Nothing Special   »   [go: up one dir, main page]

Next Article in Journal
Contextual Inferences, Nonlocality, and the Incompleteness of Quantum Mechanics
Previous Article in Journal
Estimating the Epicenter of a Future Strong Earthquake in Southern California, Mexico, and Central America by Means of Natural Time Analysis and Earthquake Nowcasting
You seem to have javascript disabled. Please note that many of the page functionalities won't work as expected without javascript enabled.
 
 
Font Type:
Arial Georgia Verdana
Font Size:
Aa Aa Aa
Line Spacing:
Column Width:
Background:
Article

Finite Element Iterative Methods for the 3D Steady Navier--Stokes Equations †

School of Mathematics and Statistics, Xi’an Jiaotong University, Xi’an 710049, China
This work is partially supported by the NSF of China under grant Nos. 11771348.
Entropy 2021, 23(12), 1659; https://doi.org/10.3390/e23121659
Submission received: 11 November 2021 / Revised: 4 December 2021 / Accepted: 5 December 2021 / Published: 9 December 2021

Abstract

:
In this work, a finite element (FE) method is discussed for the 3D steady Navier–Stokes equations by using the finite element pair X h × M h . The method consists of transmitting the finite element solution ( u h , p h ) of the 3D steady Navier–Stokes equations into the finite element solution pairs ( u h n , p h n ) based on the finite element space pair X h × M h of the 3D steady linearized Navier–Stokes equations by using the Stokes, Newton and Oseen iterative methods, where the finite element space pair X h × M h satisfies the discrete inf-sup condition in a 3D domain Ω . Here, we present the weak formulations of the FE method for solving the 3D steady Stokes, Newton and Oseen iterative equations, provide the existence and uniqueness of the FE solution ( u h n , p h n ) of the 3D steady Stokes, Newton and Oseen iterative equations, and deduce the convergence with respect to ( σ , h ) of the FE solution ( u h n , p h n ) to the exact solution ( u , p ) of the 3D steady Navier–Stokes equations in the H 1 L 2 norm. Finally, we also give the convergence order with respect to ( σ , h ) of the FE velocity u h n to the exact velocity u of the 3D steady Navier–Stokes equations in the L 2 norm.

1. Introduction

The incompressible Navier–Stokes equations reflect the basic mechanical law of viscous fluid flow, which have important implications in fluid mechanics. This problem is one of the main systems studied in pipe flow, flow around airfoils, blood flow, weather and convective heat transfer inside industrial furnaces. Therefore, solving the 3D steady Navier–Stokes equations is of great significance and application value in the field of scientific research and engineering application. Lots of works are devoted to this problem, and the finite element methods, finite volume methods and finite difference methods are the most successful methods. There are many scholars who have studied the numerical methods of the Navier–Stokes equations; see, for example, the monographs of Temam [1], Girault and Raviart [2], Quarteroni and Valli [3], Glowinski [4], Elman et al. [5], Heywood and Rannacher [6,7,8,9], Layton [10], and He et al. [11,12,13,14,15]. An important area that is left out is the development of high order spectral volume and spectral difference methods advanced by Kannan et al. [16,17,18,19,20,21,22,23] and Sun et al. [24]. In recent years, the weak Galerkin method [25] and virtual element method [26,27,28] have also made great contributions to solve the Navier–Stokes equations. Chen et al. in [29] proposed a dimension splitting method for the 3D steady Navier–Stokes equations and in [30], proposed a dimension splitting and characteristic projection method for the 3D time-dependent Navier–Stokes equations, giving some numerical examples to verify the effectiveness of the algorithm. However, the results of the numerical analysis are not given in their papers. Much more numerical methods for the Navier–Stokes equations can be found in [31,32,33,34,35], and the references therein. Despite the considerable increase in the available computing power in recent decades, there are still some difficulties in solving the 3D steady Navier–Stokes equations under the uniqueness condition, that is, how to overcome the divergence free constraint and the nonlinearity of the steady Navier–Stokes equations in the 3D space.
Recently, He and Li [11] and Zhang et al. [36] made a great effort to overcome the difficulties mentioned above in solving the 2D steady Navier–Stokes equations; they used the finite element pair X h × M h , satisfying the discrete inf-sup condition in a 2D domain Ω , which overcomes the difficulty of divergence free constraint, using the Oseen, Newton and Stokes iterative finite element methods to overcome the difficulty of nonlinearity of the steady Navier–Stokes equations in the 2D space.
Furthermore, in order to overcome the difficulties mentioned above in solving the 3D steady Navier–Stokes equations, Xu and He [37] and He [38] used the finite element pair X h × M h , satisfying the discrete inf-sup condition in a 2D/3D domain Ω , which overcomes the difficulty of divergence free constraint, using the Stokes, Newton and Oseen iterative finite element methods to overcome the difficulty of nonlinearity of the steady Navier–Stokes equations in the 2D/3D space. However, in [37,38], they provided some poor stability and convergence results under the strong stability and convergence conditions. For the Stokes iterative finite element method, the stability result is ν ˜ u h n 0 , Ω 2 F 1 , Ω and the convergence result is ν ˜ ( u h n u h ) 0 , Ω ( 3 σ ) n F 1 , Ω under the strong stability and convergence condition 0 < σ 1 4 . For the Newton iterative finite element method, the stability result is ν ˜ u h n 0 , Ω 4 3 F 1 , Ω and the convergence result is ν ˜ ( u h n u h ) 0 , Ω ( 9 5 σ ) 2 n 1 F 1 , Ω under the strong stability and convergence condition 0 < σ 1 3 .
In this paper, we use the finite element solution ( u h n , p h n ) of the 3D steady Stokes, Newton and Oseen iterative equations (the 3D steady linearized Navier–Stokes equations) to approximate the solution ( u , p ) of the 3D steady Navier–Stokes equations. For the Stokes iterative finite element method, the stability result is ν ˜ u h n 0 , Ω 6 5 F 1 , Ω and the convergence result is ν ˜ ( u h n u h ) 0 , Ω ( 11 5 σ ) n σ F 1 , Ω under the weak stability and convergence condition 0 < σ 2 5 ; for the Newton iterative finite element method, the stability result is ν ˜ u h n 0 , Ω 6 5 F 1 , Ω and the convergence result is ν ˜ ( u h n u h ) 0 , Ω σ 2 n F 1 , Ω under the strong stability and convergence condition 0 < σ 5 11 . Compared with the results of [37,38], we obtain better stability and convergence results of the finite element iterative solution ( u h n , p h n ) of of the 3D steady Navier–Stokes equations under the weak stability and convergence condition.
The paper is structured as follows: some preliminaries on the 3D Navier–Stokes equations are recalled, and the uniform regularity results with respect to ν of the solution ( u , p ) and the uniqueness condition are reduced in Section 2. The mixed finite element methods for the 3D steady Navier–Stokes equations and the Oseen iterative equations are designed, and the existence, uniqueness and stability of the finite element solution ( u h , p h ) and ( u h n , p h n ) on the above equations based on the finite element space pair X h × M h are proved in Section 3. Moreover, the uniform optimal error estimates of the mixed finite element solution ( u h , p h ) with respect to the exact solution ( u , p ) of the 3D steady Navier–Stokes equations is provided in Section 4. The Oseen iterative finite element method is designed, and the uniform optimal error estimates of the Oseen iterative finite element solution ( u h n , p h n ) with respect to the exact solution ( u , p ) of the 3D steady Navier–Stokes equations are proven in Section 5. The Newton iterative finite element method is designed, and the uniform optimal error estimates of the Oseen iterative finite element solution ( u h n , p h n ) with respect to the exact solution ( u , p ) of the 3D steady Navier–Stokes equations are proven in Section 6. The Stokes iterative finite element method is designed and the uniform optimal error estimates of the Oseen iterative finite element solution ( u h n , p h n ) with respect to the exact solution ( u , p ) of the 3D steady Navier–Stokes equations are proven in Section 7. Finally, some conclusions of the Oseen, Newton and Stokes iterative finite element methods are provided in Section 8.

