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Iterative schemes for the non-homogeneous Navier–Stokes equations based on the finite element approximation
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Iterative schemes for the non-homogeneous Navier–Stokes equations based on the finite element approximation✩ Kun Wang ∗ College of Mathematics and Statistics, Chongqing University, Chongqing 401331, PR China Institute of Computing and Data Sciences, Chongqing University, Chongqing 400044, PR China
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Article history: Received 20 June 2015 Received in revised form 8 October 2015 Accepted 9 November 2015 Available online xxxx
In this paper, we consider the stability and convergence of three iterative schemes for the non-homogeneous steady Navier–Stokes equations. As a nonlinear problem, we will get a nonlinear discrete system if approximating the non-homogeneous Navier–Stokes equations. After proving the stability and error estimates of the finite element method for the non-homogeneous Navier–Stokes equations, three iterative schemes are investigated for solving the resulted nonlinear discrete system. The stability and convergence conditions for these iterative methods are also analyzed, respectively. Furthermore, new results for the stop criterion are proved. Finally, we show some numerical experiments to illustrate the theoretical prediction. © 2015 Elsevier Ltd. All rights reserved.
Keywords: Navier–Stokes equations Non-homogeneous boundary condition Stability and convergence analysis Iterative scheme Finite element method
1. Introduction Let us consider the stationary Navier–Stokes equations in two dimension
−ν 1u + (u · ∇)u + ∇ p = f ,
div u = 0 in Ω ,
(1)
u|∂ Ω = g ,
(2)
where u = u(x) = (u1 (x), u2 (x)) is the velocity, p = p(x) is the pressure, f = f (x) = (f1 (x), f2 (x)) is the imposed body force, ν > 0 is the viscosity and Ω is a bounded domain in R2 which has a Lipschitz continuous boundary ∂ Ω and satisfies the additional condition stated in (A1) below, g is a given function. We assume that ν = 1/Re, where Re is the Reynolds number. The non-homogeneous Navier–Stokes equations (1)–(2) can be used to describe many classical models in the practical, such as, the lid-driven cavity flow, the backward step flow, the flow over a circular cylinder and so on. But to the best of our knowledge, the approximation analysis of this problem is rare. Because the problem (1)–(2) is nonlinear, a typical numerical solution procedure will involve two steps: spatial discretization and iterative scheme solving the discretized nonlinear system. Consequently, the stability and error are also determined by these two parts. As a popular discretization method, the finite element approximation has been widely used to solve the homogeneous Navier–Stokes equations. Lots of references have been reported to address the problem related to the spatial discretization [1–14], such as the large eddy simulation (LES) method [12], the defect correction method [9,14] and the
✩ This work is supported by the Natural Science Foundation of China (No. 11201506), and Project No. 106112015CDJXY100007 supported by the Fundamental Research Funds for the Central Universities. ∗ Correspondence to: College of Mathematics and Statistics, Chongqing University, Chongqing 401331, PR China. E-mail address: [email protected].
http://dx.doi.org/10.1016/j.camwa.2015.11.011 0898-1221/© 2015 Elsevier Ltd. All rights reserved.
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techniques developed recently based on the variational multiscale method [2,6] and so on. But there are few works devoted to the finite element analysis for the non-homogeneous Navier–Stokes equations. On the other hand, it is well-known that the Reynolds number plays a very important role in searching the iterative scheme for the nonlinear discretized system from the Navier–Stokes equations. The numerical algorithm will diverge even with a very fine spatial mesh if the stability and convergence conditions of the iterative scheme for solving the nonlinear discretized system are too strong. There are many works devoted to this topic for the homogeneous Navier–Stokes equations [14–23]. But how does this idea extend to the non-homogeneous problem? In this paper, after proving the stability and error estimates of the finite element approximation for the non-homogeneous Navier–Stokes equations, we investigate the iterative scheme for solving the generated nonlinear discretization system. Besides the stability and convergence conditions, the complete error estimates are also given. Moreover, we analyze a new kind of stop criterion and some new results are got. The remainder of this paper is organized as follows: in Section 2, we introduce the functional setting which will be frequently used. Then, based on the stability and error estimates of the finite element method, we consider the iterative schemes for solving the nonlinear discretization system in Section 3. In Section 4, we investigate the stability and convergence for these iterative methods. Section 5 reports numerical simulations which confirm the theoretical predictions presented in Section 4. Finally, conclusions are provided in Section 6. 2. Functional setting We will introduce some functional setting and theorems in this section which will be frequently used in the following. First, we introduce the Hilbert spaces: X = H01 (Ω )2 ,
Y = L2 ( Ω ) 2 ,
M = L20 (Ω ) =
q ∈ L2 (Ω );
Ω
qdx = 0 .
Let ∥ · ∥i denote the norm of the Sobolev space H (Ω ) or H (Ω ) for i = 1, 2, (·, ·) and | · | be the inner product and norm of L2 (Ω ) or L2 (Ω )2 , respectively. The spaces H01 (Ω ) and X are equipped with their usual scalar product and norm i
((u, v)) = (∇ u, ∇v),
i
2
∥u∥ = ((u, u))1/2 .
Moreover, define the closed subset V of X by V = {v ∈ X ; div v = 0}, set the closed subset H of Y by H = {v ∈ Y ; div v = 0, v · n|∂ Ω = 0}, and denote the Stokes operator by A = −P ∆, where P is the L2 -orthogonal projection of Y onto H. The assumptions on the domain Ω are as follows (see [1,3,16]): (A1). Assume that Ω is sufficiently smooth and f ∈ L2 (Ω )2 , so that the unique solution (v, q) ∈ (X , M ) of the steady Stokes problem
−1v + ∇ q = f ,
div v = 0 in Ω , v|∂ Ω = 0
exists and satisfies
∥v∥2 + ∥q∥1 ≤ c |f |, where c is a general positive constant. Based on the assumption (A1), it follows that
∥v∥22 ≤ c |Av|2 ,
v ∈ D(A) = H 2 (Ω )2 ∩ X ,
γ0 |v|2 ≤ ∥v∥2 ,
∀v ∈ X ,
γ0 ∥v∥2 ≤ |Av|2 ,
(3)
v ∈ D(A),
where γ0 is a positive constant depending on Ω . Define the continuous bilinear forms a(·, ·) on X × X and d(·, ·) on X × M, respectively, by a(u, v) = ((u, v)) ∀u, v ∈ X ,
d(v, q) = (div v, q) ∀v ∈ X , q ∈ M ,
and the trilinear form b(·, ·, ·) on X × X × X by b(u, v, w) = ((u · ∇)v, w) +
=
1 2
1 2
((div u)v, w) 1
((u · ∇)v, w) − ((u · ∇)w, v) ∀u, v, w ∈ X . 2
(4)
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It is classical that the bilinear form a(·, ·) is continuous and coercive on X × X ; and d(·, ·) is continuous on X × M and satisfies the inf–sup condition, i.e., there exists a positive constant β > 0, such that (see [1,13,24]) sup
v∈X ,v̸=0
|d(u, p)| ≥ β|q| ∀q ∈ M . ∥v∥
(5)
For the trilinear form b(·, ·, ·), we have (see [1,13,25]) b(u, v, w) = −b(u, w, v) ∀u, v, w ∈ X , |b(u, v, w)| ≤ N ∥u∥ ∥v∥ ∥w∥ ∀u, v, w ∈ X ,
|b(u, v, w)| ≤ c0 ∥u∥ ∥v∥ |b(u, v, w)| ≤ c0 ∥u∥
1/2
1/2
|Av|
1/2
|Au|
1/2
(6) (7)
|w| ∀u, v ∈ X , w ∈ Y ,
(8)
∥v∥ |w| ∀u, v ∈ X , w ∈ Y .
(9)
Let ψ be any vector function belonging to H 1 (Ω ) such that div ψ = 0,
ψ = g on ∂ Ω .
(10)
Setting u˜ = u − ψ,
(11)
it follows that u˜ |∂ Ω = 0. With the above notations, the variational formulation of the Navier–Stokes equations (1)–(2) is: search (˜u, p) ∈ (X , M ), such that
ν a(˜u, v) − d(v, p) + d(˜u, q) + b(˜u, u˜ , v) + b(ψ, u˜ , v) + b(˜u, ψ, v) = (f˜ , v),
(12)
for all (v, q) ∈ (X , M ), where f˜ = f + ν 1ψ − (ψ · ∇)ψ . Lemma 2.1 ([13]). For any ϵ > 0, there exists some ψ(ϵ) satisfying (10) and
∥ψ∥ ≤ ϵ.
(13)
For the problem (12), we have the following existence, uniqueness and stability results: Theorem 2.1. Assume f ∈ H −1 (Ω ), g ∈ H 1/2 (Ω ) such that ∂ Ω g · nd∂ Ω = 0 with n being unit out normal, if the assumption (A1) holds, and ν and f satisfy the following inequality
N ∥f˜ ∥−1
(ν − N ∥ψ∥)2
< 1,
(14)
then there exists a unique solution of the variational problem (12), which satisfies
∥f˜ ∥−1 , ν − N ∥ψ∥ |Au˜ | + ∥p∥1 ≤ c |f˜ |. ∥˜u∥ ≤
(15) (16)
Remark 1. The result (13) in Lemma 2.1 indicates that ∥ψ∥ can be taken as any small value if necessary, and this function can be constructed by setting it to have a bubble near the boundary but decay fast in the domain. Thus, ν − N ∥ψ∥ > 0 in Theorem 2.1 can be always guaranteed by choosing a suitable ψ . Proof. The existence and uniqueness of the solution of the problem (12) have been proved, the reader is referred to Chapter II of [13] for the details. Next, we consider the stability results. Taking (v, q) = (˜u, p) in (12), we obtain
ν∥˜u∥2 + b(˜u, ψ, u˜ ) = (f˜ , u˜ ). Thanks to
|b(˜u, ψ, u˜ )| ≤ N ∥ψ∥ ∥˜u∥2 , |(f˜ , u˜ )| ≤ ∥f˜ ∥−1 ∥˜u∥, it yields
(ν − N ∥ψ∥)∥˜u∥ ≤ ∥f˜ ∥−1 ,
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which suggests
∥˜u∥ ≤
∥f˜ ∥−1 . ν − N ∥ψ∥
Furthermore, setting (v, q) = (Au˜ , 0) in (12), it holds
ν|Au˜ |2 + b(˜u, u˜ n , Au˜ ) + b(ψ, u˜ , Au˜ ) + b(˜u, ψ, Au˜ ) = (f˜ , Au˜ ).
