(u · ∇)u],Φh − ΓhΦh)∣∣
≤ ch2(∥∥f∥∥1 + ‖∇[(u · ∇)u]‖0)‖Φh‖1
≤ ch2(∥∥f∥∥1 + ‖u‖1/20 ‖u‖3/22 + ‖u‖21,4
)‖e‖0.
|b(u; e,Φ − ΓhΦh) + b(e;u,Φ − ΓhΦh)| ≤ c‖u‖2‖e‖1(‖Φh − ΓhΦh‖0 + ‖Φ −Φh‖0)
≤ ch2(‖u‖2 +∥∥p∥∥1
)‖Φ‖1≤ ch2(‖u‖2 +
∥∥p∥∥1)‖e‖0,
|b(e; e,ΓhΦh)| = |b(e; e,ΓhΦh −Φh) + b(e; e,Φh)|
≤ c(‖e‖0,4‖e‖1‖ΓhΦh −Φh‖0,4 + ‖e‖21‖Φh‖1
)
≤ ch‖e‖1/40 ‖e‖7/41 ‖∇Φh‖0,4 + c‖e‖21‖Φh‖1≤ ch2(‖u‖2 +
∥∥p∥∥1)‖e‖0.
(4.16)
Combining all these inequalities with (4.15) yields (4.13).
In this paper, we have obtained optimal and convergent results of the stabilized mixedfinite volume method for the stationary Navier-Stokes equations approximated by the loworder finite elements. Furthermore, we could apply the same technique presented to developand obtain the corresponding results of other (stabilized) mixed finite volume methods intwo or three dimensions.
Acknowledgments
This research was supported in part by the NSF of China (no. 11071193 and 10971124),Program for New Century Excellent Talents in University, Natural Science New Star ofScience and Technologies Research Plan in Shaanxi Province of China (no. 2011kjxx12),Research Program of Education Department of Shaanxi Province (no. 11JK0490), the projectsponsored by SRF for ROCS, SEM, and Key project of Baoji University of Arts and Science(no. ZK11157).
14 Mathematical Problems in Engineering
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