2. Preliminaries and the 3D Steady Linearized Navier–Stokes Equations

In this section, we first recall the regularity results on the Stokes equations with the Dirichlet boundary condition in a bounded convex polyhedron Ω R 3 . Then, we consider the 3D steady Navier–Stokes equations and define the iterative solution ( u n , p n ) by the 3D steady Oseen iterative equations (the 3D steady linearized Navier–Stokes equations) and obtain the regularity results of the Oseen iterative solution ( u n , p n ) and the error bound of ( u n , p n ) to ( u , p ) .
First, we consider the 3D steady Stokes equations in Ω with the Dirichlet boundary condition:
Δ ˜ u + ˜ p = F , in Ω ,
˜ · u = 0 , in Ω ,
u = 0 , on Ω ,
where u = ( u 1 , u 2 , u 3 ) represents the velocity, p the pressure with and Ω p ( x , y , z ) d x d y d z = 0 , F = ( F 1 , F 2 , F 3 ) the external volumetric force on the fluid. Additionally, we introduce the following notations: ˜ = ( 1 , 2 , 3 ) = ( x , y , z ) , Δ ˜ = ˜ · ˜ and ˜ u = ( i u j ) 3 × 3 .
Using the Green formula, we deduce the weak formulation of the 3D steady Stokes Equations (1)–(3): find ( u , p ) X × M such that for each ( v , q ) X × M there holds
G ( ( u , p ) , ( v , q ) ) A ( u , v ) d ( v , p ) d ( u , q ) = F ˜ ( v , q ) ( F , v ) Ω ,
where X = H 0 1 ( Ω ) 3 , M = L 0 2 ( Ω ) and d ( v , q ) = ( ˜ · v , q ) Ω .
In order to consider the existence and the uniqueness of the solution ( u , p ) X × M , we recall the inf-sup condition of d ( v , p ) in [1,2].
Lemma 1.
There exists a positive constant β such that for each p M , there exists a u ˜ X such that
d ( u ˜ , p ) = p 0 , Ω 2 , ˜ u ˜ 0 , Ω β 1 p 0 , Ω ,
or
sup v X d ( v , p ) ˜ v 0 , Ω β p 0 , Ω p M .
Next, we need to recall the general Lax–Milgram theorem.
General Lax–Milgram theorem. For the weak formulation, find ( u , p ) X × M such that for each ( v , q ) X × M , there holds
G ˜ ( ( u , p ) , ( v , q ) ) = F ˜ ( v , q ) ,
if there holds the following condition:
| F ˜ ( v , q ) | F ˜ ( X × M ) ( v X + q M ) ,
| G ˜ ( ( u , p ) , ( v , q ) ) | α ( u X + p M ) ( v X + q M ) ,
β ( u X + p M ) sup ( v , q ) X × M G ˜ ( ( u , p ) , ( v , q ) ) v X + q M ,
β ( v X + q M ) sup ( u , p ) X × M G ˜ ( ( u , p ) , ( v , q ) ) u X + p M ,
Then, (7) admits a unique solution ( u , p ) X × M , satisfying
β ( u X + p M ) F ˜ ( X × M ) .
Now, we give the existence, uniqueness and stability of the solution ( u , p ) X × M for the 3D steady Stokes equations.
Lemma 2.
If F H 1 ( Ω ) 3 , then (4) admits a unique solution ( u , p ) X × M , satisfying the following bound:
˜ u 0 , Ω F 1 , Ω , β p 0 , Ω 2 F 1 , Ω .
Proof. 
First, we easily prove the following inequalities
| F ˜ ( v , q ) | = | ( F , v ) Ω | F 1 , Ω ˜ v 0 , Ω ,
G ( ( u , p ) , ( v , q ) ) 3 ( ˜ u 0 , Ω + p 0 , Ω ) ( ˜ v 0 , Ω + q 0 , Ω ) ,
for any ( u , p ) , ( v , q ) X × M . Using Lemma 1, for each ( u , p ) X × M , we set ε = β 2 and ( v ˜ , q ˜ ) = ( u ε u ˜ , p ) , where u ˜ satisfies
d ( u ˜ , p ) = p 0 , Ω 2 , ˜ u ˜ 0 , Ω β 1 p 0 , Ω .
Thus, we have
sup ( v , q ) X × M G ( ( u , p ) , ( v , q ) ) ˜ v 0 , Ω + q 0 , Ω G ( ( u , p ) , ( v ˜ , q ˜ ) ) ˜ v ˜ 0 , Ω + q ˜ 0 , Ω = ( ˜ u , ˜ ( u ε u ˜ ) ) Ω + ε d ( u ˜ , p ) ˜ ( u ε u ˜ ) 0 , Ω + p 0 , Ω ˜ u 0 , Ω 2 ε β 1 p 0 , Ω ˜ u 0 , Ω + ε p 0 , Ω 2 ˜ u 0 , Ω + ( 1 + ε β 1 ) p 0 , Ω 0.5 ˜ u 0 , Ω 2 + 0.5 ε p 0 , Ω 2 ˜ u 0 , Ω + ( 1 + ε β 1 ) p 0 , Ω β 1 ( ˜ u 0 , Ω + p 0 , Ω ) .
Since G ( ( u , p ) , ( v , q ) ) = G ( ( v , q ) , ( u , p ) ) , there holds
sup ( u , p ) X × M G ( ( u , p ) , ( v , q ) ) ˜ u 0 , Ω + p 0 , Ω = sup ( u , p ) X × M G ( ( v , q ) , ( u , p ) ) ˜ u 0 , Ω + p 0 , Ω β 1 ( ˜ v 0 , Ω + q 0 , Ω ) .
Thus, using the general Lax–Milgram theorem, (4) admits a unique solution ( u , p ) X × M .
Taking ( v , q ) = ( u , p ) in (4), we obtain
˜ u 0 , Ω F 1 , Ω .
Using again Lemma 1 and (4) with q = 0 , we have
β p 0 , Ω ˜ u 0 , Ω + F 1 , Ω 2 F 1 , Ω .
Combining (19) with (20) yields (13). The proof ends. □
Proof. 
We introduce the subspace V of X as follows:
V = { v X ; d ( v , q ) = 0 , q M } .
Thus, we deduce from (4) that u V satisfies
A ( u , v ) = ( F , v ) Ω , v V ,
where A ( u , v ) satisfies
| A ( u , v ) | ˜ u 0 , Ω ˜ v 0 , Ω , | A ( u , u ) | ˜ u 0 , Ω 2 , u , v V .
Using the Lax–Miligram theorem, (21) admits a unique solution u V such that
˜ u 0 , Ω F 1 , Ω ,
Now, we introduce a Polar set
V 0 = { g X ; < g , v > = 0 v V } ,
and define two dual operators B v M and B q X such that
d ( v , q ) = ( B v , q ) Ω = < v , B q > ( v , q ) X × M .
Thus, referring to [1,2], we know that the inf-sup condition (6) implies that B is a isomorphic operator from M onto V 0 . Moreover, we deduce from (21) that Δ ˜ u F V 0 . Thus, there exists a unique p M such that Δ ˜ u F = B p or
A ( u , v ) d ( v , p ) = ( F , v ) Ω , v X .
Due to u V , there holds d ( u , q ) = 0 for each q M . Thus, we have proved that ( u , p ) X × M is a unique solution of (4). Using (19) and (20), we show that ( u , p ) X × M satisfies (13). The proof ends. □
Recalling Temam [1], if F ( L 2 ( Ω ) ) 3 , there holds the following regularity result of the solution ( u , p ) for the Stokes equations:
u 2 , Ω + p 1 , Ω c 0 F 0 , Ω .
Next, we consider the 3D steady Navier–Stokes equations with the Dirichlet boundary condition in a bounded domain Ω :
ν Δ ˜ u + ˜ p + B ( u , u ) = F , in Ω ,
˜ · u = 0 , in Ω ,
u = 0 , on Ω ,
with Ω p ( x , y , z ) d x d y d z = 0 , where B ( u , v ) = ( u · ˜ ) v + 1 2 ˜ · u v .
Using the Green formula, we deduce the weak formulation of the 3D steady Navier–Stokes Equations (26)–(28): We find ( u , p ) X × M such that for each ( v , q ) X × M there holds
ν A ( u , v ) d ( v , p ) d ( u , q ) + ( B ( u , u ) , v ) Ω = ( F , v ) Ω ,
where X = H 0 1 ( Ω ) 3 , M = L 0 2 ( Ω ) .
Here and hereafter, some positive constants N, β , γ Ω and c 1 and some inequalities are stated as follows:
( B ( u , v ) , w ) Ω = 0.5 ( u · ˜ v , w ) Ω 0.5 ( u · ˜ w , v ) Ω ,
| ( B ( u , v ) , w ) Ω | N ˜ u 0 , Ω ˜ v 0 , Ω ˜ w 0 , Ω u , v , w X , B ( u , v ) 0 , Ω N ˜ u 0 , Ω ˜ v 0 , Ω 1 2 v 2 , Ω 1 2 , u X , v ( H 0 1 ( Ω ) H 2 ( Ω ) ) 3 ,
B ( v , w ) 0 , Ω N ˜ v 0 , Ω ˜ w 0 , Ω 1 2 w 2 , Ω 1 2 , v X , w ( H 0 1 ( Ω ) H 2 ( Ω ) ) 3 ,
ϕ L 2 ( Ω ) γ Ω ˜ ϕ 0 , Ω ϕ H 0 1 ( Ω ) ,
ϕ L ( Ω ) c 1 ˜ ϕ 0 , Ω 1 2 ϕ 2 , Ω 1 2 ϕ ( H 2 ( Ω ) H 0 1 ( Ω ) ) ,
ϕ L 6 ( Ω ) c 1 ˜ ϕ 0 , Ω ϕ H 0 1 ( Ω ) ,
where N = ( c 1 + c 1 3 2 ) ( 1 + γ Ω 1 2 ) .
In fact, using (33)–(35) and the Green formula, we have
( B ( u , v ) , w ) Ω = 0.5 ( u · ˜ v , w ) Ω + 0.5 ( u · ˜ v , w ) Ω + 0.5 ( ˜ · u v , w ) Ω = 0.5 ( u · ˜ v , w ) Ω + 0.5 i , j = 1 3 ( ˜ i ( u i v j ) , w j ) Ω = 0.5 ( u · ˜ v , w ) Ω 0.5 i , j = 1 3 ( u i ˜ i w j , v j ) Ω = 0.5 ( u · ˜ v , w ) Ω 0.5 ( u · ˜ w , v ) Ω = ( u , B ( v , w ) ) Ω , | ( B ( u , v ) , w ) Ω | 0.5 u L 3 ( ˜ v 0 , Ω w L 6 + ˜ w 0 , Ω v L 6 ) 0.5 u L 2 1 2 u L 6 1 2 ( ˜ v 0 , Ω w L 6 + ˜ w 0 , Ω v L 6 ) γ Ω 1 2 c 1 3 2 ˜ u 0 , Ω ˜ v 0 , Ω ˜ w 0 , Ω u , v , w X , B ( u , v ) 0 , Ω u L 6 ˜ v L 3 + ˜ u 0 , Ω v L u L 6 ˜ v L 2 1 2 ˜ v L 6 1 2 + c 1 ˜ u 0 , Ω ˜ v 0 , Ω 1 2 v 2 , Ω 1 2 ( c 1 + c 1 3 2 ) ˜ u 0 , Ω ˜ v 0 , Ω 1 2 v 2 , Ω 1 2 , B ( v , w ) 0 , Ω 0.5 ( ˜ v 0 , Ω w L + ˜ w L 3 v L 6 ) 0.5 c 1 ˜ v 0 , Ω ˜ w L 2 1 2 w 2 , Ω 1 2 + 0.5 c 1 3 2 ˜ v 0 , Ω ˜ w 0 , Ω 1 2 w 2 , Ω 1 2 0.5 ( c 1 + c 1 3 2 ) ˜ u 0 , Ω ˜ v 0 , Ω 1 2 v 2 , Ω 1 2 ,