(17)
Using (8)–(9), we have
|b(˜u, u˜ , Au˜ )| ≤ c0 ∥˜u∥3/2 |Au˜ |3/2 , |b(ψ, u˜ , Au˜ ) + b(˜u, ψ, Au˜ )| ≤ 2c0 ∥ψ∥ ∥˜u∥1/2 |Au˜ |3/2 , which combining with (17) yields
ν|Au˜ | ≤ |f˜ | + c0 ∥˜u∥3/2 |Au˜ |1/2 + 2c0 ∥ψ∥ ∥˜u∥1/2 |Au˜ |1/2 ≤
ν 2
|Au˜ | + |f˜ | +
c02
ν
∥˜u∥3 +
4c02
ν
∥ψ∥2 ∥˜u∥.
Applying (15) and Lemma 2.1 in the above inequality, we can easily derive
|Au˜ | ≤ c |f˜ |. Finally, using (8)–(9), (12), Lemma 2.1 and the above inequality, we obtain
∥p∥1 ≤ ν|Au˜ | + c0 ∥˜u∥3/2 |Au˜ |1/2 + 2c0 ∥˜u∥1/2 |Au˜ |1/2 ∥ψ∥ + |f˜ | ≤ c |f˜ |. The proof is completed. 3. Iterative schemes based on finite element approximation Supposing 0 < h < 1 is the mesh size, we introduce the finite dimensional subspaces (Xh , Mh ) ⊂ (X , M ) for the velocity and pressure, respectively. The partitioning of Ω is assumed to be uniformly regular in the usual sense. the reader is referred to [26] for further details of the finite element discretization. Let the L2 -orthogonal projection operator Ph : Y → Xh be defined as:
(Ph v, vh ) = (v, vh ) ∀v ∈ Y , vh ∈ Xh , and the discrete analogue Ah = −Ph ∆h of the Stokes operator A be 1/2
1/2
(−∆h uh , vh ) = (Ah uh , Ah vh ) = ((uh , vh )) ∀uh , vh ∈ Xh . Similar to (3)–(4), it holds
γ0 |vh |2 ≤ ∥vh ∥2 ,
γ0 ∥vh ∥2 ≤ |Ah vh |2 ,
∀vh ∈ Vh .
(18)
We also make further assumptions for the finite element spaces (Xh , Mh ) (see [1,3,13,21]): (A2). For each v ∈ (H 2 (Ω ))2 ∩ V and q ∈ H 1 (Ω ) ∩ M, there exist approximations πh v ∈ Xh and ρh q ∈ Mh , such that
∥v − πh v∥ ≤ ch∥v∥2 ,
|q − ρh q| ≤ ch∥q∥1 ,
(19)
together with the inverse inequality
∥vh ∥ ≤ ch−1 |vh |,
v h ∈ Xh ,
(20)
and the so-called inf–sup inequality, i.e., for each qh ∈ Mh , there exists vh ∈ Xh , vh ̸= 0, such that sup
vh ∈Xh ,vh ̸=0
d(vh , qh )
∥vh ∥
≥ βh |qh |,
(21)
where βh is a positive constant depending on Ω . Moreover, let (v, q) ∈ (X , M ) be given, we define the Galerkin projection (Rh (v, q), Qh (v, q)) ∈ (Xh , Mh ) by a(Rh , vh ) − d(vh , Qh ) + d(Rh , qh ) = a(v, vh ) − d(vh , q) + d(v, qh ),
(22)
∀(vh , qh ) ∈ (Xh , Mh ), the following properties are classical for ∀(v, q) ∈ (X , M ) |Rh − v| + h∥Rh − v∥ ≤ ch(∥v∥ + |q|), |Rh − v| + h(∥Rh − v∥ + |Qh − q|) ≤ ch (|Av| + ∥q∥1 ). 2
(23) (24)
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Let (˜uh , ph ) denote the finite element solution of the non-homogeneous Navier–Stokes equations. The finite element approximation of the Navier–Stokes equations is to find (˜uh , ph ) ∈ (Xh , Mh ), such that
ν a(˜uh , vh ) − d(vh , ph ) + d(˜uh , qh ) + b(˜uh , u˜ h , vh ) + b(ψ, u˜ h , vh ) + b(˜uh , ψ, vh ) = (f˜ , vh ),
(25)
for all (vh , qh ) ∈ (Xh , Mh ). Before considering the iterative schemes, we derive first some regularities for the finite element solution of the nonhomogeneous Navier–Stokes equations (25). Theorem 3.1. Under the assumptions of Theorem 2.1, if (A2) and the uniqueness condition (14) are valid, then there exists a unique solution for the finite element approximation problem (25) which satisfies the following regularities:
∥˜uh ∥ ≤
∥f˜ ∥−1 , ν − N ∥ψ∥
|Ah u˜ h | + ∥p∥1 ≤ c |f˜ |.
(26)
Proof. The proof of Theorem 3.1 is similar to that of Theorem 2.1, which is omitted here. Theorem 3.2. Under the assumptions of Theorem 2.1, if (A2) and the uniqueness condition (14) are valid, then, the finite element solution of (25) satisfies the following error estimates
|˜u − u˜ h | ≤ ch2 , ∥˜u − u˜ h ∥ + |p − ph | ≤ ch.
(27) (28)
Proof. Applying the Galerkin projection in (12), we have for ∀(vh , qh ) ∈ (Xh , Mh )
ν a(Rh , vh ) − d(vh , Qh ) + d(ν Rh , qh ) + b(˜u, u˜ , vh ) + b(ψ, u˜ , vh ) + b(˜u, ψ, vh ) = (f˜ , vh ),
(29)
where (Rh , Qh ) = (Rh (ν u˜ , p), Q (ν u˜ , p)). Subtracting (25) from (29) and setting (˜eh , ηh ) = (Rh − u˜ h , Qh − ph ), we get
ν a(˜eh , vh ) − d(vh , ηh ) + d(ν e˜ h , qh ) + b(˜u − u˜ h , u˜ , vh ) + b(˜uh , u˜ − u˜ h , vh ) + b(ψ, u˜ − u˜ h , vh ) + b(˜u − u˜ h , ψ, vh ) = 0, Taking (vh , qh ) = (˜eh , ν
−1
∀(vh , qh ) ∈ (Xh , Mh ).
(30)
ηh ) in (30) and using (6), it yields
ν∥˜eh ∥ + b(˜eh , u˜ , e˜ h ) + b(˜u − Rh , u˜ , e˜ h ) + b(˜uh , u˜ − Rh , e˜ h ) 2
+ b(ψ, u˜ − Rh , e˜ h ) + b(Rh − u˜ h , ψ, e˜ h ) + b(˜eh , ψ, e˜ h ) = 0.
(31)
Due to
|b(˜eh , u˜ , e˜ h )| ≤ N ∥˜u∥ ∥˜eh ∥2 , |b(˜u − Rh , u˜ , e˜ h ) + b(˜uh , u˜ − Rh , e˜ h )| ≤ c0 (|Au˜ | + |Ah u˜ h |)|˜u − Rh | ∥˜eh ∥, |b(ψ, u˜ − Rh , e˜ h ) + b(Rh − u˜ h , ψ, e˜ h )| ≤ 2c0 |Aψ| |˜u − Rh | ∥˜eh ∥, |b(˜eh , ψ, e˜ h )| ≤ N ∥ψ∥ ∥˜eh ∥2 , combining these inequalities with (31), and using Theorems 2.1 and 3.1, it follows
(ν − N ∥ψ∥) 1 −
N ∥f˜ ∥−1
(ν − N ∥ψ∥)2
∥˜eh ∥ ≤ ch2 .
(32)
Applying (14), (23)–(24) and the triangle inequality, we can easily get (27). Using (21) and (30), we obtain
|ηh | ≤ c (ν∥˜eh ∥ + N (∥˜u∥ + ∥˜uh ∥)∥˜u − u˜ h ∥ + c0 ∥ψ∥ ∥˜u − u˜ h ∥),
(33)
considering (27), (23)–(24), Theorems 2.1 and 3.1, and the triangle inequality, we derive (28). The proof is completed. Since (25) is a discrete nonlinear system, the solution is usually sought by an iterative scheme. For the Navier–Stokes equations with the homogeneous boundary condition, the authors investigated the iterative schemes in [16]. Here, by extending the idea to the non-homogeneous problem, we consider the following three iterative methods: find (˜unh , pnh ) ∈ (Xh , Mh ), such that Oseen scheme:
ν a(˜unh , vh ) − d(vh , pnh ) + d(˜unh , qh ) + b(˜unh−1 , u˜ nh , vh ) + b(ψ, u˜ nh , vh ) + b(˜unh , ψ, vh ) = (f˜ , vh ),
(34)
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Newton scheme:
ν a(˜unh , vh ) − d(vh , pnh ) + d(˜unh , qh ) + b(˜unh−1 , u˜ nh , vh ) + b(˜unh , u˜ hn−1 , vh ) + b(ψ, u˜ nh , vh ) + b(˜unh , ψ, vh ) = b(˜unh−1 , u˜ hn−1 , vh ) + (f˜ , vh ),
(35)
Stokes scheme:
ν a(˜unh , vh ) − d(vh , pnh ) + d(˜unh , qh ) + b(˜unh−1 , u˜ nh−1 , vh ) + b(ψ, u˜ nh , vh ) + b(˜unh , ψ, vh ) = (f˜ , vh ),
(36)
with (u0h , p0h ) ∈ (Xh , Mh ) being the solution of the discrete Stokes problem:
ν a(˜u0h , vh ) − d(vh , p0h ) + d(˜u0h , qh ) = (fˆ , vh ),
(37)
for all (vh , qh ) ∈ (Xh , Mh ), where fˆ = f + ν 1ψ . 4. Stability and convergence analysis In this section, we will analyze the stability and convergence of the iterative schemes (34)–(36) for the non-homogeneous Navier–Stokes equations. 4.1. Stability analysis In this subsection, we will provide the stabilities of the schemes. By applying the similar process as the proof of Theorem 2.1 and that in [16,23], we can derive easily the following theorems, and the detail is omitted here. Lemma 4.1. Under the assumptions of Theorem 2.1, suppose that (A1) and (A2) are valid, if the uniqueness condition (14) holds, then the solution (˜unh , pnh ) ∈ (Xh , Mh ) generated by the Oseen scheme (34) satisfies
∥˜unh ∥ ≤
∥f˜ ∥−1 , ν − N ∥ψ∥
|Ah u˜ nh | ≤ c |f˜ |,
∀n > 0.
