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

In this work, a finite element (FE) method is discussed for the 3D steady Navier–Stokes equations by using the finite element pair Xh×Mh. The method consists of transmitting the finite element solution (uh,ph) of the 3D steady Navier–Stokes equations into the finite element solution pairs (uhn,phn) based on the finite element space pair Xh×Mh of the 3D steady linearized Navier–Stokes equations by using the Stokes, Newton and Oseen iterative methods, where the finite element space pair Xh×Mh 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 (uhn,phn) of the 3D steady Stokes, Newton and Oseen iterative equations, and deduce the convergence with respect to (σ,h) of the FE solution (uhn,phn) to the exact solution (u,p) of the 3D steady Navier–Stokes equations in the H1−L2 norm. Finally, we also give the convergence order with respect to (σ,h) of the FE velocity uhn to the exact velocity u of the 3D steady Navier–Stokes equations in the L2 norm.


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 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 n h 0,Ω ≤ 2 F −1,Ω and the convergence result is ν ∇ (u n h − u h ) 0,Ω ≤ (3σ) 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 n h 0,Ω ≤ 4 3 F −1,Ω and the convergence result is ν ∇ (u n h − 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 n h , p n h ) 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 n h 0,Ω ≤ 6 5 F −1,Ω and the convergence result is ν ∇ (u n h − 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 n h 0,Ω ≤ 6 5 F −1,Ω and the convergence result is ν ∇ (u n h − 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 n h , p n h ) 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 n h , p n h ) 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 n h , p n h ) 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 n h , p n h ) 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 n h , p n h ) 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.

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: where u = (u 1 , u 2 , u 3 ) represents the velocity, p the pressure with and Ω p(x, y, z)dxdydz = 0, F = (F 1 , F 2 , F 3 ) the external volumetric force on the fluid. Additionally, we introduce the following notations: Using the Green formula, we deduce the weak formulation of the 3D steady Stokes where X = H 1 0 (Ω) 3 , M = L 2 0 (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 aũ ∈ X such that d(ũ, p) = p 2 0,Ω , ∇ũ 0,Ω ≤ β −1 p 0,Ω , Next, we need to recall the general Lax-Milgram theorem.
Proof. We introduce the subspace V of X as follows: Thus, we deduce from (4) that u ∈ V satisfies where A(u, v) satisfies Using the Lax-Miligram theorem, (21) admits a unique solution u ∈ V such that Now, we introduce a Polar set and define two dual operators Bv ∈ M and B q ∈ X such that 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 −∆u − F ∈ V 0 . Thus, there exists a unique p ∈ M such that −∆u − F = B p or 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.
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): where u n = (u n 1 , u n 2 , u n 3 ) = (w n , u n 3 ). Using the Green formula, we deduce the weak formulation of the 3D steady Oseen iterative Equations (41) or where 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
then there exists a β 1 > 0 such that where c 2 ≥ √ 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 n h , p n h ) 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 b h of H 1 (Ω) based on P 1 and P b 1 elements as follows: 1 is a bubble element on K and satisfies P b 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 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 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: • The inverse inequality holds: • There exists a constant β 0 > 0 such that 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: 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: Proof. We introduce the subspace V h of X h as follows: Thus, we deduce from (69) that u h ∈ V h satisfies Using the Lax-Miligram theorem with X = V h ,Ã(u, v) = A(u h , v h ) and (22), we show that (69) admits a unique solution u h ∈ V h , satisfying Now, we introduce a Polar set and define two dual operators B h v h ∈ M h and B h q h ∈ X h such that 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 0 h . Moreover, we deduce from (71) where the discrete Laplace operator −∆ h u h ∈ X h is defined as 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 Combining (74) with (72) yields (70). The proof ends.
Due to (68), we can consider the weak formulation of the general Stokes equations:

Lemma 5. If F ∈ X and X h × M h satisfies the discrete inf-sup condition (68), the bilinear form
then (75) admits a unique solution (u h , p h ) ∈ X h × M h such that where c 4 ≥ √ 3.
Proof. First, we deduce from (75) that u h ∈ V h satisfies Using (76) and (77) and the Lax-Miligram theorem with X = V h andÃ(u, v) = A(u h , v h ), we show that (79) or (80) admits a unique solution u h ∈ V h satisfying Next, (80) showsÃ h u h − F ∈ V 0 h . Thus, referring to [1,2], the inf-sup condition (68) implies that B h is an isomorphic operator from M h onto V 0 h . Thus, there exists a unique 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 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: 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: Proof. We set a bounded convex subset K of X h × M h as and define a map T : Due to (w h , r h ) ∈ K, we deduce from (30) and (31) and the uniqueness condition that for each u h , v h ∈ X h . Using Lemma 5, we show that (87) or (88) admits a unique solution 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 (92) and using (30) and (31), we deduce Thanks to 0 < σ < 1, (93) yields w h = 0 or u 1 h = u 2 h . Next, using (68) and (93) with q h = 0, we deduce r h = 0. The proof ends.
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 .
The proof ends.

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: Proof. First, we deduce from (4) and (69) that Using (103) with q h = 0 and (68), we deduce 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 Combining (104) and (105) and (66), we deduce 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 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 Using again (25), we have Taking (v, q) = (u − u h , p − p h ) in (110), using (103) and (65) and (66) and (111), we deduce Combining (112) with (106), we obtain (102). The proof ends.

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 −1 h = 0, we define the Oseen iterative FE solution (u n h , p n h ) of the 3D steady Navier-Stokes equations: or where In order to prove the existence, uniqueness and stability of the solution (u n h , p n h ) based on (134) or (135), we consider the continuous and elliptic condition of the bilinear form for each u h , v h ∈ X h .
Proof. Using (30) and (31), we have which are (136) and (137) and Proof. For n = 0 and u −1 Using Lemma 4, we show the existence and uniqueness of the solution (u 0 h , p 0 h ) satisfying (138). Moreover, using (140) and (84), we easily show that (u 0 Using again Lemma 4, Lemma 6 and (141), we have 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 n h , p n h ) ∈ X × M which satisfies Next, it follows from (134) and (84) that (144) and using (30) and (31) which, with the induction assumption for n − 1, yields Finally, using (31), (145) and Lemma 6, we obtain Combining (146) and (145) and using the induction assumption for n − 1 yields (139). Hence, Lemma 11 holds for n. The proof ends.

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 −1 h = 0, we define the Newton iterative FE solution (u n h , p n h ) of the 3D steady Navier-Stokes equations: find (u n h , p n h ) ∈ X h × M h such that for each (v h , q h ) ∈ X h × M h there holds or whereF In order to prove the existence, uniqueness and stability of the solution (u n h , p n h ) based on (150) or (151), we consider the continuous of the linear formF(v h ) and the continuous and elliptic condition of the bilinear form A n−1 (u h , v h ).
Furthermore, we obtain the existence, uniqueness, stability and convergence results of the solution (u n h , p n h ) based on (150) or (151).