which yield (30)–(32).
Now we discuss the existence, uniqueness and regularity results of the solution ( u , p ) based on (29).
Theorem 1.
If F H 1 ( Ω ) 3 and the uniqueness index σ = N F 1 , Ω ν 2 satisfies the uniqueness condition 0 < σ < 1 , then the 3D steady Navier–Stokes equations admit a unique solution ( u , p ) satisfying the following bound:
ν ˜ u 0 , Ω F 1 , Ω , β p 0 , Ω 3 F 1 , Ω ,
and if F L 2 ( Ω ) 3 , then there holds the following regularity result:
ν u 2 , Ω + p 1 , Ω C 0 F 0 , Ω ,
where C 0 = 2 c 0 + c 0 2 γ Ω .
Proof. 
For the weak formulation (29), the existence of the solution ( u , p ) satisfying (36) can be proved by (30)–(31), the uniqueness condition and the Galerkin spectral method referring to [1] or the Galerkin finite element method referring to Section 3. Now, we let ( u 1 , p 1 ) X × M and ( u 2 , p 2 ) X × M be the solutions of (29). Then, ( w , η ) = ( u 1 u 2 , p 1 p 2 ) satisfies the following relation
ν A ( w , v ) d ( v , η ) d ( w , q ) + ( B ( w , u 1 ) , v ) Ω + ( B ( u 2 , w ) , v ) Ω = 0 ,
By taking ( v , q ) = ( w , η ) in (38) and using (30)–(31) and (36), we obtain
ν ˜ w 0 , Ω 2 = ( B ( w , u 1 ) , w ) Ω N ˜ u 1 0 , Ω ˜ w 0 , Ω 2 σ ν ˜ w 0 , Ω 2 ,
which, with the uniqueness condition 0 < σ < 1 , yields w = 0 . Using again Lemma 1 and (38) with w = 0 , we deduce η = 0 . Thus, the uniqueness of the solution ( u , p ) of (29) is proved. Moreover, if F ( L 2 ( Ω ) ) 3 , we deduce from (25), (32) and (36) that
ν u 2 , Ω + p 1 , Ω c 0 F 0 , Ω + c 0 B ( u , u ) 0 , Ω c 0 F 0 , Ω + c 0 N ˜ u 0 , Ω 3 2 u 2 , Ω 1 2 ν 2 u 2 , Ω + c 0 F 0 , Ω + 0.5 c 0 2 N 2 ν 1 ˜ u 0 , Ω 3 ν 2 u 2 , Ω + c 0 F 0 , Ω + 0.5 c 0 2 ( N 2 ν 4 F 1 , Ω 2 ) F 1 , Ω ν 2 u 2 , Ω + c 0 F 0 , Ω + 0.5 c 0 2 γ Ω F 0 , Ω ,
which yield (37). The proof ends. □
Setting u 1 = 0 , we define the iterative solution ( u n , p n ) by the 3D steady Oseen iterative equations (the 3D steady linearized Navier–Stokes equations):
ν Δ ˜ u n + ˜ p n + B ( u n 1 , u n ) = F , in Ω ,
˜ · u n = 0 , in Ω ,
u n = 0 , on Ω ,
where u n = ( u 1 n , u 2 n , u 3 n ) = ( w n , u 3 n ) . Using the Green formula, we deduce the weak formulation of the 3D steady Oseen iterative Equations (41)–(43): we find ( u , p ) X × M such that for each ( v , q ) X × M there holds
ν A ( u n , v ) d ( v , p n ) d ( u n , q ) + ( B ( u n 1 , u n ) , v ) Ω = ( F , v ) Ω ,
or
G n 1 ( ( u n , p n ) , ( v , q ) ) = ( F , v ) Ω ,
where
G n 1 ( ( u , p ) , ( v , q ) ) = A n 1 ( u , v ) d ( v , p ) d ( u , q ) , A n 1 ( u , v ) = ν A ( u , v ) + ( B ( u n 1 , u ) , v ) Ω .
In order to prove the existence, uniqueness and stability of the solution ( u n , p n ) based on (45), we consider the inf-sup condition of the general bilinear form G n 1 ( ( u , p ) , ( v , q ) ) .
Lemma 3.
If the bilinear form A n 1 ( u , v ) satisfies
A n 1 ( u , v ) c 2 ν ˜ u 0 , Ω ˜ v 0 , Ω ,
A n 1 ( u , u ) ν ˜ u 0 , Ω 2 ,
then there exists a β 1 > 0 such that
G n 1 ( ( u , p ) , ( v , q ) ) c 2 ( ν ˜ u 0 , Ω + 1 ν p 0 , Ω ) ( ν ˜ v 0 , Ω + 1 ν q 0 , Ω ) ,
β 1 ( ν ˜ u 0 , Ω + 1 ν p 0 , Ω ) sup ( v , q ) X × M G n 1 ( ( u , p ) , ( v , q ) ) ν ˜ v 0 , Ω + 1 ν q 0 , Ω ,
β 1 ( ν ˜ v 0 , Ω + 1 ν q 0 , Ω ) sup ( u , p ) X × M G n 1 ( ( u , p ) , ( v , q ) ) ν ˜ u 0 , Ω + 1 ν p 0 , Ω ,
where c 2 3 .
Proof. 
First, using (46) and (47), we deduce that G n 1 ( ( u , p ) , ( v , q ) ) satisfies (48) and the following inequality
G n 1 ( ( u , p ) , ( u , p ) ) ν ˜ u 0 , Ω 2 ,
for any ( u , p ) , ( v , q ) X × M .
Using Lemma 1, for each ( u , p ) X × M , we set ε = c 2 2 β 0 2 and ( v ˜ , q ˜ ) = ( u ν 1 ε u ˜ , p ) , where u ˜ satisfies
d ( u ˜ , p ) = p 0 , Ω 2 , ˜ u ˜ 0 , Ω β 0 1 p 0 , Ω .
Thus, we have
sup ( v , q ) X × M G n 1 ( ( u , p ) , ( v , q ) ) ν ˜ v 0 , Ω + 1 ν q 0 , Ω G n 1 ( ( u , p ) , ( v ˜ , q ˜ ) ) ν v ˜ 0 , Ω + 1 ν q ˜ 0 , Ω ν ˜ u 0 , Ω 2 c 2 ε ˜ u 0 , Ω ˜ u ˜ 0 , Ω + ε ν 1 d ( u ˜ , p ) ν ˜ ( u ε ν 1 u ˜ ) 0 , Ω + 1 ν q ˜ 0 , Ω ν ˜ u 0 , Ω 2 c 2 ε β 0 1 ˜ u 0 , Ω p 0 , Ω + ε ν 1 p 0 , Ω 2 ν ˜ u 0 , Ω + ( 1 + ε β 0 1 ) 1 ν p 0 , Ω 0.5 ν ˜ u 0 , Ω 2 + 0.5 ε ν 1 p 0 , Ω 2 ν ˜ u 0 , Ω + ( 1 + ε β 0 1 ) 1 ν p 0 , Ω β 1 ( ν ˜ u 0 , Ω + 1 ν p 0 , Ω ) ,
which is (49). Similarly, we can prove (50). The proof ends. □
Furthermore, we obtain the following regularity and convergence results of the Oseen iterative solution ( u n , p n ) .
Theorem 2.
If F H 1 ( Ω ) 3 and 0 < σ < 1 , then (41)–(43) admits a unique solution ( u n , p n ) X × M such that
ν ˜ u n 0 , Ω F 1 , Ω , β p n 0 , Ω 3 F 1 , Ω
and
ν ˜ ( u n u ) 0 , Ω σ n + 1 F 1 , Ω , β p n p 0 , Ω 3 σ n + 1 F 1 , Ω .
Furthermore, if F L 2 ( Ω ) 3 , then there holds the following regularity result:
ν u n 2 , Ω + p n 1 , Ω C 0 F 0 , Ω .
Proof. 
For n = 0 and u 1 = 0 , we deduce from (44) that ( u 0 , p 0 ) X × M satisfies
ν A ( u 0 , v ) d ( v , p 0 ) d ( ν u 0 , q ) = ( F , v ) Ω ,
for each ( v , q ) X × M . Using Lemma 2, we show the existence and uniqueness of the solution ( u 0 , p 0 ) satisfying (53) of (56). Moreover, using (25), (31) and Lemma 1, we easily show that ( u 0 , p 0 ) X × M satisfies (54) and (55). Thus, we show that Theorem 2 holds for n = 0 .
Now, assuming that the conclusions of Theorem 2 hold for n 1 , we want to prove that Theorem 2 holds for n. Using (31) and the induction assumption for n 1 , we deduce
| A n 1 ( u , v ) | ν u X v X + N u n 1 X u X v X 2 ν u X v X ,
A n 1 ( u , u ) ν u X 2 ,
for any u , v X . Using Lemma 2 and the general Lax–Miligram theorem, we deduce that (41)–(43) admits a unique solution ( u n , p n ) X × M which satisfies
ν u n X F 1 , Ω .
Using again Lemma 1, (59), (45) with q = 0 and the induction assumption for n 1 , we deduce
β p n 0 , Ω sup v X d ( v , p n ) v X F 1 , Ω + ν u n X + N u n 1 X u n X 2 F 1 , Ω + N ν 2 F 1 , Ω 2 3 F 1 , Ω .
Thus, (59) and (60) yield (53).
Using again (25), (32), (53) and the induction assumptions on n 1 , we have
ν u n 2 , Ω + p n 1 , Ω c 0 F 0 , Ω + c 0 B ( u n 1 , u n ) 0 , Ω c 0 F 0 , Ω + 0.5 c 0 N ( u n 1 X 1 2 u n 1 2 , Ω 1 2 u n X + u n 1 X u n X 1 2 u n 2 , Ω 1 2 ) c 0 F 0 , Ω + 0.25 ν u n 2 , Ω + 0.25 ν u n 1 2 , Ω + 1 8 ν 1 N 2 c 0 2 ( u n 1 X u n X 2 + u n 1 X 2 u n X ) c 0 F 0 , Ω + 1 4 ν u n 2 , Ω + 1 4 C 0 F 0 , Ω + 1 4 ν 4 N 2 c 0 2 F 1 , Ω 3 ,
which, with the uniqueness condition, imply (55).
Next, it follows from (44) and (29) that
ν A ( u n u , v ) + ( B ( u n 1 u , u n ) , v ) Ω + ( B ( u , u n u ) , v ) Ω d ( v , p n p ) d ( u n u , q ) = 0 ,
Taking ( v , q ) = ( u n u , p n + p ) X × M in (62) and using (30) and (31) yields
ν u n u X N u n 1 u X u n X N ν 2 F 1 , Ω ν u n 1 u X σ ν u n 1 u X ,
which, with the induction assumption for n 1 , yield
ν u n u X σ n + 1 F 1 , Ω .
Finally, using (31), (62) and Lemma 1, we obtain
β p n p 0 , Ω ν u n u X + N u n 1 u X u n X + N u n u X u X ν u n u X + N ν 2 F 1 , Ω ν ( u n u X + ν u n 1 u X ) .
Combining (63) and (64) and using the induction assumption for n 1 yields (54). Hence, Theorem 2 holds for n. □