(38)
Lemma 4.2. Under the assumptions of Theorem 2.1, suppose that (A1) and (A2) are valid, if it holds the inequality N ∥f˜ ∥−1
(ν − N ∥ψ∥)2
<
1 3
,
(39)
then the solution (˜unh , pnh ) ∈ (Xh , Mh ) generated by the Newton scheme (35) satisfies
∥˜unh ∥ ≤
4∥f˜ ∥−1 3(ν − N ∥ψ∥)
,
|Ah u˜ nh | ≤ c |f˜ |,
∀n > 0.
(40)
Lemma 4.3. Under the assumptions of Theorem 2.1, suppose that assumptions (A1) and (A2) are valid, if it holds the inequality N ∥f˜ ∥−1
(ν − N ∥ψ∥)
2
<
1 4
,
(41)
then the solution (˜unh , pnh ) ∈ (Xh , Mh ) generated by the Stokes scheme (36) satisfies
∥˜unh ∥ ≤
2∥f˜ ∥−1
ν − N ∥ψ∥
,
|Ah u˜ nh | ≤ c |f˜ |,
∀n > 0.
(42)
4.2. Error estimates Next, we will consider the convergence of the iterative schemes proposed in Section 3. Setting (˜en , ηn ) = (˜uh −˜unh , ph −pnh ), and subtracting (34)–(36) from (25), respectively, we have the following error equations: find (˜en , ηn ) ∈ (Xh , Mh ), such that for all (vh , qh ) ∈ (Xh , Mh ), it satisfies respectively, Oseen scheme:
ν a(˜en , vh ) − d(vh , ηn ) + d(˜en , qh ) + b(˜en−1 , u˜ h , vh ) + b(˜unh−1 , e˜ n , vh ) + b(ψ, e˜ n , vh ) + b(˜en , ψ, vh ) = 0,
(43)
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Newton scheme:
ν a(˜en , vh ) − d(vh , ηn ) + d(˜en , qh ) + b(˜en , u˜ nh−1 , vh ) + b(˜uhn−1 , e˜ n , vh ) + b(ψ, e˜ n , vh ) + b(˜en , ψ, vh ) = −b(˜en−1 , e˜ n−1 , vh ),
(44)
Stokes scheme:
ν a(˜en , vh ) − d(vh , ηn ) + d(˜en , qh ) + b(˜en−1 , u˜ h , vh ) + b(˜unh−1 , e˜ n−1 , vh ) + b(ψ, e˜ n , vh ) + b(˜en , ψ, vh ) = 0.
(45)
Lemma 4.4. Under the assumptions of Lemma 4.1, we have the following estimates for the error (˜en , ηn ) related to the Oseen scheme for all n > 0
∥˜en ∥ ≤ c |ηn | ≤ c
N ∥f˜ ∥−1
n
(ν − N ∥ψ∥) N ∥f˜ ∥ n −1 2
(ν − N ∥ψ∥)2
,
(46)
.
(47)
Proof. Taking (vh , ph ) = (˜en , ηn ) in (43) and using (6), we get
ν∥˜en ∥2 + b(˜en−1 , u˜ h , e˜ n ) + b(˜en , ψ, e˜ n ) = 0. Due to
|b(˜en−1 , u˜ h , e˜ n )| ≤ N ∥˜en−1 ∥ ∥˜uh ∥ ∥˜en ∥, |b(˜en , ψ, e˜ n )| ≤ N ∥ψ∥ ∥˜en ∥2 , it follows
(ν − N ∥ψ∥)∥˜en ∥ ≤ N ∥˜uh ∥ ∥˜en−1 ∥ ≤
N ∥f˜ ∥−1
ν − N ∥ψ∥
∥˜en−1 ∥,
∀n > 0.
(48)
Subtracting (37) from (25), taking (vh , ph ) = (˜e0 , η0 ) and using Lemma 2.1 and Theorem 3.1, it holds
ν∥˜e0 ∥ ≤ N ∥uh ∥2 ≤ N (∥˜uh ∥ + ∥ψ∥)2 ≤ c . Therefore, (48) yields
∥˜en ∥ ≤
N ∥f˜ ∥ N ∥f˜ ∥ n n −1 −1 0 n−1 ∥˜ e ∥ ≤ c , ∥˜ e ∥ ≤ (ν − N ∥ψ∥)2 (ν − N ∥ψ∥)2 (ν − N ∥ψ∥)2 N ∥f˜ ∥−1
∀n > 0,
which suggests (46). Moreover, from (43) we can get by using Theorem 3.1, Lemma 4.1, (6) and the above estimate
|ηn | ≤ ν∥˜en ∥ + N ∥˜en−1 ∥ ∥˜uh ∥ + N ∥˜unh−1 ∥ ∥˜en ∥ + 2N ∥ψ∥ ∥˜en ∥ N ∥f˜ ∥−1 N ∥f˜ ∥−1 ≤ ∥˜en−1 ∥ + ν + + 2N ∥ψ∥ ∥˜en ∥ ν − N ∥ψ∥ ν − N ∥ψ∥ N ∥f˜ ∥ n −1 ≤c , (ν − N ∥ψ∥)2 which suggests (47). The proof is completed. Applying the similar process as that in the above theorem and in [16,23], we can easily derive the following results. Lemma 4.5. Under the assumptions of Lemma 4.2, we have the following estimates for the error (˜en , ηn ) related to the Newton scheme for all n > 0
∥˜en ∥ ≤ c
|ηn | ≤ c
9N ∥f˜ ∥−1
2n −1
5(ν − N ∥ψ∥)2 9N ∥f˜ ∥−1 5(ν − N ∥ψ∥)2
2n −1
,
(49)
.
(50)
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Lemma 4.6. Under the assumptions of Lemma 4.3, we have the following estimates for the error (˜en , ηn ) related to the Stokes scheme for all n > 0
∥˜en ∥ ≤ c |ηn | ≤ c
3N ∥f˜ ∥−1
n
(ν − N ∥ψ∥) n 3N ∥f˜ ∥ −1 2
(ν − N ∥ψ∥)2
,
(51)
.
(52)
Now, we arrive at the main results: Theorem 4.1. Under the assumptions of Lemmas 4.1–4.3, for all n > 0, we have
n ∥f˜ ∥−1 , (ν − N ∥ψ∥)2 n ∥f˜ ∥−1 ∥u − unh ∥ + |p − pnh | ≤ ch + c , (ν − N ∥ψ∥)2 |u − unh | ≤ ch2 + c
(53) (54)
for the Oseen scheme, and
|u − unh | ≤ ch2 + c ∥u − unh ∥ + |p − pnh | ≤ ch + c
9∥f˜ ∥−1
2n −1
5(ν − N ∥ψ∥)2 9∥f˜ ∥−1
2n −1
5(ν − N ∥ψ∥)2
,
(55)
,
(56)
for the Newton scheme, and
|u − unh | ≤ ch2 + c ∥u − unh ∥ + |p − pnh | ≤ ch + c
3∥f˜ ∥−1
n
(ν − N ∥ψ∥) n 3∥f˜ ∥ −1 2
(ν − N ∥ψ∥)2
,
(57)
,
(58)
for the Stokes iterative scheme, respectively, where f˜ = f + ν 1ψ − (ψ · ∇)ψ . Proof. Since u − unh = (˜u + ψ) − (˜unh + ψ) = u˜ − u˜ nh , the above theorem is obviously from Lemmas 4.1–4.6. In Theorem 4.1, the complete error estimates are derived for solving the non-homogeneous Navier–Stokes equations with the finite element method based on three iterative schemes. We can see that the errors consisted of two parts: spatial discretization and solving the discretized nonlinear system iteratively. Only both of the errors are controlled well, we can get a good approximation. Furthermore, the error generated by solving the discretized nonlinear system iteratively indicates that different schemes converge based on different assumptions of the Reynolds number. 4.3. Stop criterion The relationships between the Reynolds number and the convergence of the iterative schemes are shown in Lemmas 4.4– 4.6, which indicate that the Newton scheme converges fastest among these schemes, but the Oseen scheme has the weakest assumption on the Reynolds number and is the most efficient for solving the high Reynolds number problems. These results also suggest that it is enough to ensure the error generated by the iterative schemes smaller than that generated by the N ∥f˜ ∥
−1 spacial discretization when implementing, which means to let Θ (ν−N ∥ψ∥) 2
N ∥f˜ ∥
−1 as Θ (ν−N ∥ψ∥) 2
=c
N ∥f˜ ∥−1 (ν−N ∥ψ∥)2
2
≤ ch2 hold with Θ (·) being a function, such
for the Oseen scheme. Overall iteration will cost more computational time but without N ∥f˜ ∥
−1 improvement on the approximation results. Although we have given the evaluation factor (ν−N ∥ψ∥) 2 in the theoretical prediction, usually it is not easily computed. Generally, the following strategy is used popularly as a stopping criterion:
|˜unh − u˜ nh−1 | ≤ 10−l , |˜unh | with l being a positive integer. But since the optimal value l is unknown, to ensure the error generated by the iterative schemes smaller than that generated by the spacial discretization, we always take a large l and the overall iteration may appear. Next, we will consider a new kind of stop criterion for the iterative schemes which is firstly proposed in [16].
K. Wang / Computers and Mathematics with Applications (
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Theorem 4.2. Under the assumptions of Lemmas 4.4–4.6, (˜en , ηn ) related to the iterative schemes satisfies, for all n > 0,
∥˜en ∥ + |ηn | ≤ c |˜unh − u˜ nh−1 |
(59)
for the Oseen and Stokes schemes, and
∥˜en ∥ + |ηn | ≤ c ∥˜unh − u˜ nh−1 ∥2
(60)
for the Newton scheme. Furthermore, there holds lim (∥˜unh − u˜ nh−1 ∥ + |˜unh − u˜ nh−1 |) = 0
(61)
n→∞
for the Oseen, Newton and Stokes schemes. Proof. Taking (vh , qh ) = (˜en , ηn ) in (43) and using (6), we have
ν∥˜en ∥ + b(˜en , u˜ h , e˜ n ) + b(˜en−1 − e˜ n , u˜ h , e˜ n ) + b(˜en , ψ, e˜ n ) = 0.
(62)
Thanks to
|b(˜en , u˜ h , e˜ n )| ≤ N ∥˜uh ∥ ∥˜en ∥2 , |b(˜en−1 − e˜ n , u˜ h , e˜ n )| ≤ c0 |˜en−1 − e˜ n | |Ah u˜ h | ∥˜en ∥, |b(˜en , ψ, e˜ n )| ≤ N ∥ψ∥ ∥˜en ∥2 , it follows, by using Theorem 3.1, that
(ν − N ∥ψ∥) 1 −
N ∥f˜ ∥−1
(ν − N ∥ψ∥)2
∥˜en ∥ ≤ c0 |Ah u˜ h | |˜en−1 − e˜ n | ≤ c |˜unh − u˜ nh−1 |,
∀n > 0.
