1 Introduction

Let Ω be a bounded domain in ,which is assumed to have the C² boundary or be a convex polygon. We consider the stationary Navier-Stokes problem as follows:

where u = (u₁(x),u₂(x)) represents the velocity vector,p = p(x) is the pressure,f = (f₁(x),f₂(x)) is the prescribed body force,and υ > 0 is the viscosity. Lots of works have investigated the stationary Navier-Stokes problem^{[1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15]}. There are also numerous works devoted to the development of efficient schemes for the nonstationary Navier-Stokes problem^{[16, 17, 18, 19, 20, 21, 22, 23, 24]}.

For convenience,we first introduce some notations as follows:

We denote by (·,·) the usual L² product and let . In addition,we define two continuous bilinear forms and a trilinear form which are

and for ∀u,v,w ∈ X, Evidently,the trilinear form b(·,·,·) is continuous on X × X × X,and its norm is defined as follows:

As for the above comments,the variational formulation of Problem (1) reads as follows: find (u,p) ∈ X ×M such that

Denote . Then,the existence and uniqueness result for Problem (4) is pretty classical (see Refs. ^[6] and ^[14]).

Theorem 1 Suppose that f ∈ X′ and υ satisfy 0 < σ < 1. Then,Problem (4) admits a unique solution pair (u,p) ∈ X ×M such that

The iterative methods are efficient for solving the stationary Navier-Stokes equations under some strong uniqueness conditions^{[9, 10, 25]}. As we know,there are three efficient iterative schemes for solving Problem (1) as follows:

(i) Stokes iteration,

or

(ii) Newton iteration,

or

(Ⅲ) Oseen iteration,

for all (v_h,q_h) ∈ X_h ×M_h. The initial value u_h⁰ is obtained from

The stability and convergence conditions of the Stokes and Newton iterations to solve the stationary Navier-Stokes equations are 0 < σ ≤ and 0 < σ ≤ ,respectively. Besides,the Oseen scheme is unconditionally stable and convergent under 0 < σ < 1^{[9, 25]}. Recently, the stability and convergence condition of the Newton iteration was improved^{[10, 15]},and the improved condition is 0 < σ ≤ . In the present paper,using a new proof technique,we obtain the more accurate conditions of stability and convergence of the Stokes and Newton schemes, and the new results are 0 < σ ≤ ,respectively. According to our new theory,when ,the Stokes iteration is still an efficient way to solve the stationary Navier-Stokes equations. When ,the Newton scheme still works. In addition,a numerical experiment,the double-driven cavity flow problem,is presented to validate that the Newton iteration has a larger admissible range of data than the Stokes iteration.

The rest of this paper is organized as follows. In Section 2,we show some preliminaries playing an important role in the work. In Section 3,we present our main results of stability and convergence for the Stokes and Newton iterations. Finally,a numerical test is carried out to test our established theoretical results. 2 Preliminary

Let us consider h as a real positive parameter and X_h ×M_h characterized by T_h as a finite element subspace pair of X × M ,while T_h is a uniform partition of Ω into triangles K or quadrilaterals K with the diameters bounded by h. Readers can refer to Ref. ^[6] for details. We define

to be the subspace of Xh and Ph : Y → Vh to be the L2-orthogonal projection determined by

Besides,we assume that X_h ×M_h satisfies the following properties:

(i) There exist approximations π_hv ∈ X_h and ρ_hq ∈ M_h for all v ∈ H²(Ω)² ∩ X and q ∈ H¹(Ω) ∩M such that

(ii) X_h ×M_h satisfies the so-called inf-sup condition as follows: for each q_h ∈ M_h,there exists a positive constant β only relying on Ω and satisfying

There are several finite element spaces pairs satisfying (9) and (10)^{[6, 14]}.

Define the discrete Stokes operator A_h = −P_hΔ_h by

Throughout this paper,we denote by c and C two general positive constants only depending on Ω and (f,Ω),which may stand for different values under different circumstances. In order to obtain some estimates of the trilinear form b(·,·,·) presented in the following lemma,we bring in^{[17, 23]} Using (12) and the Hölder inequality,we have some estimates for the trilinear form b.