3. The Finite Element Method for the 3D Steady Navier-Stokes Equations

In this section, we design a finite element method for the 3D steady Stokes equations, steady Navier–Stokes equations and the Oseen iterative equations. In addition, we provide the existence, uniqueness and stability of the finite element solutions u h , ( u h , p h ) and ( u h n , p h n ) on the above equations based on the finite element space pair X h × M h .
Let τ h = { K } be quasi-uniformly regular partition made of tetrahedra with diameters bounded by h of Ω . Define the finite element subspaces S h and S h b of H 1 ( Ω ) based on P 1 and P 1 b elements as follows:
S h = { v h C ( Ω ¯ ) H 1 ( Ω ) ; v h | K P 1 ( K ) , K τ h } ,
S h b = { v h C ( Ω ¯ ) H 1 ( Ω ) ; v h | K P 1 b ( K ) , K τ h } ,
where P 1 b is a bubble element on K and satisfies P 1 b ( K ) = P 1 ( K ) span b ^ ( K ) and b ^ ( K ) is a bubble function on K.
For the 3D steady Stokes equation, Navier–Stokes equations and Oseen iterative equations, we define the finite element subspace pair X h × M h of X × M as
X h = ( S h b H 0 1 ( Ω ) ) 3 , M h = S h M ,
S h b H 0 1 ( Ω ) = span { ϕ 1 , ϕ 2 , , ϕ m } , M h = S h M = span { ψ 1 , ψ 2 , , ψ l } .
Remark 1.
From [39,40,41], the finite element space pair X h × M h satisfies the discrete inf-sup condition.
We easily deduce that the above finite element spaces satisfy the following standard assumption:
  • There exist the mappings π h L ( X ; X h ) such that
    π h v v 0 , Ω + h ( π h v v ) 0 , Ω c 3 h l v H l , l = 1 , 2 ,
    for each v X ( H l ( Ω ) ) d with d = 1 or 3.
  • The L 2 -orthogonal projection operator ρ h : L 2 ( Ω ) S h satisfies:
    ϕ ρ h ϕ 0 , Ω c 3 h l ϕ H l ϕ H l ( Ω ) , l = 1 , 2 .
  • The inverse inequality holds:
    ˜ ϕ h 0 , Ω c 3 h 1 ϕ h 0 , Ω ϕ h S h .
  • There exists a constant β 0 > 0 such that
    sup v h X h d ( v h , q h ) v h X β 0 q h 0 , Ω q h M h .
FE method of the 3D Stokes equations.
Referring to the weak formulation (4), we design the FE method of the 3D steady Stokes equations as follows: find ( u h , p h ) X h × M h such that for each ( v h , q h ) X h × M h there holds
A ( u h , v h ) d ( v h , p h ) d ( u h , q h ) = ( F , v h ) Ω .
Lemma 4.
If F H 1 ( Ω ) 3 and the finite element space pair X h × M h satisfies (68), then (69) admits a unique solution ( u h , p h ) X h × M h , satisfying the following bound:
˜ u h 0 , Ω F 1 , Ω , β 0 p h 0 , Ω 2 F 1 , Ω .
Proof. 
We introduce the subspace V h of X h as follows:
V h = { v h X h ; d ( v h , q h ) = 0 , q h M h } .
Thus, we deduce from (69) that u h V h satisfies
A ( u h , v h ) = ( F , v h ) Ω , v h V h .
Using the Lax–Miligram theorem with X = V h , A ˜ ( u , v ) = A ( u h , v h ) and (22), we show that (69) admits a unique solution u h V h , satisfying
˜ u h 0 , Ω F 1 , Ω .
Now, we introduce a Polar set
V h 0 = { g X h ; < g , v h > = 0 v h V h } ,
and define two dual operators B h v h M h and B h q h X h such that
d ( v h , q h ) = ( B h v h , q h ) Ω = < v h , B h q h > ( v h , q h ) X h × M h .
Thus, referring to [1,2], we know that inf-sup condition (68) implies that B h is a isomorphic operator from M h onto V h 0 . Moreover, we deduce from (71) that Δ ˜ h u h F V h 0 . Thus, there exists a unique p h M h such that Δ ˜ h u h F = B h p h or
A ( u h , v h ) d ( v h , p h ) = ( F , v h ) Ω , v h X h ,
where the discrete Laplace operator Δ h u h X h is defined as
( Δ h u h , v h ) = A ( u h , v h ) , v h X h .
Thus, we have proved that ( u h , p h ) X h × M h is a unique solution of (69).
Using again (68) and (69) with q = 0 , we have
β 0 p h 0 , Ω ˜ u h 0 , Ω + F 1 , Ω 2 F 1 , Ω .
Combining (74) with (72) yields (70).
The proof ends. □
Due to (68), we can consider the weak formulation of the general Stokes equations: find ( u h , p h ) : X h × M h such that for each ( v h , q h ) X h × M h , there holds
A ˜ ( u h , v h ) d ( v h , p h ) d ( u h , q h ) = ( F , v h ) Ω .
Lemma 5.
If F X and X h × M h satisfies the discrete inf-sup condition (68), the bilinear form A ˜ ( u h , v h ) satisfies
A ˜ ( u h , v h ) c 4 ν u h X v h X ,
A ˜ ( v h , v h ) c 5 ν v h X 2 ,
then (75) admits a unique solution ( u h , p h ) X h × M h such that
c 5 ν u h X F 1 , Ω , β 0 p h 0 , Ω ( c 4 c 5 1 + 1 ) F 1 , Ω ,
where c 4 3 .
Proof. 
First, we deduce from (75) that u h V h satisfies
A ˜ ( u h , v h ) = ( F , v h ) Ω , v h V h ,
or
( A ˜ h u h , v h ) = ( F , v h ) Ω , v h V h .
Using (76) and (77) and the Lax–Miligram theorem with X = V h and A ˜ ( u , v ) = A ˜ ( u h , v h ) , we show that (79) or (80) admits a unique solution u h V h satisfying
c 5 ν u h X F 1 , Ω .
Next, (80) shows A ˜ h u h F V h 0 . Thus, referring to [1,2], the inf-sup condition (68) implies that B h is an isomorphic operator from M h onto V h 0 . Thus, there exists a unique p h M h such that A ˜ h u h F = B h p h or
A ˜ ( u h , v h ) d ( v h , p h ) = ( F , v h ) Ω , v h X h .
Thus, we have proved that ( u h , p h ) X h × M h is a unique solution of (75).
Using again (68), (76) and (75) with q h = 0 , we have
β 0 p h 0 , Ω c 4 ν u h X + F 1 , Ω ( 1 + c 4 c 5 1 ) F 1 , Ω .
Combining (83) with (81) yields (78).
The proof ends. □
FE method of the 3D Navier-Stokes equations.
Referring to the weak formulation (29), we design the FE method of the 3D steady Navier–Stokes equations as follows: find ( u h , p h ) X h × M h such that for each ( v h , q h ) X h × M h , there holds
ν A ( u h , v h ) d ( v h , p h ) d ( u h , q h ) + ( B ( u h , u h ) , v h ) Ω = ( F , v h ) Ω .
Lemma 6.
If F X , the finite element space pair X h × M h satisfies (68) and 0 < σ < 1 , then (84) admits a unique solution ( u h , p h ) X h × M h satisfying the following bound:
ν ˜ u h 0 , Ω F 1 , Ω , β 0 p h 0 , Ω 3 F 1 , Ω .
Proof. 
We set a bounded convex subset K of X h × M h as
K = { ( v h , q h ) X h × M h ; ν v h 0 , Ω F 1 , Ω , β 0 q h M 3 F 1 , Ω } ,
and define a map T : K X h × M h such that for each ( w h , r h ) K , ( u h , p h ) = T ( w h , r h ) X h × M h satisfies
ν A ( u h , v h ) + ( B ( w h , u h ) , v h ) Ω d ( v h , p h ) d ( u h , q h ) = ( F , v h ) Ω , ( v h , q h ) X h × M h .
or
A ˜ ( w h ; u h , v h ) d ( v h , p h ) d ( u h , q h ) = ( F , v h ) Ω , v h V h .
where
A ˜ ( w h ; u h , v h ) = ν A ( u h , v h ) + ( B ( w h , u h ) , v h ) Ω .
Due to ( w h , r h ) K , we deduce from (30) and (31) and the uniqueness condition that A ˜ ( w h ; u h , v h ) satisfies
| A ˜ ( w h ; u h , v h ) | ν | A ( u h , v h ) | + N w h X u h X v h X 2 ν u h X v h X ,
A ˜ ( w h ; u h , u h ) = ν u h X 2 ,
for each u h , v h X h . Using Lemma 5, we show that (87) or (88) admits a unique solution ( u h , p h ) satisfying
ν u h X F 1 , Ω , β 0 p h 0 , Ω 3 F 1 , Ω .
Thus, (91) shows that T is a map from K into K. Using the fixed point theorem in finite dimensional space, the map T at least has a fixed point ( u h , p h ) K such that T ( u h , p h ) = ( u h , p h ) or (84) at least admits a solution ( u h , p h ) K . Now, we assume that ( u h 1 , p h 1 ) K and ( u h 2 , p h 2 ) K satisfy (84). Then ( w h , r h ) = ( u h 1 u h 2 , p h 1 p h 2 ) satisfies
ν A ( w h , v h ) + ( B ( w h , u h 1 ) , v h ) Ω + ( B ( u h 2 , w h ) , v h ) Ω d ( v h , r h ) d ( w h , q h ) = 0 , ( v h , q h ) X h × M h .
for each ( v h , q h ) X h × M h . Taking ( v h , q h ) = ( w h , r h ) in (92) and using (30) and (31), we deduce
ν w h X 2 N w h 0 , Ω 2 u h 1 X σ ν w h X 2 .
Thanks to 0 < σ < 1 , (93) yields w h = 0 or u h 1 = u h 2 . Next, using (68) and (93) with q h = 0 , we deduce r h = 0 . The proof ends. □
Lemma 7.
If F H 1 ( Ω ) 3 , 0 < σ < 1 , the finite element space pair X h i × M h i is dense in X × M , satisfies (68) and X h i × M h i X h i + 1 × M h i + 1 with lim i h i = 0 , then the finite element solutions ( u h i , p h i ) X h i × M h i based on (84) satisfying
( u h i , p h i ) i s   w e a k   c on v e r g e n t   t o ( u , p ) in   X × M   a s   i ,
here ( u , p ) X × M satisfies (29).
Proof. 
We deduce from Lemma 5 that for each 0 < h < 1 (84) admits a unique solution ( u h , p h ) X h × M h satisfying (85). Applying the compact theorem in X × M , there exist a sequence { h i } and ( u , p ) such that (94) holds. Thanks to X ( L 2 ( Ω ) ) 3 being compact, u h i is strong convergent to u in ( L 2 ( Ω ) ) 3 .
Thus, for a fixed i 0 and i i 0 , there hold
( B ( u h i , u h i ) , v h i 0 ) Ω ( B ( u , u ) , v h i 0 ) Ω = ( B ( u h i u , u h i ) , v h i 0 ) Ω + ( B ( u , u h i u ) , v h i 0 ) Ω 0.5 u h i u L 3 ( Ω ) ( ˜ u h i 0 , Ω v h i 0 L 6 ( Ω ) + u h i L 6 ( Ω ) ˜ v h i 0 0 , Ω ) + ( u L 6 ( Ω ) ˜ v h i 0 0 , Ω + ˜ u 0 , Ω v h i 0 L 6 ( Ω ) ) u h i u L 3 ( Ω ) c 1 3 2 u h i u L 2 ( Ω ) 1 2 ˜ ( u h i u ) 0 , Ω 1 2 ˜ u h i 0 , Ω ˜ v h i 0 0 , Ω + 2 c 1 3 2 u h i u L 2 ( Ω ) 1 2 ˜ ( u h i u ) 0 , Ω 1 2 ˜ u 0 , Ω ˜ v h i 0 0 , Ω .
Thus, using (94) and (95), we deduce
lim i ν A ( u h i , v h i 0 ) = ν A ( u , v h i 0 ) ,
lim i d ( v h i 0 , p h i ) = d ( v h i 0 , p ) ,
lim i d ( u h i , q h i 0 ) = d ( u , q h i 0 ) ,
lim i ( B ( u h i , u h i ) , v h i 0 ) Ω = ( B ( u , u ) , v h i 0 ) Ω .
Setting ( u h , p h ) = ( u h i , p h i ) X h i × M h i and ( v h , q h ) = ( v h i 0 , q h i 0 ) X h i 0 × M h i 0 in (84), we obtain
ν A ( u h i , v h i 0 ) d ( v h i 0 , p h i ) d ( u h i , q h i 0 ) + ( B ( u h i , u h i ) , v h i 0 ) Ω = ( F , v h i 0 ) Ω .
Setting i in (100) and using (96)–(99), we obtain
ν A ( u , v h i 0 ) d ( v h i 0 , p ) d ( u , q h i 0 ) + ( B ( u , u ) , v h i 0 ) Ω = ( F , v h i 0 ) Ω .
Since X h i 0 × M h i 0 is dense in X × M , setting i 0 in (101), we deduce that ( u , p ) X × M satisfies (29).
The proof ends. □