(63)
Using (21), Theorem 3.1 and the above inequality, we get (59) for the Oseen scheme. On the other hand, taking (vh , qh ) = (˜en , ηn ) in (45) and using (6), we have
ν∥˜en ∥ + b(˜en , u˜ h , e˜ n ) + b(˜en−1 − e˜ n , u˜ h , e˜ n ) + b(˜unh−1 , e˜ n−1 − e˜ n , e˜ n ) + b(˜en , ψ, e˜ n ) = 0.
(64)
Similar to (63), we can derive that (59) also holds for the Stokes scheme. Moreover, setting (vh , qh ) = (˜en , ηn ) in (44), it follows
ν∥˜en ∥2 + b(˜en , u˜ h , e˜ n ) + b(˜en , ψ, e˜ n ) + b(˜unh − u˜ nh−1 , u˜ nh − u˜ hn−1 , e˜ n ) = 0.
(65)
Thanks to
|b(˜en , u˜ h , e˜ n )| ≤ N ∥˜en ∥2 , |b(˜en , ψ, e˜ n )| ≤ N ∥ψ∥ ∥˜en ∥2 , |b(˜unh − u˜ hn−1 , u˜ nh − u˜ nh−1 , e˜ n )| ≤ N ∥˜unh − u˜ nh−1 ∥2 ∥˜en ∥, we get
(ν − N ∥ψ∥) 1 −
N ∥f˜ ∥−1
(ν − N ∥ψ∥)
2
∥˜en ∥ ≤ N ∥˜unh − u˜ hn−1 ∥2 .
(66)
Applying the similar process as that in (59), we deduce (60). Finally, to prove (61), using (18) and the triangle inequality, we have −1/2
∥˜unh − u˜ nh−1 ∥ + |˜unh − u˜ nh−1 | ≤ (1 + γ0
−1/2
)∥˜unh − u˜ nh−1 ∥ ≤ (1 + γ0
)(∥˜en ∥ + ∥˜en−1 ∥).
Under the assumptions of Lemmas 4.4–4.6, respectively, there holds for the Oseen, Newton and Stokes schemes lim ∥˜en ∥ = 0,
n→∞
which indicates (61). The proof is completed. Remark 2. Theorem 4.2 indicates that the iterative errors generated by three schemes can be evaluated by the iterative N ∥f˜ ∥
−1 solution instead of the factor (ν−N ∥ψ∥) 2 , which is easily realized when implementing. Based on this results, we can use the new stop criterion
|˜unh − u˜ hn−1 | ≤ ch2
10
K. Wang / Computers and Mathematics with Applications (
(a) u1 at the line (0.5, y).
)
–
(b) u2 at the line (x, 0.5). Fig. 1. Crosscut with different stop criteria for the Oseen scheme (Re = 200). Table 1 Iteration numbers of the Oseen scheme with different stop criteria (P2 − P1 elements, h = 1/40). Re
10
50
100
200
Criterion 1 Criterion 2
5 2
8 4
10 5
13 6
for the Oseen and Stokes schemes, and
∥˜unh − u˜ hn−1 ∥ ≤ ch2 for the Newton scheme. Moreover, the new stop criteria of three iterative schemes will converge under the same assumptions of Lemmas 4.4–4.6, respectively. Remark 3. Especially, if g = 0, the problem (1)–(2) becomes the Navier–Stokes with homogeneous boundary. Therefore, the results in Theorem 4.2 also hold for this kind of problem, which is investigated in [16,23]. It is obvious that the results here are better than that in [16], some improvements are achieved. 5. Numerical experiments To confirm the theoretical results proved above, we report some computational results in this section. As a classical benchmark, the lid-driven cavity flow is also a non-homogeneous problem (see [27,28]). The following numerical simulations are all based on this model. First, we consider the standard and new stopping strategies: Criterion 1:
∥˜unh −˜unh−1 ∥ ≤ ∥˜unh ∥ n −1 n
10−6 ,
Criterion 2: |˜uh − u˜ h | ≤ h2 for the Oseen and Stokes schemes and ∥˜unh − u˜ hn−1 ∥ ≤ h2 for the Newton scheme. The results got by two kinds of stop criteria for different iterative schemes are collected in Tables 1–3. We can see that the iteration numbers demanded by applying the new stopping strategies are uniform less than that needed by using the standard one for three schemes. Furthermore, Figs. 1–3 suggest that these two stop criteria have almost the same computational accuracy. This means that the overall iteration is avoided by applying the new stop criterion, which confirms Theorem 4.2 very well. Furthermore, from Theorem 4.1 we can see that the new stop criterion can control the iterative number and avoid the overall iteration, but get almost the same computational accuracy in L2 - and H 1 -norm. Then, we verify the convergence of three iterative schemes. From the analysis above, it is obvious that the Oseen scheme has the weakest convergence condition, and the Newton scheme converges fastest among these methods. Setting h = 1/40, we show the results in Table 4 with the Reynolds number in the range of 50–1000. We can see that the computational results are consistent with the theoretical predictions. Finally, the accuracy of the Oseen scheme applying to some high Reynolds number problems is investigated by comparing with a well-known method in [27] in which a very fine mesh size is used (see Figs. 4–5). The numerical result says by itself. All of these results confirm that theoretical analysis in this paper is right and can provide good prediction.
K. Wang / Computers and Mathematics with Applications (
(a) u1 at the line (0.5, y).
)
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(b) u2 at the line (x, 0.5). Fig. 2. Crosscut with different stop criteria for the Newton scheme (Re = 100).
(a) u1 at the line (0.5, y).
(b) u2 at the line (x, 0.5). Fig. 3. Crosscut with different stop criteria for the Stokes scheme (Re = 50). Table 2 Iteration numbers of the Newton scheme with different stop criteria (P2 − P1 elements, h = 1/40). Re
10
50
80
100
Criterion 1 Criterion 2
3 2
4 2
4 3
4 3
Table 3 Iteration numbers of the Stokes scheme with different stop criteria (P2 − P1 elements, h = 1/40). Re
10
20
30
50
Criterion 1 Criterion 2
5 2
7 3
9 4
15 6
6. Conclusions Three iterative schemes for solving the nonlinear discrete system generated from the non-homogeneous Navier–Stokes equations are investigated in this paper. Including the stability and error estimates for the finite element method, we analyze
12
K. Wang / Computers and Mathematics with Applications (
(a) u1 at the line (0.5, y).
)
–
(b) u2 at the line (x, 0.5). Fig. 4. Crosscut for different schemes with Re = 1000 and h = 1/40.
(a) u1 at the line (0.5, y).
(b) u2 at the line (x, 0.5). Fig. 5. Crosscut for different schemes with Re = 2500 and h = 1/40. Table 4 Iteration numbers of different schemes (P2 − P1 elements, h = 1/40). Re
50
100
200
500
1000
Oseen scheme Newton scheme Stokes scheme
4 2 6
5 3 D
6 4 D
7 6 D
9 Da D
a
D indicates divergence or the maximum number of iteration M > 1000.
the stability and convergence conditions for the iterative schemes and show the complete error estimates which consist of two parts: spatial discretization and solving the discretized nonlinear system iteratively. These theoretical results suggest that only both of the errors are controlled well, we can get a good approximation. Furthermore, the relationship between the iteration solutions and the stop criteria is also analyzed for different methods, which reflect the character of the iterative schemes exactly. Applying the technique in [29–31], the time-dependent non-homogeneous problem will be considered in future. References [1] V. Girault, P. Raviart, Finite Element Method for Navier–Stokes Equations: Theory and Algorithms, Springer, Berlin, 1986. [2] J. Guermond, A. Marra, L. Quartapelle, Subgrid stabilized projection method for 2D unsteady flows at high Reynolds numbers, Comput. Methods Appl. Mech. Engrg. 195 (2006) 5857–5876.
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[3] Y. He, A. Wang, A simplified two-level method for the steady Navier–Stokes equations, Comput. Methods Appl. Mech. Engrg. 197 (2008) 1568–1576. [4] R. Ingram, Unconditional convergence of high-order extrapolations of the Crank–Nicolson, finite element method for the Navier–Stokes equations, Int. J. Numer. Anal. Model. 10 (2013) 257–297. [5] J. Jiao, G. Guo, Mixed spectral method for Navier–Stokes equations in an infinite strip by using generalized Lagrange functions, Int. J. Numer. Anal. Model. 9 (2012) 982–989. [6] S. Kaya, W. Layton, B. Riviére, Subgrid stabilized defect correction methods for the Navier–Stokes equations, SIAM J. Numer. Anal. 44 (2006) 1639–1654. [7] S. Kaya, B. Riviére, A two-grid stabilization method for solving the steady-state Navier–Stokes equations, Numer. Methods Partial Differential Equations 22 (2005) 728–743. [8] W. Layton, Solution algorithm for incompressible viscous flows at high Reynolds number, Vestnik Moskov. Gos. Univ. Ser. 15 (1996) 25–35. [9] W. Layton, H. Lee, J. Peterson, A defect-correction method for the incompressible Navier–Stokes equations, Appl. Math. Comput. 129 (2002) 1–19. [10] J. Li, J. Wu, Z. Chen, A. Wang, Superconvergence of stabilized low order finite volume approximation for the three-dimensional stationary Navier–Stokes equations, Int. J. Numer. Anal. Model. 9 (2012) 419–431. [11] J. Pyo, Error estimates for the second order semi-discrete stabilized gauge-uzawa method for the Navier–Stokes equations, Int. J. Numer. Anal. Model. 10 (2013) 24–41. [12] P. Sagaut, Large Eddy Simulation for Incompressible Flows, second ed., Springer, Berlin, Heidelberg, New York, 2003. [13] R. Temam, Navier–Stokes Equations, Theory and Numerical Analysis, North-Holland, Amsterdam, 1984. [14] K. Wang, A new defect correction method for the Navier–Stokes equations at high Reynolds numbers, Appl. Math. Comput. 216 (2010) 3252–3264. [15] Y. He, Euler implicit/explicit iterative scheme for the stationary Navier–Stokes equations, Numer. Math. 123 (2013) 67–96. [16] Y. He, J. Li, Convergence of three iterative methods based on the finite element discretization for the stationary Navier–Stokes equations, Comput. Methods Appl. Mech. Engrg. 198 (2009) 1351–1359. [17] Y. He, J. Li, Numerical implementation of the Crank–Nicolson/Adams–Bashforth scheme for the time-dependent Navier–Stokes equations, Internat. J. Numer. Methods Fluids 62 (2010) 647–659. [18] Y. He, J. Li, Numerical comparisons of time–space iterative method and spatial iterative methods for the stationary Navier–Stokes equations, J. Comput. Phys. 231 (2012) 6790–6800. [19] P. Huang, X. Feng, H. Su, Two-level defect-correction locally stabilized finite element method for the steady Navier–Stokes equations, Nonlinear Anal. RWA 14 (2013) 1171–1181. [20] Y. Shang, A two-level subgrid stabilized Oseen iterative method for the steady Navier–Stokes equations, J. Comput. Phys. 233 (2013) 210–226. [21] Y. Shang, Y. He, A parallel Oseen-linearized algorithm for the stationary Navier–Stokes equations, Comput. Methods Appl. Mech. Engrg. 209–212 (2012) 172–183. [22] K. Wang, Y. Wong, Error correction method for Navier–Stokes equations at high Reynolds numbers, J. Comput. Phys. 255 (2013) 245–265. [23] H. Xu, Y. He, Some iterative finite element methods for steady Navier–Stokes equations with different viscosities, J. Comput. Phys. 232 (2013) 136–152. [24] J. Heywood, R. Rannacher, Finite element approximation of the nonstationary Navier–Stokes problem. I. Regularity of solutions and second-order error estimates for spatial discretization, SIAM J. Numer. Anal. 19 (1982) 275–310. [25] J. 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. [26] P. Ciarlet, The Finite Element Method for Elliptic Problems, North-Holland, Amsterdam, 1984. [27] E. Erturk, T. Corke, C. Gokcol, Numerical solutions of 2-D steady incompressible driven cavity flow at high Reynolds numbers, Internat. J. Numer. Methods Fluids 48 (2005) 747–774. [28] U. Ghia, K. Ghia, C. Shin, High-Re solutions for incompressible flow using the Navier–Stokes equations and a multigrid method, J. Comput. Phys. 48 (1982) 387–411. [29] Y. He, Unconditional convergence of the Euler semi-implicit scheme for the three-dimensional incompressible MHD equations, IMA J. Numer. Anal. 35 (2015) 767–801. [30] Y. He, W. Sun, Stability and convergence of the Crank–Nicolson/Adams–Bashforth scheme for the time-dependent Navier–Stokes equations, SIAM J. Numer. Anal. 45 (2007) 837–869. [31] G. Zhang, Y. He, Decoupled schemes for unsteady MHD equations II: Finite element spatial discretization and numerical implementation, Comput. Math. Appl. 69 (2015) 1390–1406.
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Computers and Mathematics with Applications journal homepage: www.elsevier.com/locate/camwa
Iterative schemes for the non-homogeneous Navier–Stokes equations based on the finite element approximation✩ Kun Wang ∗ College of Mathematics and Statistics, Chongqing University, Chongqing 401331, PR China Institute of Computing and Data Sciences, Chongqing University, Chongqing 400044, PR China
article
abstract
info
Article history: Received 20 June 2015 Received in revised form 8 October 2015 Accepted 9 November 2015 Available online xxxx
In this paper, we consider the stability and convergence of three iterative schemes for the non-homogeneous steady Navier–Stokes equations. As a nonlinear problem, we will get a nonlinear discrete system if approximating the non-homogeneous Navier–Stokes equations. After proving the stability and error estimates of the finite element method for the non-homogeneous Navier–Stokes equations, three iterative schemes are investigated for solving the resulted nonlinear discrete system. The stability and convergence conditions for these iterative methods are also analyzed, respectively. Furthermore, new results for the stop criterion are proved. Finally, we show some numerical experiments to illustrate the theoretical prediction. © 2015 Elsevier Ltd. All rights reserved.
Keywords: Navier–Stokes equations Non-homogeneous boundary condition Stability and convergence analysis Iterative scheme Finite element method
1. Introduction Let us consider the stationary Navier–Stokes equations in two dimension
−ν 1u + (u · ∇)u + ∇ p = f ,
div u = 0 in Ω ,
(1)
u|∂ Ω = g ,
(2)
where u = u(x) = (u1 (x), u2 (x)) is the velocity, p = p(x) is the pressure, f = f (x) = (f1 (x), f2 (x)) is the imposed body force, ν > 0 is the viscosity and Ω is a bounded domain in R2 which has a Lipschitz continuous boundary ∂ Ω and satisfies the additional condition stated in (A1) below, g is a given function. We assume that ν = 1/Re, where Re is the Reynolds number. The non-homogeneous Navier–Stokes equations (1)–(2) can be used to describe many classical models in the practical, such as, the lid-driven cavity flow, the backward step flow, the flow over a circular cylinder and so on. But to the best of our knowledge, the approximation analysis of this problem is rare. Because the problem (1)–(2) is nonlinear, a typical numerical solution procedure will involve two steps: spatial discretization and iterative scheme solving the discretized nonlinear system. Consequently, the stability and error are also determined by these two parts. As a popular discretization method, the finite element approximation has been widely used to solve the homogeneous Navier–Stokes equations. Lots of references have been reported to address the problem related to the spatial discretization [1–14], such as the large eddy simulation (LES) method [12], the defect correction method [9,14] and the
✩ This work is supported by the Natural Science Foundation of China (No. 11201506), and Project No. 106112015CDJXY100007 supported by the Fundamental Research Funds for the Central Universities. ∗ Correspondence to: College of Mathematics and Statistics, Chongqing University, Chongqing 401331, PR China. E-mail address: [email protected].
http://dx.doi.org/10.1016/j.camwa.2015.11.011 0898-1221/© 2015 Elsevier Ltd. All rights reserved.
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K. Wang / Computers and Mathematics with Applications (
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techniques developed recently based on the variational multiscale method [2,6] and so on. But there are few works devoted to the finite element analysis for the non-homogeneous Navier–Stokes equations. On the other hand, it is well-known that the Reynolds number plays a very important role in searching the iterative scheme for the nonlinear discretized system from the Navier–Stokes equations. The numerical algorithm will diverge even with a very fine spatial mesh if the stability and convergence conditions of the iterative scheme for solving the nonlinear discretized system are too strong. There are many works devoted to this topic for the homogeneous Navier–Stokes equations [14–23]. But how does this idea extend to the non-homogeneous problem? In this paper, after proving the stability and error estimates of the finite element approximation for the non-homogeneous Navier–Stokes equations, we investigate the iterative scheme for solving the generated nonlinear discretization system. Besides the stability and convergence conditions, the complete error estimates are also given. Moreover, we analyze a new kind of stop criterion and some new results are got. The remainder of this paper is organized as follows: in Section 2, we introduce the functional setting which will be frequently used. Then, based on the stability and error estimates of the finite element method, we consider the iterative schemes for solving the nonlinear discretization system in Section 3. In Section 4, we investigate the stability and convergence for these iterative methods. Section 5 reports numerical simulations which confirm the theoretical predictions presented in Section 4. Finally, conclusions are provided in Section 6. 2. Functional setting We will introduce some functional setting and theorems in this section which will be frequently used in the following. First, we introduce the Hilbert spaces: X = H01 (Ω )2 ,
Y = L2 ( Ω ) 2 ,
M = L20 (Ω ) =
q ∈ L2 (Ω );
Ω
qdx = 0 .
Let ∥ · ∥i denote the norm of the Sobolev space H (Ω ) or H (Ω ) for i = 1, 2, (·, ·) and | · | be the inner product and norm of L2 (Ω ) or L2 (Ω )2 , respectively. The spaces H01 (Ω ) and X are equipped with their usual scalar product and norm i
((u, v)) = (∇ u, ∇v),
i
2
∥u∥ = ((u, u))1/2 .
Moreover, define the closed subset V of X by V = {v ∈ X ; div v = 0}, set the closed subset H of Y by H = {v ∈ Y ; div v = 0, v · n|∂ Ω = 0}, and denote the Stokes operator by A = −P ∆, where P is the L2 -orthogonal projection of Y onto H. The assumptions on the domain Ω are as follows (see [1,3,16]): (A1). Assume that Ω is sufficiently smooth and f ∈ L2 (Ω )2 , so that the unique solution (v, q) ∈ (X , M ) of the steady Stokes problem
−1v + ∇ q = f ,
div v = 0 in Ω , v|∂ Ω = 0
exists and satisfies
∥v∥2 + ∥q∥1 ≤ c |f |, where c is a general positive constant. Based on the assumption (A1), it follows that
∥v∥22 ≤ c |Av|2 ,
v ∈ D(A) = H 2 (Ω )2 ∩ X ,
γ0 |v|2 ≤ ∥v∥2 ,
∀v ∈ X ,
γ0 ∥v∥2 ≤ |Av|2 ,
(3)
v ∈ D(A),
where γ0 is a positive constant depending on Ω . Define the continuous bilinear forms a(·, ·) on X × X and d(·, ·) on X × M, respectively, by a(u, v) = ((u, v)) ∀u, v ∈ X ,
d(v, q) = (div v, q) ∀v ∈ X , q ∈ M ,
and the trilinear form b(·, ·, ·) on X × X × X by b(u, v, w) = ((u · ∇)v, w) +
=
1 2
1 2
((div u)v, w) 1
((u · ∇)v, w) − ((u · ∇)w, v) ∀u, v, w ∈ X . 2
(4)
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It is classical that the bilinear form a(·, ·) is continuous and coercive on X × X ; and d(·, ·) is continuous on X × M and satisfies the inf–sup condition, i.e., there exists a positive constant β > 0, such that (see [1,13,24]) sup
v∈X ,v̸=0
|d(u, p)| ≥ β|q| ∀q ∈ M . ∥v∥
(5)
For the trilinear form b(·, ·, ·), we have (see [1,13,25]) b(u, v, w) = −b(u, w, v) ∀u, v, w ∈ X , |b(u, v, w)| ≤ N ∥u∥ ∥v∥ ∥w∥ ∀u, v, w ∈ X ,
|b(u, v, w)| ≤ c0 ∥u∥ ∥v∥ |b(u, v, w)| ≤ c0 ∥u∥
1/2
1/2
|Av|
1/2
|Au|
1/2
(6) (7)
|w| ∀u, v ∈ X , w ∈ Y ,
(8)
∥v∥ |w| ∀u, v ∈ X , w ∈ Y .
(9)
Let ψ be any vector function belonging to H 1 (Ω ) such that div ψ = 0,
ψ = g on ∂ Ω .