4. Uniform Error Estimates of FE Solutions

In this section, we provide the error estimate of ( u h , p h ) with respect to ( u , p ) .
For the FE solution ( u h , p h ) X h × M h of the Stokes equations, there hold the following error estimates.
Lemma 8.
If F L 2 ( Ω ) and X h satisfy (65), then the FE solution ( u h , p h ) of the Stokes equations satisfies the following error estimates:
u u h 0 , Ω + h u u h X C h 2 F 0 , Ω , β 0 p p h 0 , Ω C h F 0 , Ω .
Proof. 
First, we deduce from (4) and (69) that
A ( u u h , v h ) d ( v h , p p h ) d ( u u h , q h ) = 0 ( v h , q h ) X h × M h .
Using (103) with q h = 0 and (68), we deduce
β 0 ρ h p p h 0 , Ω u u h X + 3 p ρ h p 0 , Ω .
Next, taking ( v h , q h ) = ( π h u u h , ρ h p + p h ) in (103), using (104), (65) and (66) and (25), we deduce
1 2 u u h X 2 + 1 2 π h u u h X 2 = 1 2 π h u u X 2 + d ( π h u u h , p ρ h p ) + d ( u π h u , ρ h p p h ) 1 2 π h u u X 2 + 3 ( π h u u h X p ρ h p 0 , Ω + u π h u X ρ h p p h 0 , Ω ) 1 2 π h u u X 2 + 3 π h u u h X p ρ h p 0 , Ω + 3 β 0 1 u π h u X ( u u h X + 3 ρ h p p 0 , Ω ) 1 2 π h u u h X 2 + 1 4 u u h X 2 + c ( p ρ h p 0 , Ω 2 + u π h u X 2 ) 1 2 π h u u h X 2 + 1 4 u u h X 2 + c h 2 ( u 2 , Ω 2 + p 1 , Ω 2 ) 1 2 π h u u h X 2 + 1 4 u u h X 2 + c h 2 F 0 , Ω 2 .
Combining (104) and (105) and (66), we deduce
u u h X + β 0 p p h 0 , Ω C h F 0 , Ω .
In order to estimate the L 2 ( Ω ) bound of the error estimate u u h , we consider the dual equation of the 3D Stokes equations with the Dirichlet boundary condition
Δ ˜ ϕ + ˜ ψ = u u h , in Ω ,
˜ · ϕ = 0 , in Ω ,
ϕ = 0 , on Ω .
Using the Green formula, we deduce the weak formulation of dual Equations (107) and (109): find ϕ X such that for each v X there holds
A ( ϕ , v ) d ( v , ψ ) d ( ϕ , q ) = ( u u h , v ) Ω .
Using again (25), we have
ϕ 2 , Ω + ψ 1 , Ω c 0 u u h 0 , Ω .
Taking ( v , q ) = ( u u h , p p h ) in (110), using (103) and (65) and (66) and (111), we deduce
u u h 0 , Ω 2 = A ( u u h , ϕ ) d ( u u h , ψ ) d ( ϕ , p p h ) = A ( u u h , ϕ π h ϕ ) d ( u u h , ψ ρ h ψ ) d ( ϕ π h ϕ , p p h ) u u h X ϕ π h ϕ X + 3 ( u u h X ψ ρ h ψ 0 , Ω + ϕ π h ϕ X p p h 0 , Ω ) c h u u h X ϕ 2 , Ω + c h u u h X ψ 1 , Ω + c h ϕ 2 , Ω p p h 0 , Ω c h ( u u h X + p p h 0 , Ω ) ( ϕ 2 , Ω + ψ 1 , Ω ) c h ( u u h X + p p h 0 , Ω ) u u h 0 , Ω .
Combining (112) with (106), we obtain (102).
The proof ends. □
Thanks to (68), we can define the Stokes projection ( R h , Q h ) : V × M X h × M h such that for each ( u , p ) V × M , ( R h , Q h ) = ( R h ( u , p ) , Q h ( u , p ) ) X h × M h satisfies
( R h , v h ) Ω d ( v h , Q h ) d ( R h , q h ) = ( u , v h ) Ω d ( v h , p ) d ( u , q h ) ,
for each ( v h , q h ) X h × M h . From Lemma 4 and Lemma 8, we have
R h X 3 ( u X + p 0 , Ω ) , β 0 Q h 0 , Ω ( 1 + 3 ) ( u X + p 0 , Ω ) ( u , p ) V × M , R h ( u , p ) u 0 , Ω + h R h ( u , p ) u X + h β 0 Q h ( u , p ) p 0 , Ω
c 6 h 2 ( u 2 , Ω + p 1 , Ω ) ( u , p ) ( V ( H 2 ( Ω ) ) 3 ) × ( M H 1 ( Ω ) ) .
For the FE solution ( u h , p h ) X h × M h of the Navier–Stokes equations, there hold the following error estimates.
Lemma 9.
If F L 2 ( Ω ) and X h × M h satisfies (65)–(68), 0 < σ < 1 and σ h 1 2 1 σ , then the FE solution ( u h , p h ) satisfies the following error estimates:
ν u u h X 2 c 6 C 0 h F 0 , Ω , β 0 p p h 0 , Ω 4 c 6 C 0 h F 0 , Ω , ν u u h 0 , Ω C 1 σ h 2 F 0 , Ω .
Proof. 
First, we deduce from (4), (69) and (113) that
ν A ( R h u h , v h ) d ( v h , Q h p h ) d ( R h u h , q h ) + ( B ( u u h , u ) , v h ) Ω + ( B ( u h , u u h ) , v h ) Ω = 0 ,
or
ν A ( R h u h , v h ) d ( v h , Q h p h ) d ( R h u h , q h ) + ( B ( R h u h , u ) , v h ) Ω + ( B ( u h , R h u h ) , v h ) Ω = ( B ( R h u , u ) , v h ) Ω + ( B ( u h , R h u ) , v h ) Ω ( v h , q h ) X h × M h ,
where ( R h , Q h ) = ( R h ( u , p ) , Q h ( u , p ) ) . Using (118) with q h = 0 , (31), (68) and the uniqueness condition 0 < σ < 1 , we deduce
β 0 Q h p h 0 , Ω ν R h u h X + N R h u h X ( u X + u h X ) + N R h u 0 , Ω 1 2 R h u X 1 2 ( u X + u h X ) 3 ν R h u h X + 2 σ c 6 h 3 2 ( ν u 2 , Ω + p 1 , Ω ) 3 ν R h u h X + 2 σ c 6 h 3 2 C 0 F 0 , Ω .
Next, taking ( v h , q h ) = ( R h u h , Q h + p h ) in (118), using (119), (65) and (66), (30) and (31) and (37), we deduce
ν ( 1 σ ) R h u h X 2 = ( B ( u R h , u ) , R h u h ) Ω ( B ( u h , u R h ) , R h u h ) Ω 0.5 u R h L 3 ( ˜ u 0 , Ω R h u h L 6 + ˜ ( R h u u h ) 0 , Ω u L 6 ) + u R h L 3 ( u h L 6 ˜ ( R h u h ) 0 , Ω + ˜ u h 0 , Ω R h u u h L 6 ) 2 c 1 3 2 u R h 0 , Ω 1 2 u R h X 1 2 ( u X + u h X ) R h u h X 2 c 1 3 2 c 6 h 3 2 ν 1 ( ν u 2 , Ω + p 1 , Ω ) ( u x + u h X ) R h u h X N c 6 h 3 2 ν 2 ( ν u 2 , Ω + p 1 , Ω ) ( ν u x + ν u h X ) R h u h X 2 σ c 6 h 3 2 C 0 F 0 , Ω R h u h X .
Combining (119) and (120) and (65) and (66), we deduce
ν R h u h X 2 c 6 C 0 h 3 2 σ 1 σ F 0 , Ω , β 0 Q h p h 0 , Ω 5 c 6 C 0 h 3 2 σ 1 σ F 0 , Ω .
Combining (121) with (115) and using the stability condition σ h 1 2 1 σ , we obtain
ν u u h X C h F 0 , Ω , β 0 p p h 0 , Ω C h F 0 , Ω .
In order to estimate the L 2 ( Ω ) bound of the error estimate u u h , we consider the dual equation of the 3D Navier–Stokes equations with the Dirichlet boundary condition
ν Δ ˜ ϕ + ˜ ψ + B ( u , ϕ ) B ( u h , ϕ ) = u u h , in Ω ,
˜ · ϕ = 0 , in Ω ,
ϕ = 0 , on Ω ,
where ( B ( v , u ) , ϕ ) Ω = ( v , B ( u , ϕ ) ) Ω .
Using the Green formula, we deduce the weak formulation of the dual Equations (123) and (125): Find ( ϕ , ψ ) X × M such that for each ( v , q ) X × M there holds
ν A ( ϕ , v ) d ( v , ψ ) d ( ϕ , q ) + ( B ( v , u ) , ϕ ) Ω + ( B ( u h , v ) , ϕ ) Ω = ( u u h , v ) Ω .
Setting
A ˜ ( ϕ , v ) = ν A ( ϕ , v ) + ( B ( v , u ) , ϕ ) Ω + ( B ( u h , v ) , ϕ ) Ω ,
we deduce from (30) and (31) that
| A ˜ ( ϕ , v ) | ν ϕ X v X + N v X ϕ X ( u X + u h X ) 3 ν ϕ X v X ,
A ˜ ( ϕ , ϕ ) = ν A ( ϕ , ϕ ) + ( B ( ϕ , u ) , ϕ ) Ω ( 1 σ ) ν ϕ X 2 .
Using Lemma 3 with A n 1 ( u , v ) = A ˜ ( ϕ , v ) , we show that (126) admits a unique solution ( ϕ , ψ ) X × M . Using (126)–(128) and Lemma 1, we have
ν ( 1 σ ) ϕ X u u h 1 , Ω ,
β ψ 0 , Ω u u h 1 , Ω + 3 ν ϕ X ( 1 + 3 1 σ ) u u h 1 , Ω .
Using again (25), (30)–(32) and (126), we have
ν ϕ 2 , Ω + ψ 1 , Ω c 0 u u h 0 , Ω + c 0 B ( u , ϕ ) 0 , Ω + c 0 B ( u h , ϕ ) 0 , Ω c 0 u u h 0 , Ω + 2 c 0 N ( u X + u h X ) ϕ X 1 2 ϕ 2 , Ω 1 2 1 2 ν ϕ 2 , Ω + c 0 u u h 0 , Ω + 4 c 0 2 N 2 ν 1 ( u X 2 + u h X 2 ) ϕ X 1 2 ν ϕ 2 , Ω + c 0 u u h 0 , Ω + 4 c 0 2 N 2 ν 4 F 1 , Ω 2 ν ϕ X 1 2 ν ϕ 2 , Ω + c 0 u u h 0 , Ω + 4 c 0 2 γ Ω 1 1 σ u u h 0 , Ω ,
which yields
ν ϕ 2 , Ω + ψ 1 , Ω 2 c 0 u u h 0 , Ω + 8 c 0 2 γ Ω ( 1 σ ) 1 u u h 0 , Ω .
Taking ( v , q ) = ν ( u u h , p + p h ) in (126), using (118), (132) and (65) and (66), we deduce
ν u u h 0 , Ω 2 = ν 2 A ( u u h , ϕ ) ν d ( u u h , ψ ) ν d ( ϕ , p p h ) + ν ( B ( u u h , u ) , ϕ ) Ω + ν ( B ( u h , u u h ) , ϕ ) Ω = ν 2 A ( u u h , ϕ π h ϕ ) ν d ( u u h , ψ ρ h ψ ) ν d ( ϕ π h ϕ , p p h ) + ν ( B ( u u h , u ) , ϕ π h ϕ ) Ω + ν ( B ( u h , u u h ) , ϕ π h ϕ ) Ω ν 2 u u h X ϕ π h ϕ X + 3 ( ν u u h X ψ ρ h ψ 0 , Ω + ν ϕ π h ϕ X p p h 0 , Ω ) + N u u h X ( u X + u h X ) ν ϕ π h ϕ X 3 c 3 h ( ν u u h X + p p h 0 , Ω ) ( ν ϕ 2 , Ω + ψ 1 , Ω ) + N ν u u h X ( u X + u h X ) c 3 h ϕ 2 , Ω C 1 σ h ( ν u u h X + p p h 0 , Ω ) u u h 0 , Ω .
Combining (133) with (122), we obtain (116).
The proof ends. □

5. Oseen Iterative FE Method

Referring to the nonlinearity of the weak formulation (84), we design the Oseen iterative FE method of the 3D steady Navier–Stokes equations as follows. Setting u h 1 = 0 , we define the Oseen iterative FE solution ( u h n , p h n ) of the 3D steady Navier–Stokes equations: find ( u h n , p h n ) X h × M h such that for each ( v h , q h ) X h × M h there holds
ν A ( u h n , v h ) d ( v h , p h n ) d ( u h n , q h ) + ( B ( u h n 1 , u h n ) , v h ) Ω = ( F , v h ) Ω ,
or
A n 1 ( u h n , v h ) d ( v h , p h n ) d ( u h n , q h ) = ( F , v h ) Ω ,
where
A n 1 ( u h , v h ) = ν A ( u h , v h ) + ( B ( u h n 1 , u h ) , v h ) Ω .
In order to prove the existence, uniqueness and stability of the solution ( u h n , p h n ) based on (134) or (135), we consider the continuous and elliptic condition of the bilinear form A n 1 ( u h , v h ) .
Lemma 10.
If ν u h n 1 X F 1 , Ω and 0 < σ < 1 , then the bilinear form A n 1 ( u h , v h ) satisfies
A n 1 ( u h , v h ) 2 ν u h X v h X ,
A n 1 ( u h , u h ) = ν u h X 2 ,
for each u h , v h X h .
Proof. 
Using (30) and (31), we have
A n 1 ( u h , v h ) ν u h X v h X + N u h n 1 X u h X v h X 2 ν u h X v h X , A n 1 ( u h , u h ) = ν u h X 2 + ( B ( u h n 1 , u h ) , u h ) Ω = ν u h X 2 ,
which are (136) and (137). The proof ends. □
Furthermore, we obtain the existence, uniqueness, stability and convergence results of the solution ( u h n , p h n ) based on (134) or (135).
Lemma 11.
If F H 1 ( Ω ) 3 and 0 < σ < 1 , X h × M h satisfies (68), then (134) or (135) admits a unique solution ( u h n , p h n ) X h × M h such that
ν u h n X F 1 , Ω , β 0 p h n 0 , Ω 3 F 1 , Ω ,
and
ν u h n u h X σ n + 1 F 1 , Ω , β 0 p h n p h 0 , Ω 3 σ n + 1 F 1 , Ω .
Proof. 
For n = 0 and u h 1 = 0 , we deduce from (134) that ( u h 0 , p h 0 ) X h × M h satisfies
ν A ( u h 0 , v h ) d ( v h , p h 0 ) d ( ν u h 0 , q h ) = ( F , v h ) Ω ,
for each ( v h , q h ) X h × M h .
Using Lemma 4, we show the existence and uniqueness of the solution ( u h 0 , p h 0 ) satisfying (138). Moreover, using (140) and (84), we easily show that ( u h 0 , p h 0 ) X h × M h satisfies
ν A ( u h 0 u h , v h ) d ( v h , p h 0 p h ) d ( ν u h 0 ν u h , q h ) = ( B ( u h , u h ) , v h ) Ω .
Using again Lemma 4, Lemma 6 and (141), we have
ν u h 0 u h X N u h X 2 σ F 1 , Ω , β 0 p h 0 p h 2 N u h X 2 2 σ F 1 , Ω .
Thus, we show that Lemma 11 holds for n = 0 .
Now, assuming that the conclusions of Lemma 11 hold for n 1 , we want to prove that Lemma 11 holds for n. Using the induction assumption for n 1 , Lemma 10 and Lemma 5, we deduce that (135) admits a unique solution ( u h n , p h n ) X × M which satisfies
ν u h n X F 1 , Ω , β 0 p h n 0 , Ω 3 F 1 , Ω .
Next, it follows from (134) and (84) that
ν A ( u h n u h , v h ) + ( B ( u h n 1 u h , u h n ) , v h ) Ω + ( B ( u h , u h n u h ) , v h ) Ω d ( v h , p h n p h ) d ( u h n u h , q h ) = 0 .
Taking ( v h , q h ) = ( u h n u h , p h n + p h ) X h × M h in (144) and using (30) and (31) yields
ν u h n u h X N u h n 1 u h X u h n X N ν 2 F 1 , Ω ν u h n 1 u h X σ ν u h n 1 u h X ,
which, with the induction assumption for n 1 , yields
ν u h n u h X σ n + 1 F 1 , Ω .
Finally, using (31), (145) and Lemma 6, we obtain
β 0 p h n p h 0 , Ω ν u h n u h X + N u h n 1 u h X u h n X + N u h n u h X u h X ν u h n u h X + N ν 2 F 1 , Ω ν ( u h n u h X + ν u h n 1 u h X ) 2 ν u h n u h X + σ ν u h n 1 u h X 3 σ n + 1 F 1 , Ω .
Combining (146) and (145) and using the induction assumption for n 1 yields (139). Hence, Lemma 11 holds for n. The proof ends. □
Finally, by combining Lemma 11 with Lemma 9, we obtain the convergence result of the Oseen iterative FE solution ( u h n , p h n ) with respect to the exact solution ( u , p ) of the 3D steady Navier–Stokes equations.
Theorem 3.
If F L 2 ( Ω ) and X h × M h satisfies (65)–(68), 0 < σ < 1 and σ h 1 2 1 σ , then the Oseen iterative FE solution ( u h n , p h n ) satisfies the following error estimates:
ν u u h n X σ n + 1 F 1 , Ω + C h F 0 , Ω ,
β 0 p p h n 0 , Ω 3 σ n + 1 F 1 , Ω + C h F 0 , Ω ,
ν u u h n 0 , Ω σ n + 1 γ Ω F 1 , Ω + C 1 σ h 2 F 0 , Ω .

6. Newton Iterative FE Method

In this section, referring to the nonlinearity of the weak formulation (84), we design the Newton iterative FE method of the 3D steady Navier–Stokes equations as follows. Setting u h 1 = 0 , we define the Newton iterative FE solution ( u h n , p h n ) of the 3D steady Navier–Stokes equations: find ( u h n , p h n ) X h × M h such that for each ( v h , q h ) X h × M h there holds
ν A ( u h n , v h ) d ( v h , p h n ) d ( u h n , q h ) + ( B ( u h n 1 , u h n ) , v h ) Ω + ( B ( u h n , u h n 1 ) , v h ) Ω ( B ( u h n 1 , u h n 1 ) , v h ) Ω = ( F , v h ) Ω ,
or
A n 1 ( u h n , v h ) d ( v h , p h n ) d ( u h n , q h ) = F ˜ ( v h ) ,
where
F ˜ ( v h ) = ( F , v h ) Ω + ( B ( u h n 1 , u h n 1 ) , v h ) Ω , A n 1 ( u h , v h ) = ν A ( u h , v h ) + ( B ( u h n 1 , u h ) , v h ) Ω + ( B ( u h , u h n 1 ) , v h ) Ω .
In order to prove the existence, uniqueness and stability of the solution ( u h n , p h n ) based on (150) or (151), we consider the continuous of the linear form F ˜ ( v h ) and the continuous and elliptic condition of the bilinear form A n 1 ( u h , v h ) .
Lemma 12.
If ν u h n 1 X 6 5 F 1 , Ω and 0 < σ 5 11 , then the bilinear form A n 1 ( u h , v h ) satisfies
| F ˜ ( v h ) | 2 F 1 , Ω v h X ,
A n 1 ( u h , v h ) 3 ν u h X v h X ,
A n 1 ( u h , u h ) σ ν u h X 2 ,
for each u h , v h X h .
Proof. 
Using (30) and (31), we have
| F ˜ ( v h ) | F 1 , Ω v h X + N u h n 1 X 2 v h X F 1 , Ω v h X ( 1 + σ 36 25 ) 2 F 1 , Ω v h X , A n 1 ( u h , v h ) ν u h X v h X + 2 N u h n 1 X u h X v h X 23 11 ν u h X v h X , A n 1 ( u h , u h ) = ν u h X 2 + ( B ( u h , u h n 1 ) , u h ) Ω σ ν u h X 2 ,
which are (152)–(154). The proof ends. □
Furthermore, we obtain the existence, uniqueness, stability and convergence results of the solution ( u h n , p h n ) based on (150) or (151).
Lemma 13.
If F H 1 ( Ω ) 3 and 0 < σ 5 11 , X h × M h satisfies (68), then (150) or (151) admits a unique solution ( u h n , p h n ) X h × M h such that
ν u h n X 6 5 F 1 , Ω ,
β 0 p h n 0 , Ω 3 F 1 , Ω ,
and
ν u h n u h X σ 2 n F 1 , Ω ,
β 0 p h n p h 0 , Ω 3 σ 2 n F 1 , Ω .
Proof. 
For n = 0 and u h 1 = 0 , we deduce from (150) that ( u h 0 , p h 0 ) X h × M h satisfies
ν A ( u h 0 , v h ) d ( v h , p h 0 ) d ( ν u h 0 , q h ) = ( F , v h ) Ω ,
for each ( v h , q h ) X h × M h .
Using Lemma 4, we show the existence and uniqueness of the solution ( u h 0 , p h 0 ) satisfying (155) and (156). Moreover, using (159) and (84), we easily show that ( u h 0 , p h 0 ) X h × M h satisfies
ν A ( u h 0 u h , v h ) d ( v h , p h 0 p h ) d ( ν u h 0 ν u h , q h ) = ( B ( u h , u h ) , v h ) Ω .
Using again Lemma 4, Lemma 6 and (160), we have
ν u h 0 u h X N u h X 2 σ F 1 , Ω ,
β 0 p h 0 p h ν u h 0 u h X + N u h X 2 2 σ F 1 , Ω ,
which yield (157) and (158) for n = 0 . Thus, we show that Lemma 13 holds for n = 0 .
Now, assuming that the conclusions of Lemma 13 hold for n 1 , we want to prove that Lemma 13 holds for n.
Next, we deduce from (150), (161), (30) and (31) and the induction assumption for n 1 that ( u h n , p h n ) and ( e n , η n ) = ( u h n u h , p h n p h ) satisfy
ν A ( u h n , v h ) d ( v h , p h n ) d ( u h n , q h ) + ( B ( u h n , u h n ) , v h ) Ω ( B ( u h n u h n 1 , u h n u h n 1 ) , v h ) Ω = 0 ,
ν A ( u h n , u h n ) ( B ( u h n u h n 1 , u h n u h n 1 ) , u h n ) Ω = ( F , u h n ) Ω ,
ν A ( e n , v h ) d ( v h , η n ) d ( e n , q h ) + ( B ( e n , u h n 1 ) , v h ) Ω + ( B ( u h n 1 , e n ) , v h ) Ω ( B ( e n 1 , e n 1 ) , v h ) Ω = 0 ,
which yield
( 1 σ ) ν e 1 X ν ( 1 N ν 1 u h 0 X ) e 1 X N e 0 X 2 N ν 2 σ 2 F 1 , Ω 2 σ 3 F 1 , Ω ( 1 σ ) 5 6 σ 2 F 1 , Ω ,
ν u h 1 u h 0 X ν ( e 1 X + e 0 X ) ( 1 + 5 6 σ ) σ F 1 , Ω F 1 , Ω ,
ν u h 1 X N u h 1 u h 0 X 2 + F 1 , Ω F 1 , Ω + N ν 2 ( 1 + 5 6 σ ) 2 σ 2 F 1 , Ω 2 [ 1 + ( 1 + 5 6 σ ) 2 σ 3 ] F 1 , Ω 6 5 F 1 , Ω ,
β 0 p h 1 0 , Ω ν u h 1 X + F 1 , Ω + N u h 0 X ( u h 1 X + u h 1 u h 0 X ) 3 F 1 , Ω ,
β 0 η 1 0 , Ω ν e 1 X + N ( 2 u h 0 X e 1 X + e 0 X 2 ) 3 σ 2 F 1 , Ω ,
for n = 1 and
σ ν e n X ν ( 1 N ν 1 u h n 1 X ) e n X
N e n 1 | X 2 N ν 2 ( σ 2 n 1 ) 2 F 1 , Ω 2 σ σ 2 n F 1 , Ω ,
ν u h n u h n 1 X ν e n e n 1 X ( σ 2 n + σ 2 n 1 ) F 1 , Ω , ν u h n X N u h n u h n 1 X 2 + F 1 , Ω
F 1 , Ω + σ ( σ 2 n + σ 2 n 1 ) 2 F 1 , Ω 6 5 F 1 , Ω ,
β 0 p h n 0 , Ω ν u h n X + F 1 , Ω + N u h n 1 X ( u h n X + u h n u h n 1 X ) 3 F 1 , Ω ,
β 0 η n 0 , Ω ν e n X + 2 N u h n 1 X e n X + N e n 1 X 2 3 σ 2 n F 1 , Ω ,
for n 2 . Thus, (166)–(175) shows that (155)–(158) hold for n.
The proof ends. □
Finally, by combining Lemma 13 with Lemma 9, we obtain the convergence result of the Newton iterative FE solution ( u h n , p h n ) with respect to the exact solution ( u , p ) of the 3D steady Navier–Stokes equations.
Theorem 4.
If F L 2 ( Ω ) and X h × M h satisfies (65)–(68) and 0 < σ 5 11 , then the Newton iterative FE solution ( u h n , p h n ) satisfies the following error estimates:
ν u u h n X σ 2 n F 1 , Ω + C h F 0 , Ω ,
β 0 p p h n 0 , Ω 3 σ 2 n F 1 , Ω + C h F 0 , Ω ,
ν u u h n 0 , Ω σ 2 n γ Ω F 1 , Ω + C h 2 F 0 , Ω .