(10)
Setting u˜ = u − ψ,
(11)
it follows that u˜ |∂ Ω = 0. With the above notations, the variational formulation of the Navier–Stokes equations (1)–(2) is: search (˜u, p) ∈ (X , M ), such that
ν a(˜u, v) − d(v, p) + d(˜u, q) + b(˜u, u˜ , v) + b(ψ, u˜ , v) + b(˜u, ψ, v) = (f˜ , v),
(12)
for all (v, q) ∈ (X , M ), where f˜ = f + ν 1ψ − (ψ · ∇)ψ . Lemma 2.1 ([13]). For any ϵ > 0, there exists some ψ(ϵ) satisfying (10) and
∥ψ∥ ≤ ϵ.
(13)
For the problem (12), we have the following existence, uniqueness and stability results: Theorem 2.1. Assume f ∈ H −1 (Ω ), g ∈ H 1/2 (Ω ) such that ∂ Ω g · nd∂ Ω = 0 with n being unit out normal, if the assumption (A1) holds, and ν and f satisfy the following inequality
N ∥f˜ ∥−1
(ν − N ∥ψ∥)2
< 1,
(14)
then there exists a unique solution of the variational problem (12), which satisfies
∥f˜ ∥−1 , ν − N ∥ψ∥ |Au˜ | + ∥p∥1 ≤ c |f˜ |. ∥˜u∥ ≤
(15) (16)
Remark 1. The result (13) in Lemma 2.1 indicates that ∥ψ∥ can be taken as any small value if necessary, and this function can be constructed by setting it to have a bubble near the boundary but decay fast in the domain. Thus, ν − N ∥ψ∥ > 0 in Theorem 2.1 can be always guaranteed by choosing a suitable ψ . Proof. The existence and uniqueness of the solution of the problem (12) have been proved, the reader is referred to Chapter II of [13] for the details. Next, we consider the stability results. Taking (v, q) = (˜u, p) in (12), we obtain
ν∥˜u∥2 + b(˜u, ψ, u˜ ) = (f˜ , u˜ ). Thanks to
|b(˜u, ψ, u˜ )| ≤ N ∥ψ∥ ∥˜u∥2 , |(f˜ , u˜ )| ≤ ∥f˜ ∥−1 ∥˜u∥, it yields
(ν − N ∥ψ∥)∥˜u∥ ≤ ∥f˜ ∥−1 ,
4
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which suggests
∥˜u∥ ≤
∥f˜ ∥−1 . ν − N ∥ψ∥
Furthermore, setting (v, q) = (Au˜ , 0) in (12), it holds
ν|Au˜ |2 + b(˜u, u˜ n , Au˜ ) + b(ψ, u˜ , Au˜ ) + b(˜u, ψ, Au˜ ) = (f˜ , Au˜ ).
(17)
Using (8)–(9), we have
|b(˜u, u˜ , Au˜ )| ≤ c0 ∥˜u∥3/2 |Au˜ |3/2 , |b(ψ, u˜ , Au˜ ) + b(˜u, ψ, Au˜ )| ≤ 2c0 ∥ψ∥ ∥˜u∥1/2 |Au˜ |3/2 , which combining with (17) yields
ν|Au˜ | ≤ |f˜ | + c0 ∥˜u∥3/2 |Au˜ |1/2 + 2c0 ∥ψ∥ ∥˜u∥1/2 |Au˜ |1/2 ≤
ν 2
|Au˜ | + |f˜ | +
c02
ν
∥˜u∥3 +
4c02
ν
∥ψ∥2 ∥˜u∥.
Applying (15) and Lemma 2.1 in the above inequality, we can easily derive
|Au˜ | ≤ c |f˜ |. Finally, using (8)–(9), (12), Lemma 2.1 and the above inequality, we obtain
∥p∥1 ≤ ν|Au˜ | + c0 ∥˜u∥3/2 |Au˜ |1/2 + 2c0 ∥˜u∥1/2 |Au˜ |1/2 ∥ψ∥ + |f˜ | ≤ c |f˜ |. The proof is completed. 3. Iterative schemes based on finite element approximation Supposing 0 < h < 1 is the mesh size, we introduce the finite dimensional subspaces (Xh , Mh ) ⊂ (X , M ) for the velocity and pressure, respectively. The partitioning of Ω is assumed to be uniformly regular in the usual sense. the reader is referred to [26] for further details of the finite element discretization. Let the L2 -orthogonal projection operator Ph : Y → Xh be defined as:
(Ph v, vh ) = (v, vh ) ∀v ∈ Y , vh ∈ Xh , and the discrete analogue Ah = −Ph ∆h of the Stokes operator A be 1/2
1/2
(−∆h uh , vh ) = (Ah uh , Ah vh ) = ((uh , vh )) ∀uh , vh ∈ Xh . Similar to (3)–(4), it holds
γ0 |vh |2 ≤ ∥vh ∥2 ,
γ0 ∥vh ∥2 ≤ |Ah vh |2 ,
∀vh ∈ Vh .
(18)
We also make further assumptions for the finite element spaces (Xh , Mh ) (see [1,3,13,21]): (A2). For each v ∈ (H 2 (Ω ))2 ∩ V and q ∈ H 1 (Ω ) ∩ M, there exist approximations πh v ∈ Xh and ρh q ∈ Mh , such that
∥v − πh v∥ ≤ ch∥v∥2 ,
|q − ρh q| ≤ ch∥q∥1 ,
(19)
together with the inverse inequality
∥vh ∥ ≤ ch−1 |vh |,
v h ∈ Xh ,
(20)
and the so-called inf–sup inequality, i.e., for each qh ∈ Mh , there exists vh ∈ Xh , vh ̸= 0, such that sup
vh ∈Xh ,vh ̸=0
d(vh , qh )
∥vh ∥
≥ βh |qh |,
(21)
where βh is a positive constant depending on Ω . Moreover, let (v, q) ∈ (X , M ) be given, we define the Galerkin projection (Rh (v, q), Qh (v, q)) ∈ (Xh , Mh ) by a(Rh , vh ) − d(vh , Qh ) + d(Rh , qh ) = a(v, vh ) − d(vh , q) + d(v, qh ),
(22)
∀(vh , qh ) ∈ (Xh , Mh ), the following properties are classical for ∀(v, q) ∈ (X , M ) |Rh − v| + h∥Rh − v∥ ≤ ch(∥v∥ + |q|), |Rh − v| + h(∥Rh − v∥ + |Qh − q|) ≤ ch (|Av| + ∥q∥1 ). 2
(23) (24)
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Let (˜uh , ph ) denote the finite element solution of the non-homogeneous Navier–Stokes equations. The finite element approximation of the Navier–Stokes equations is to find (˜uh , ph ) ∈ (Xh , Mh ), such that
ν a(˜uh , vh ) − d(vh , ph ) + d(˜uh , qh ) + b(˜uh , u˜ h , vh ) + b(ψ, u˜ h , vh ) + b(˜uh , ψ, vh ) = (f˜ , vh ),
(25)
for all (vh , qh ) ∈ (Xh , Mh ). Before considering the iterative schemes, we derive first some regularities for the finite element solution of the nonhomogeneous Navier–Stokes equations (25). Theorem 3.1. Under the assumptions of Theorem 2.1, if (A2) and the uniqueness condition (14) are valid, then there exists a unique solution for the finite element approximation problem (25) which satisfies the following regularities:
∥˜uh ∥ ≤
∥f˜ ∥−1 , ν − N ∥ψ∥
|Ah u˜ h | + ∥p∥1 ≤ c |f˜ |.
(26)
Proof. The proof of Theorem 3.1 is similar to that of Theorem 2.1, which is omitted here. Theorem 3.2. Under the assumptions of Theorem 2.1, if (A2) and the uniqueness condition (14) are valid, then, the finite element solution of (25) satisfies the following error estimates
|˜u − u˜ h | ≤ ch2 , ∥˜u − u˜ h ∥ + |p − ph | ≤ ch.
(27) (28)
Proof. Applying the Galerkin projection in (12), we have for ∀(vh , qh ) ∈ (Xh , Mh )
ν a(Rh , vh ) − d(vh , Qh ) + d(ν Rh , qh ) + b(˜u, u˜ , vh ) + b(ψ, u˜ , vh ) + b(˜u, ψ, vh ) = (f˜ , vh ),
(29)
where (Rh , Qh ) = (Rh (ν u˜ , p), Q (ν u˜ , p)). Subtracting (25) from (29) and setting (˜eh , ηh ) = (Rh − u˜ h , Qh − ph ), we get
ν a(˜eh , vh ) − d(vh , ηh ) + d(ν e˜ h , qh ) + b(˜u − u˜ h , u˜ , vh ) + b(˜uh , u˜ − u˜ h , vh ) + b(ψ, u˜ − u˜ h , vh ) + b(˜u − u˜ h , ψ, vh ) = 0, Taking (vh , qh ) = (˜eh , ν
−1
∀(vh , qh ) ∈ (Xh , Mh ).
(30)
ηh ) in (30) and using (6), it yields
ν∥˜eh ∥ + b(˜eh , u˜ , e˜ h ) + b(˜u − Rh , u˜ , e˜ h ) + b(˜uh , u˜ − Rh , e˜ h ) 2
+ b(ψ, u˜ − Rh , e˜ h ) + b(Rh − u˜ h , ψ, e˜ h ) + b(˜eh , ψ, e˜ h ) = 0.
(31)
Due to
|b(˜eh , u˜ , e˜ h )| ≤ N ∥˜u∥ ∥˜eh ∥2 , |b(˜u − Rh , u˜ , e˜ h ) + b(˜uh , u˜ − Rh , e˜ h )| ≤ c0 (|Au˜ | + |Ah u˜ h |)|˜u − Rh | ∥˜eh ∥, |b(ψ, u˜ − Rh , e˜ h ) + b(Rh − u˜ h , ψ, e˜ h )| ≤ 2c0 |Aψ| |˜u − Rh | ∥˜eh ∥, |b(˜eh , ψ, e˜ h )| ≤ N ∥ψ∥ ∥˜eh ∥2 , combining these inequalities with (31), and using Theorems 2.1 and 3.1, it follows
(ν − N ∥ψ∥) 1 −
N ∥f˜ ∥−1
(ν − N ∥ψ∥)2
∥˜eh ∥ ≤ ch2 .