7. Stokes Iterative FE Method

In this section, referring to the nonlinearity of the weak formulation (84), we design the Stokes iterative FE method of the 3D steady Navier–Stokes equations as follows. Setting u h 1 = 0 , we define the Stokes iterative FE solution ( u h n , p h n ) of the 3D steady Navier–Stokes equations: find ( u h n , p h n ) X h × M h such that for each ( v h , q h ) X h × M h there holds
ν A ( u h n , v h ) d ( v h , p h n ) d ( u h n , q h ) + ( B ( u h n 1 , u h n 1 ) , v h ) Ω = ( F , v h ) Ω ,
or
A n 1 ( u h n , v h ) d ( v h , p h n ) d ( u h n , q h ) = F ˜ ( v h ) ,
where
F ˜ ( v h ) = ( F , v h ) Ω ( B ( u h n 1 , u h n 1 ) , v h ) Ω , A n 1 ( u h , v h ) = ν A ( u h , v h ) .
In order to prove the existence, uniqueness and stability of the solution ( u h n , p h n ) based on (179) or (180), we consider the continuous condition of the linear form F ˜ ( v h ) and the continuous and elliptic conditions of the bilinear form A n 1 ( u h , v h ) .
Lemma 14.
If ν u h n 1 X 6 5 F 1 , Ω and 0 < σ 2 5 , then the bilinear form A n 1 ( u h , v h ) satisfies
| F ˜ ( v h ) | 2 F 1 , Ω v h X ,
A n 1 ( u h , v h ) ν u h X v h X ,
A n 1 ( u h , u h ) = ν u h X 2 ,
for each u h , v h X h .
Proof. 
The proof of (181)–(183) is very simple and can be omitted.
Furthermore, we obtain the existence, uniqueness, stability and convergence results of the solution ( u h n , p h n ) based on (179) or (180). □
Lemma 15.
If F H 1 ( Ω ) 3 and 0 < σ 2 5 , X h × M h satisfy (68), then (179) or (180) admits a unique solution ( u h n , p h n ) X h × M h such that
ν u h n X 6 5 F 1 , Ω , β 0 p h n 0 , Ω 3 F 1 , Ω ,
ν u h n u h n 1 X ( 12 5 σ ) n 1 σ F 1 , Ω ( n 1 ) ,
and
ν u h n u h X ( 11 5 σ ) n σ F 1 , Ω ,
β 0 p h n p h 0 , Ω 3 ( 11 5 σ ) n σ F 1 , Ω .
Proof. 
For n = 0 and u h 1 = 0 , we deduce from (179) that ( u h 0 , p h 0 ) X h × M h satisfies
ν A ( u h 0 , v h ) d ( v h , p h 0 ) d ( ν u h 0 , q h ) = ( F , v h ) Ω ,
for each ( v h , q h ) X h × M h .
Using Lemma 4, we show the existence and uniqueness of the solution ( u h 0 , p h 0 ) satisfying (184) and (185). Moreover, using (188) and (84), we easily show that ( u h 0 , p h 0 ) X h × M h satisfies
ν A ( u h 0 u h , v h ) d ( v h , p h 0 p h ) d ( ν u h 0 ν u h , q h ) = ( B ( u h , u h ) , v h ) Ω .
Using again Lemma 4, Lemma 6 and (189), we have
ν u h 0 u h X N u h X 2 σ F 1 , Ω ,
β 0 p h 0 p h ν u h 0 u h X + N u h X 2 2 σ F 1 , Ω .
Thus, we show that Lemma 15 holds for n = 0 .
Now, assuming that the conclusions of Lemma 15 hold for 0 , , n 1 , we want to prove that Lemma 15 holds for n.
Next, we deduce from (179), (190), (30) and (31) and the induction assumption for 0 , , n 1 that ( u h n , p h n ) and ( e n , η n ) = ( u h n u h , p h n p h ) satisfy
ν A ( u h n u h n 1 , v h ) d ( v h , p h n p h n 1 ) d ( u h n u h n 1 , q h )
+ ( B ( u h n 1 u h n 2 , u h n 1 ) , v h ) Ω + ( B ( u h n 2 , u h n 1 u h n 2 ) , v h ) Ω = 0 ,
ν A ( u h n , u h n ) ( B ( u h n 1 , u h n u h n 1 ) , u h n ) Ω = ( F , u h n ) Ω ,
ν A ( e n , v h ) d ( v h , η n ) d ( e n , q h ) + ( B ( e n 1 , u h n 1 ) , v h ) Ω + ( B ( u h , e n 1 ) , v h ) Ω = 0 ,
which yield
ν e 1 X N ( u h 0 X + u h X ) e 0 X
2 N ν 2 F 1 , Ω e 0 X 2 σ 2 F 1 , Ω ,
ν u h 1 u h 0 X N u h 0 X 2 σ F 1 , Ω , ν u h 1 X N u h 1 u h 0 X u h 0 X + F 1 , Ω F 1 , Ω + N ν 2 σ F 1 , Ω 2
( 1 + σ 2 ) F 1 , Ω 6 5 F 1 , Ω ,
β 0 p h 1 0 , Ω ν u h 1 X + F 1 , Ω + N u h 0 X 2 3 F 1 , Ω ,
β 0 η 1 0 , Ω ν e 1 X + N ( u h 0 X + u h X ) e 0 X 3 σ 2 F 1 , Ω ,
for n = 1 and
ν u h n u h n 1 X N ( u h n 1 X + u h n 2 X ) u h n 1 u h n 2 X
12 5 σ u h n 1 u h n 2 X ( 12 5 σ ) n 1 σ F 1 , Ω , ν u n X N u h n 1 X u h n u h n 1 X + F 1 , Ω
F 1 , Ω + 6 5 σ ( 12 5 σ ) n 1 σ F 1 , Ω 6 5 F 1 , Ω ,
β 0 p h n 0 , Ω ν u h n X + F 1 , Ω + N u h n 1 X 2 3 F 1 , Ω ,
ν e n X N ( u h X + u h n 1 X ) e n 1 X 11 5 σ e n 1 X ( 11 5 σ ) n σ F 1 , Ω ,
β 0 η n 0 , Ω ν e n X + 2 N ( u h n 1 X + u h X ) e n 1 X 3 ( 11 5 σ ) n σ F 1 , Ω ,
for n 2 . Thus, (195)–(204) shows that (184)–(187) hold for n.
The proof ends. □
Finally, by combining Lemma 15 with Lemma 9, we obtain the convergence result of the Stokes iterative FE solution ( u h n , p h n ) with respect to the exact solution ( u , p ) of the 3D steady Navier–Stokes equations.
Theorem 5.
If F L 2 ( Ω ) and X h × M h satisfies (65)–(68) and 0 < σ 2 5 , then the Stokes FE solution ( u h n , p h n ) satisfies the following error estimates:
ν u u h n X ( 11 5 σ ) n σ F 1 , Ω + C h F 0 , Ω ,
β 0 p p h n 0 , Ω 3 ( 11 5 σ ) n σ F 1 , Ω + C h F 0 , Ω ,
ν u u h n 0 , Ω ( 11 5 σ ) n σ γ Ω F 1 , Ω + C h 2 F 0 , Ω .

8. Conclusions

Remark 2.
From Theorems 3–5, we find that the three iterative FE methods are H 1 -uniform stable and first-order convergent with respect to ( ν , 1 σ ) ; the three iterative FE methods are L 2 -uniform stable and second-order convergent with respect to ν; the Stokes iterative FE method is simpler than the Oseen iterative FE method; the Oseen iterative FE method is simpler than the Newton iterative FE method; the convergence of the Newton iterative FE method is better than that of the Oseen iterative FE method; and the convergence of the Oseen finite element iterative method is better than the Stokes iterative FE method in the case of 0 < σ 2 5 .
Remark 3.
From Theorems 3 and 4, we find that the two iterative FE methods are H 1 -uniform stable and convergent with respect to ( ν , 1 σ ) ; the three iterative FE methods are L 2 -uniform stable and second-order convergent with respect to ν; the Oseen iterative FE method is simpler than the Newton iterative FE method; and the convergence of the Newton iterative FE method is better than the the Oseen iterative FE method in the case of 2 5 < σ 5 11 .
Remark 4.
From Theorem 3, we find that the Oseen iterative FE method is H 1 -uniform stable and convergent with respect to ( ν , 1 σ ) and the Oseen iterative FE method is L 2 -uniform stable and second-order convergent with respect to ν in the case of 5 11 < σ < 1 .