(32)
Applying (14), (23)–(24) and the triangle inequality, we can easily get (27). Using (21) and (30), we obtain
|ηh | ≤ c (ν∥˜eh ∥ + N (∥˜u∥ + ∥˜uh ∥)∥˜u − u˜ h ∥ + c0 ∥ψ∥ ∥˜u − u˜ h ∥),
(33)
considering (27), (23)–(24), Theorems 2.1 and 3.1, and the triangle inequality, we derive (28). The proof is completed. Since (25) is a discrete nonlinear system, the solution is usually sought by an iterative scheme. For the Navier–Stokes equations with the homogeneous boundary condition, the authors investigated the iterative schemes in [16]. Here, by extending the idea to the non-homogeneous problem, we consider the following three iterative methods: find (˜unh , pnh ) ∈ (Xh , Mh ), such that Oseen scheme:
ν a(˜unh , vh ) − d(vh , pnh ) + d(˜unh , qh ) + b(˜unh−1 , u˜ nh , vh ) + b(ψ, u˜ nh , vh ) + b(˜unh , ψ, vh ) = (f˜ , vh ),
(34)
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Newton scheme:
ν a(˜unh , vh ) − d(vh , pnh ) + d(˜unh , qh ) + b(˜unh−1 , u˜ nh , vh ) + b(˜unh , u˜ hn−1 , vh ) + b(ψ, u˜ nh , vh ) + b(˜unh , ψ, vh ) = b(˜unh−1 , u˜ hn−1 , vh ) + (f˜ , vh ),
(35)
Stokes scheme:
ν a(˜unh , vh ) − d(vh , pnh ) + d(˜unh , qh ) + b(˜unh−1 , u˜ nh−1 , vh ) + b(ψ, u˜ nh , vh ) + b(˜unh , ψ, vh ) = (f˜ , vh ),
(36)
with (u0h , p0h ) ∈ (Xh , Mh ) being the solution of the discrete Stokes problem:
ν a(˜u0h , vh ) − d(vh , p0h ) + d(˜u0h , qh ) = (fˆ , vh ),
(37)
for all (vh , qh ) ∈ (Xh , Mh ), where fˆ = f + ν 1ψ . 4. Stability and convergence analysis In this section, we will analyze the stability and convergence of the iterative schemes (34)–(36) for the non-homogeneous Navier–Stokes equations. 4.1. Stability analysis In this subsection, we will provide the stabilities of the schemes. By applying the similar process as the proof of Theorem 2.1 and that in [16,23], we can derive easily the following theorems, and the detail is omitted here. Lemma 4.1. Under the assumptions of Theorem 2.1, suppose that (A1) and (A2) are valid, if the uniqueness condition (14) holds, then the solution (˜unh , pnh ) ∈ (Xh , Mh ) generated by the Oseen scheme (34) satisfies
∥˜unh ∥ ≤
∥f˜ ∥−1 , ν − N ∥ψ∥
|Ah u˜ nh | ≤ c |f˜ |,
∀n > 0.
(38)
Lemma 4.2. Under the assumptions of Theorem 2.1, suppose that (A1) and (A2) are valid, if it holds the inequality N ∥f˜ ∥−1
(ν − N ∥ψ∥)2
<
1 3
,
(39)
then the solution (˜unh , pnh ) ∈ (Xh , Mh ) generated by the Newton scheme (35) satisfies
∥˜unh ∥ ≤
4∥f˜ ∥−1 3(ν − N ∥ψ∥)
,
|Ah u˜ nh | ≤ c |f˜ |,
∀n > 0.
(40)
Lemma 4.3. Under the assumptions of Theorem 2.1, suppose that assumptions (A1) and (A2) are valid, if it holds the inequality N ∥f˜ ∥−1
(ν − N ∥ψ∥)
2
<
1 4
,
(41)
then the solution (˜unh , pnh ) ∈ (Xh , Mh ) generated by the Stokes scheme (36) satisfies
∥˜unh ∥ ≤
2∥f˜ ∥−1
ν − N ∥ψ∥
,
|Ah u˜ nh | ≤ c |f˜ |,
∀n > 0.
(42)
4.2. Error estimates Next, we will consider the convergence of the iterative schemes proposed in Section 3. Setting (˜en , ηn ) = (˜uh −˜unh , ph −pnh ), and subtracting (34)–(36) from (25), respectively, we have the following error equations: find (˜en , ηn ) ∈ (Xh , Mh ), such that for all (vh , qh ) ∈ (Xh , Mh ), it satisfies respectively, Oseen scheme:
ν a(˜en , vh ) − d(vh , ηn ) + d(˜en , qh ) + b(˜en−1 , u˜ h , vh ) + b(˜unh−1 , e˜ n , vh ) + b(ψ, e˜ n , vh ) + b(˜en , ψ, vh ) = 0,
(43)
K. Wang / Computers and Mathematics with Applications (
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Newton scheme:
ν a(˜en , vh ) − d(vh , ηn ) + d(˜en , qh ) + b(˜en , u˜ nh−1 , vh ) + b(˜uhn−1 , e˜ n , vh ) + b(ψ, e˜ n , vh ) + b(˜en , ψ, vh ) = −b(˜en−1 , e˜ n−1 , vh ),
(44)
Stokes scheme:
ν a(˜en , vh ) − d(vh , ηn ) + d(˜en , qh ) + b(˜en−1 , u˜ h , vh ) + b(˜unh−1 , e˜ n−1 , vh ) + b(ψ, e˜ n , vh ) + b(˜en , ψ, vh ) = 0.
(45)
Lemma 4.4. Under the assumptions of Lemma 4.1, we have the following estimates for the error (˜en , ηn ) related to the Oseen scheme for all n > 0
∥˜en ∥ ≤ c |ηn | ≤ c
N ∥f˜ ∥−1
n
(ν − N ∥ψ∥) N ∥f˜ ∥ n −1 2
(ν − N ∥ψ∥)2
,
(46)
.
(47)
Proof. Taking (vh , ph ) = (˜en , ηn ) in (43) and using (6), we get
ν∥˜en ∥2 + b(˜en−1 , u˜ h , e˜ n ) + b(˜en , ψ, e˜ n ) = 0. Due to
|b(˜en−1 , u˜ h , e˜ n )| ≤ N ∥˜en−1 ∥ ∥˜uh ∥ ∥˜en ∥, |b(˜en , ψ, e˜ n )| ≤ N ∥ψ∥ ∥˜en ∥2 , it follows
(ν − N ∥ψ∥)∥˜en ∥ ≤ N ∥˜uh ∥ ∥˜en−1 ∥ ≤
N ∥f˜ ∥−1
ν − N ∥ψ∥
∥˜en−1 ∥,
∀n > 0.
(48)
Subtracting (37) from (25), taking (vh , ph ) = (˜e0 , η0 ) and using Lemma 2.1 and Theorem 3.1, it holds
ν∥˜e0 ∥ ≤ N ∥uh ∥2 ≤ N (∥˜uh ∥ + ∥ψ∥)2 ≤ c . Therefore, (48) yields
∥˜en ∥ ≤
N ∥f˜ ∥ N ∥f˜ ∥ n n −1 −1 0 n−1 ∥˜ e ∥ ≤ c , ∥˜ e ∥ ≤ (ν − N ∥ψ∥)2 (ν − N ∥ψ∥)2 (ν − N ∥ψ∥)2 N ∥f˜ ∥−1
∀n > 0,
which suggests (46). Moreover, from (43) we can get by using Theorem 3.1, Lemma 4.1, (6) and the above estimate
|ηn | ≤ ν∥˜en ∥ + N ∥˜en−1 ∥ ∥˜uh ∥ + N ∥˜unh−1 ∥ ∥˜en ∥ + 2N ∥ψ∥ ∥˜en ∥ N ∥f˜ ∥−1 N ∥f˜ ∥−1 ≤ ∥˜en−1 ∥ + ν + + 2N ∥ψ∥ ∥˜en ∥ ν − N ∥ψ∥ ν − N ∥ψ∥ N ∥f˜ ∥ n −1 ≤c , (ν − N ∥ψ∥)2 which suggests (47). The proof is completed. Applying the similar process as that in the above theorem and in [16,23], we can easily derive the following results. Lemma 4.5. Under the assumptions of Lemma 4.2, we have the following estimates for the error (˜en , ηn ) related to the Newton scheme for all n > 0
∥˜en ∥ ≤ c
|ηn | ≤ c
9N ∥f˜ ∥−1
2n −1
5(ν − N ∥ψ∥)2 9N ∥f˜ ∥−1 5(ν − N ∥ψ∥)2
2n −1
,
(49)
.
(50)
8
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Lemma 4.6. Under the assumptions of Lemma 4.3, we have the following estimates for the error (˜en , ηn ) related to the Stokes scheme for all n > 0
∥˜en ∥ ≤ c |ηn | ≤ c
3N ∥f˜ ∥−1
n
(ν − N ∥ψ∥) n 3N ∥f˜ ∥ −1 2
(ν − N ∥ψ∥)2
,
(51)
.
(52)
Now, we arrive at the main results: Theorem 4.1. Under the assumptions of Lemmas 4.1–4.3, for all n > 0, we have
n ∥f˜ ∥−1 , (ν − N ∥ψ∥)2 n ∥f˜ ∥−1 ∥u − unh ∥ + |p − pnh | ≤ ch + c , (ν − N ∥ψ∥)2 |u − unh | ≤ ch2 + c
(53) (54)
for the Oseen scheme, and
|u − unh | ≤ ch2 + c ∥u − unh ∥ + |p − pnh | ≤ ch + c
9∥f˜ ∥−1
2n −1
5(ν − N ∥ψ∥)2 9∥f˜ ∥−1
2n −1
5(ν − N ∥ψ∥)2
,
(55)
,
(56)
for the Newton scheme, and
|u − unh | ≤ ch2 + c ∥u − unh ∥ + |p − pnh | ≤ ch + c
3∥f˜ ∥−1
n
(ν − N ∥ψ∥) n 3∥f˜ ∥ −1 2
(ν − N ∥ψ∥)2
,
(57)
,
(58)
for the Stokes iterative scheme, respectively, where f˜ = f + ν 1ψ − (ψ · ∇)ψ . Proof. Since u − unh = (˜u + ψ) − (˜unh + ψ) = u˜ − u˜ nh , the above theorem is obviously from Lemmas 4.1–4.6. In Theorem 4.1, the complete error estimates are derived for solving the non-homogeneous Navier–Stokes equations with the finite element method based on three iterative schemes. We can see that the errors consisted of two parts: spatial discretization and solving the discretized nonlinear system iteratively. Only both of the errors are controlled well, we can get a good approximation. Furthermore, the error generated by solving the discretized nonlinear system iteratively indicates that different schemes converge based on different assumptions of the Reynolds number. 4.3. Stop criterion The relationships between the Reynolds number and the convergence of the iterative schemes are shown in Lemmas 4.4– 4.6, which indicate that the Newton scheme converges fastest among these schemes, but the Oseen scheme has the weakest assumption on the Reynolds number and is the most efficient for solving the high Reynolds number problems. These results also suggest that it is enough to ensure the error generated by the iterative schemes smaller than that generated by the N ∥f˜ ∥
−1 spacial discretization when implementing, which means to let Θ (ν−N ∥ψ∥) 2
N ∥f˜ ∥
−1 as Θ (ν−N ∥ψ∥) 2
=c
N ∥f˜ ∥−1 (ν−N ∥ψ∥)2
2
≤ ch2 hold with Θ (·) being a function, such
for the Oseen scheme. Overall iteration will cost more computational time but without N ∥f˜ ∥
−1 improvement on the approximation results. Although we have given the evaluation factor (ν−N ∥ψ∥) 2 in the theoretical prediction, usually it is not easily computed. Generally, the following strategy is used popularly as a stopping criterion:
|˜unh − u˜ nh−1 | ≤ 10−l , |˜unh | with l being a positive integer. But since the optimal value l is unknown, to ensure the error generated by the iterative schemes smaller than that generated by the spacial discretization, we always take a large l and the overall iteration may appear. Next, we will consider a new kind of stop criterion for the iterative schemes which is firstly proposed in [16].