Funding

This research received no external funding.

Institutional Review Board Statement

Not applicable.

Informed Consent Statement

Not applicable.

Data Availability Statement

Not applicable.

Conflicts of Interest

The authors declare no conflict of interest.

References

  1. Temam, R. Navier-Stokes Equations, Theory and Numerical Analysis, 3rd ed.; Elsevier: Amsterdam, The Netherlands; New York, NY, USA; Oxford, UK, 1984. [Google Scholar]
  2. Girault, V.; Raviart, P.A. Finite Element Methods for Navier-Stokes Equations. Theory and Algorithms; Springer: Berlin/Heidelberg, Germany, 1986. [Google Scholar]
  3. Quarteroni, A.; Valli, A. Numerical Approximation of Partial Differential Equations; Springer: Berlin/Heidelberg, Germany, 1997. [Google Scholar]
  4. Glowinski, R. Finite element methods for incompressible viscous flow. In Handbook of Numerical Analysis; Ciarlet, P.G., Lions, J.L., Eds.; Numerical Methods for Fluids (Part 3); Elsevier Science Publisher: Amsterdam, The Netherlands, 2003; Volume IX. [Google Scholar]
  5. Elman, H.C.; Silvester, D.J.; Wathen, A.J. Finite Elements and Fast Iterative Solvers: With Applications in Incompressible Fluid Dynamics; Oxford University Press: Oxford, UK, 2005. [Google Scholar]
  6. Heywood, J.G.; Rannacher, R. Finite element approximation of the nonstationary Navier-Stokes problem. I. Regularity of solutions and second-order spatial discretization. SIAM J. Numer. Anal. 1982, 19, 275–311. [Google Scholar] [CrossRef]
  7. Heywood, J.G.; Rannacher, R. Finite element approximation of the nonstationary Navier-Stokes problem. II. Stability of solutions and error estimates uniform in time. SIAM J. Numer. Anal. 1986, 23, 750–777. [Google Scholar] [CrossRef] [Green Version]
  8. Heywood, J.G.; Rannacher, R. Finite element approximation of the nonstationary Navier-Stokes problem, III. Smoothing property and higher order estimates for spatial discretization. SIAM J. Numer. Anal. 1988, 25, 489–512. [Google Scholar] [CrossRef]
  9. Heywood, J.G.; Rannacher, R. Finite element approximation of the nonstationary Navier-Stokes problem, IV: Error analysis for second order time discretizafion. SIAM J. Numer. Anal. 1990, 27, 353–384. [Google Scholar] [CrossRef]
  10. Layton, W. A two level discretization method for the Navier-Stokes equations. Comput. Math. Appl. 1993, 26, 33–38. [Google Scholar] [CrossRef] [Green Version]
  11. He, Y.; Li, J. Convergence of three iterative methods based on the finite element discretization for the stationary Navier-Stokes equations. Comput. Methods Appl. Mech. Eng. 2009, 198, 1351–1359. [Google Scholar] [CrossRef]
  12. He, Y.; Li, K. Two-level stabilized finite element methods for the steady Navier-Stokes equations. Computing 2005, 74, 337–351. [Google Scholar] [CrossRef]
  13. He, Y.; Wang, A. A simplified two-level method for the steady Navier-Stokes equations. Comput. Methods Appl. Mech. Eng. 2008, 197, 1568–1576. [Google Scholar] [CrossRef]
  14. He, Y.; Wang, A.; Mei, L. Stabilized finite element methods for the stationary Navier-Stokes equations. J. Eng. Math. 2005, 51, 367–380. [Google Scholar] [CrossRef]
  15. He, Y.; Wang, A.; Chen, Z.; Li, K. An optimal nonlinear Galerkin method with mixed finite elements for the steady Navier-Stokes equations. Numer. Methods Partial Differ. Equ. 2003, 19, 762–775. [Google Scholar] [CrossRef]
  16. Kannan, R.; Wang, Z. A study of viscous flux formulations for a p-multigrid spectral volume Navier Stokes solver. J. Sci. Comput. 2009, 41, 165–199. [Google Scholar] [CrossRef]
  17. Kannan, R.; Wang, Z. The direct discontinuous Galerkin (DDG) viscous flux scheme for the high order spectral volume method. Comput. Fluids 2010, 39, 2007–2021. [Google Scholar] [CrossRef]
  18. Kannan, R.; Wang, Z. LDG2: A variant of the LDG flux formulation for the spectral volume method. J. Sci. Comput. 2011, 46, 314–328. [Google Scholar] [CrossRef]
  19. Kannan, R.; Wang, Z. Curvature and entropy based wall boundary condition for the high order spectral volume Euler solver. Comput. Fluids 2011, 44, 79–88. [Google Scholar] [CrossRef]
  20. Kannan, R. A high order spectral volume formulation for solving equations containing higher spatial derivative terms: Formulation and analysis for third derivative spatial terms using the LDG discretization procedure. Commun. Comput. Phys. 2011, 10, 1257–1279. [Google Scholar] [CrossRef]
  21. Kannan, R. A high order spectral volume method for elastohydrodynamic lubrication problems: Formulation and application of an implicit p-multigrid algorithm for line contact problems. Comput. Fluids 2011, 48, 44–53. [Google Scholar] [CrossRef]
  22. Kannan, R. A high order spectral volume formulation for solving equations containing higher spatial derivative terms II: Improving the third derivative spatial discretization using the LDG2 method. Commun. Comput. Phys. 2012, 12, 767–788. [Google Scholar] [CrossRef]
  23. Kannan, R.; Harrand, V.; Lee, M.; Przekwas, A.J. Highly scalable computational algorithms on emerging parallel machine multicore architectures: Development and implementation in CFD context. Int. J. Numer. Methods Fluids 2013, 73, 869–882. [Google Scholar] [CrossRef]
  24. Sun, Y.; Wang, Z.; Liu, Y. High-order multidomain spectral difference method for the Navier-Stokes equations on unstructured hexahedral grids. Commun. Comput. Phys. 2007, 2, 310–333. [Google Scholar]
  25. Hu, X.; Mu, L.; Ye, X. A weak Galerkin finite element method for the Navier-Stokes equations on polytopal meshes. J. Comput. Appl. Math. 2019, 362, 614–625. [Google Scholar] [CrossRef]
  26. Irisarri, D.; Hauke, G. Stabilized virtual element methods for the unsteady incompressible Navier-Stokes equations. Calcolo 2019, 56, 1–21. [Google Scholar] [CrossRef]
  27. Gatica, G.N.; Munar, M.; Sequeira, F.A. A mixed virtual element method for the Navier-Stokes equations. Math. Models Methods Appl. Sci. 2018, 28, 2719–2762. [Google Scholar] [CrossRef]
  28. Liu, X.; Chen, Z. The nonconforming virtual element method for the Navier-Stokes equations. Adv. Comput. Math. 2019, 45, 51–74. [Google Scholar] [CrossRef]
  29. Chen, H.; Li, K.; Wang, S. A dimension split method for the incompressible Navier-Stokes equations in three dimensions. Int. J. Numer. Methods Fluids 2013, 73, 409–435. [Google Scholar] [CrossRef]
  30. Chen, H.; Li, K.; Chu, Y.; Chen, Z.; Yang, Y. A dimension splitting and characteristic projection method for three-dimensional incompressible flow. Discret. Contin. Dyn. Syst. B 2019, 24, 127–147. [Google Scholar] [CrossRef] [Green Version]
  31. Huang, P.; Feng, X.; He, Y. Two-level defect-correction Oseen iterative stabilized finite element methods for the stationary Navier-Stokes equations. Appl. Math. Model. 2013, 37, 728–741. [Google Scholar] [CrossRef]
  32. Huang, P.; Feng, X.; Liu, D. Two-level stabilized method based on three corrections for the stationary Navier-Stokes equations. Appl. Numer. Math. 2012, 62, 988–1001. [Google Scholar] [CrossRef]
  33. Li, J. Investigations on two kinds of two-level stabilized finite element methods for the stationary Navier-Stokes equations. Appl. Math. Comput. 2006, 182, 1470–1481. [Google Scholar] [CrossRef]
  34. Song, L.; Su, H.; Feng, X. Recovery-based error estimator for stabilized finite element method for the stationary Navier-Stokes problem. SIAM J. Sci. Comput. 2016, 38, A3758–A3772. [Google Scholar] [CrossRef]
  35. Zhang, Y.; He, Y. A two-level finite element method for the stationary Navier-Stokes equations based on a stabilized local projection. Numer. Methods Partial Differ. Equ. 2011, 27, 460–477. [Google Scholar] [CrossRef]
  36. Zhang, G.; Dong, X.; An, Y.; Liu, H. New conditions of stability and convergence of Stokes and Newton iterations for Navier-Stokes equations. Appl. Math. Mech. 2015, 36, 863–872. [Google Scholar] [CrossRef]
  37. Xu, H.; He, Y. Some iterative finite element methods for steady Navier-Stokes equations with different viscosities. J. Comput. Phys. 2013, 232, 136–152. [Google Scholar] [CrossRef]
  38. He, Y. Stability and convergence of iterative methods related to viscosities for the 2D/3D steady Navier-Stokes equations. J. Math. Anal. Appl. 2015, 423, 1129–1149. [Google Scholar] [CrossRef]
  39. Bercovier, J.; Pironneau, O. Error estimates for finite element solution of the Stokes problem in the primitive variables. Numer. Math. 1979, 33, 211–224. [Google Scholar] [CrossRef]
  40. Hecht, F. New development in freefem++. J. Numer. Math. 2012, 20, 251–265. [Google Scholar] [CrossRef]
  41. Shen, J. On error estimates of the penalty method for the unsteady Navier-Stokes euqations. SIAM J. Numer. Anal. 1995, 32, 386–403. [Google Scholar] [CrossRef]
Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Share and Cite

MDPI and ACS Style

He, Y. Finite Element Iterative Methods for the 3D Steady Navier--Stokes Equations. Entropy 2021, 23, 1659. https://doi.org/10.3390/e23121659

AMA Style

He Y. Finite Element Iterative Methods for the 3D Steady Navier--Stokes Equations. Entropy. 2021; 23(12):1659. https://doi.org/10.3390/e23121659

Chicago/Turabian Style

He, Yinnian. 2021. "Finite Element Iterative Methods for the 3D Steady Navier--Stokes Equations" Entropy 23, no. 12: 1659. https://doi.org/10.3390/e23121659

APA Style

He, Y. (2021). Finite Element Iterative Methods for the 3D Steady Navier--Stokes Equations. Entropy, 23(12), 1659. https://doi.org/10.3390/e23121659

Note that from the first issue of 2016, this journal uses article numbers instead of page numbers. See further details here.

Article Metrics

Back to TopTop