K. Wang / Computers and Mathematics with Applications (
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Theorem 4.2. Under the assumptions of Lemmas 4.4–4.6, (˜en , ηn ) related to the iterative schemes satisfies, for all n > 0,
∥˜en ∥ + |ηn | ≤ c |˜unh − u˜ nh−1 |
(59)
for the Oseen and Stokes schemes, and
∥˜en ∥ + |ηn | ≤ c ∥˜unh − u˜ nh−1 ∥2
(60)
for the Newton scheme. Furthermore, there holds lim (∥˜unh − u˜ nh−1 ∥ + |˜unh − u˜ nh−1 |) = 0
(61)
n→∞
for the Oseen, Newton and Stokes schemes. Proof. Taking (vh , qh ) = (˜en , ηn ) in (43) and using (6), we have
ν∥˜en ∥ + b(˜en , u˜ h , e˜ n ) + b(˜en−1 − e˜ n , u˜ h , e˜ n ) + b(˜en , ψ, e˜ n ) = 0.
(62)
Thanks to
|b(˜en , u˜ h , e˜ n )| ≤ N ∥˜uh ∥ ∥˜en ∥2 , |b(˜en−1 − e˜ n , u˜ h , e˜ n )| ≤ c0 |˜en−1 − e˜ n | |Ah u˜ h | ∥˜en ∥, |b(˜en , ψ, e˜ n )| ≤ N ∥ψ∥ ∥˜en ∥2 , it follows, by using Theorem 3.1, that
(ν − N ∥ψ∥) 1 −
N ∥f˜ ∥−1
(ν − N ∥ψ∥)2
∥˜en ∥ ≤ c0 |Ah u˜ h | |˜en−1 − e˜ n | ≤ c |˜unh − u˜ nh−1 |,
∀n > 0.
(63)
Using (21), Theorem 3.1 and the above inequality, we get (59) for the Oseen scheme. On the other hand, taking (vh , qh ) = (˜en , ηn ) in (45) and using (6), we have
ν∥˜en ∥ + b(˜en , u˜ h , e˜ n ) + b(˜en−1 − e˜ n , u˜ h , e˜ n ) + b(˜unh−1 , e˜ n−1 − e˜ n , e˜ n ) + b(˜en , ψ, e˜ n ) = 0.
(64)
Similar to (63), we can derive that (59) also holds for the Stokes scheme. Moreover, setting (vh , qh ) = (˜en , ηn ) in (44), it follows
ν∥˜en ∥2 + b(˜en , u˜ h , e˜ n ) + b(˜en , ψ, e˜ n ) + b(˜unh − u˜ nh−1 , u˜ nh − u˜ hn−1 , e˜ n ) = 0.
(65)
Thanks to
|b(˜en , u˜ h , e˜ n )| ≤ N ∥˜en ∥2 , |b(˜en , ψ, e˜ n )| ≤ N ∥ψ∥ ∥˜en ∥2 , |b(˜unh − u˜ hn−1 , u˜ nh − u˜ nh−1 , e˜ n )| ≤ N ∥˜unh − u˜ nh−1 ∥2 ∥˜en ∥, we get
(ν − N ∥ψ∥) 1 −
N ∥f˜ ∥−1
(ν − N ∥ψ∥)
2
∥˜en ∥ ≤ N ∥˜unh − u˜ hn−1 ∥2 .
(66)
Applying the similar process as that in (59), we deduce (60). Finally, to prove (61), using (18) and the triangle inequality, we have −1/2
∥˜unh − u˜ nh−1 ∥ + |˜unh − u˜ nh−1 | ≤ (1 + γ0
−1/2
)∥˜unh − u˜ nh−1 ∥ ≤ (1 + γ0
)(∥˜en ∥ + ∥˜en−1 ∥).
Under the assumptions of Lemmas 4.4–4.6, respectively, there holds for the Oseen, Newton and Stokes schemes lim ∥˜en ∥ = 0,
n→∞
which indicates (61). The proof is completed. Remark 2. Theorem 4.2 indicates that the iterative errors generated by three schemes can be evaluated by the iterative N ∥f˜ ∥
−1 solution instead of the factor (ν−N ∥ψ∥) 2 , which is easily realized when implementing. Based on this results, we can use the new stop criterion
|˜unh − u˜ hn−1 | ≤ ch2
10
K. Wang / Computers and Mathematics with Applications (
(a) u1 at the line (0.5, y).
)
–
(b) u2 at the line (x, 0.5). Fig. 1. Crosscut with different stop criteria for the Oseen scheme (Re = 200). Table 1 Iteration numbers of the Oseen scheme with different stop criteria (P2 − P1 elements, h = 1/40). Re
10
50
100
200
Criterion 1 Criterion 2
5 2
8 4
10 5
13 6
for the Oseen and Stokes schemes, and
∥˜unh − u˜ hn−1 ∥ ≤ ch2 for the Newton scheme. Moreover, the new stop criteria of three iterative schemes will converge under the same assumptions of Lemmas 4.4–4.6, respectively. Remark 3. Especially, if g = 0, the problem (1)–(2) becomes the Navier–Stokes with homogeneous boundary. Therefore, the results in Theorem 4.2 also hold for this kind of problem, which is investigated in [16,23]. It is obvious that the results here are better than that in [16], some improvements are achieved. 5. Numerical experiments To confirm the theoretical results proved above, we report some computational results in this section. As a classical benchmark, the lid-driven cavity flow is also a non-homogeneous problem (see [27,28]). The following numerical simulations are all based on this model. First, we consider the standard and new stopping strategies: Criterion 1:
∥˜unh −˜unh−1 ∥ ≤ ∥˜unh ∥ n −1 n
10−6 ,
Criterion 2: |˜uh − u˜ h | ≤ h2 for the Oseen and Stokes schemes and ∥˜unh − u˜ hn−1 ∥ ≤ h2 for the Newton scheme. The results got by two kinds of stop criteria for different iterative schemes are collected in Tables 1–3. We can see that the iteration numbers demanded by applying the new stopping strategies are uniform less than that needed by using the standard one for three schemes. Furthermore, Figs. 1–3 suggest that these two stop criteria have almost the same computational accuracy. This means that the overall iteration is avoided by applying the new stop criterion, which confirms Theorem 4.2 very well. Furthermore, from Theorem 4.1 we can see that the new stop criterion can control the iterative number and avoid the overall iteration, but get almost the same computational accuracy in L2 - and H 1 -norm. Then, we verify the convergence of three iterative schemes. From the analysis above, it is obvious that the Oseen scheme has the weakest convergence condition, and the Newton scheme converges fastest among these methods. Setting h = 1/40, we show the results in Table 4 with the Reynolds number in the range of 50–1000. We can see that the computational results are consistent with the theoretical predictions. Finally, the accuracy of the Oseen scheme applying to some high Reynolds number problems is investigated by comparing with a well-known method in [27] in which a very fine mesh size is used (see Figs. 4–5). The numerical result says by itself. All of these results confirm that theoretical analysis in this paper is right and can provide good prediction.
K. Wang / Computers and Mathematics with Applications (
(a) u1 at the line (0.5, y).
)
–
11
(b) u2 at the line (x, 0.5). Fig. 2. Crosscut with different stop criteria for the Newton scheme (Re = 100).
(a) u1 at the line (0.5, y).
(b) u2 at the line (x, 0.5). Fig. 3. Crosscut with different stop criteria for the Stokes scheme (Re = 50). Table 2 Iteration numbers of the Newton scheme with different stop criteria (P2 − P1 elements, h = 1/40). Re
10
50
80
100
Criterion 1 Criterion 2
3 2
4 2
4 3
4 3
Table 3 Iteration numbers of the Stokes scheme with different stop criteria (P2 − P1 elements, h = 1/40). Re
10
20
30
50
Criterion 1 Criterion 2
5 2
7 3
9 4
15 6
6. Conclusions Three iterative schemes for solving the nonlinear discrete system generated from the non-homogeneous Navier–Stokes equations are investigated in this paper. Including the stability and error estimates for the finite element method, we analyze
12
K. Wang / Computers and Mathematics with Applications (
(a) u1 at the line (0.5, y).
)
–
(b) u2 at the line (x, 0.5). Fig. 4. Crosscut for different schemes with Re = 1000 and h = 1/40.
(a) u1 at the line (0.5, y).
(b) u2 at the line (x, 0.5). Fig. 5. Crosscut for different schemes with Re = 2500 and h = 1/40. Table 4 Iteration numbers of different schemes (P2 − P1 elements, h = 1/40). Re
50
100
200
500
1000
Oseen scheme Newton scheme Stokes scheme
4 2 6
5 3 D
6 4 D
7 6 D
9 Da D
a
D indicates divergence or the maximum number of iteration M > 1000.
the stability and convergence conditions for the iterative schemes and show the complete error estimates which consist of two parts: spatial discretization and solving the discretized nonlinear system iteratively. These theoretical results suggest that only both of the errors are controlled well, we can get a good approximation. Furthermore, the relationship between the iteration solutions and the stop criteria is also analyzed for different methods, which reflect the character of the iterative schemes exactly. Applying the technique in [29–31], the time-dependent non-homogeneous problem will be considered in future. References [1] V. Girault, P. Raviart, Finite Element Method for Navier–Stokes Equations: Theory and Algorithms, Springer, Berlin, 1986. [2] J. Guermond, A. Marra, L. Quartapelle, Subgrid stabilized projection method for 2D unsteady flows at high Reynolds numbers, Comput. Methods Appl. Mech. Engrg. 195 (2006) 5857–5876.
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