Lemma 1 The trilinear form b satisfies the following estimates:

The finite element Galerkin approximation of Problem (4) based on X_h ×M_h is to find (u_h,p_h) ∈ X_h ×M_h such that

We also learn the following stability and error estimate from Ref. ^[9]: 3 Main results

In this section,we show our main results of stability and convergence for the Stokes and Newton iterations. Denoting (eⁿ,ηⁿ) = (u_h−u_hⁿ,p_h−p_hⁿ),the error equations about the Stokes iteration and the Newton iteration are

respectively. First,we give the new sufficient condition of stability and convergence for the Stokes iteration in the following.

Theorem 2 Under the condition 0 < σ ≤ ,the Stokes iteration is stable and convergent,namely,for n ≥ 0,

where the constant C_* > 2 only depends on Ω.

Proof We first prove (19) by the method of induction. It is obvious that . Setting v_h = u_h¹,q = p_h¹ in (6) as n = 1 gives

Next,by the equation with respect to u_h¹ − u_h⁰

we have Combining (23) with (24) yields

We suppose (J = 0,1,· · · ,n − 1) for n ≥ 2 and prove . Setting v_h = u_hⁿ and q = p_hⁿ in (6) gives By the above equation,we have Combining (25) with (27),we have

Taking v_h = A_hu_hⁿ,q_h = 0 in (5),using (13) and Young’s inequality,we derive

Denoting C_* = 2(1 + c),we have It is evident that . Therefore,we suppose . Then,by (28),we obtain

Thus,by induction we prove the stability (20).

Taking v_h=u_h−u_hⁿ,q_h=p_h−p_hⁿ in the error equation (17),using (3),(16),and (19),we have

Using the error equation (17),(3),the inf-sup condition (10),(16),(19),and (21),we have

We complete this proof.

Next,we provide the new results of stability and convergence for the Newton iteration.

Theorem 3 Under the condition 0 < σ ≤ ,the Newton iteration is stable and convergent,namely,for n ≥ 0,

where the constant C_** > 4 only depends on Ω.

Proof We first prove (29) using the method of induction. It is evident that ,and u_h¹ − u_h⁰ satisfies

It follows that using (3), Setting v_h = u_h¹,q_h = p_h¹ in (8) as n = 1,we have

Using (33),we deduce from the above equation that . We suppose (J = 0,1,· · · ,n−1) for n ≥ 2 and prove . By taking v_h = u_hⁿ,q_h = p_hⁿ in (8) and using (3),we derive that We know that u_hⁿ − u_h^n-1 satisfies

It follows that by (3),

Thus,by our assumption,we deduce that Combining (34) with (35),we arrive at

Choosing v_h = A_hu_hⁿ,q_h = 0 in (7),and using (13) and (14),we get

By Young’s inequality and (29),we derive that where .We suppose .Then,via (36),we obtain

By the induction,we prove (30).

Taking v_h = u_h−u_hⁿ,q_h=p_h−p_hⁿ in the error equation (18),by simple calculation,we have

Setting q_h = 0 in (18) and using (3),the inf-sup condition (10),(29),and (31),we get

We complete this proof.

Remark 1 From the above two theorems,we can see that the Stokes iteration is the simplest scheme among the three schemes. However,it has the least admissible range of data. The Newton iteration has a moderate admissible range of data,and it has a second-order convergent rate with respect to the iterative step. 4 Numerical example

In this section,a numerical experiment is shown. We consider the double-driven cavity flow problem presented in Fig. 1. The mini mixed finite element spaces^[6] are used on grids whose mesh size is h = 1/64. The iterative schemes are designed by L²-norm of difference in successive iterations with the tolerance of 1.0×10⁻⁶. All the linear equations are solved by UMFPACK routine.

Figures 2-3 describe the velocity vectors and the pressure contours with υ = 1/50 and υ = 1/100,respectively,by the usage of the Stokes iteration and the Newton iteration. We can see from Fig. 2 that the two schemes can both converge under the large viscosity υ and obtain the same numerical results. The Newton iteration consumes less time than the Stokes scheme,because the former has a second-order convergence rate with respect to the iterative step. However,when υ = 1/100,the Stokes iterative method cannot work. The Newton scheme can still solve this model (see Fig. 3). Figure 4 depicts the relationship of the iterative error with the iterative step m. We can see that,when the two scheme are both effective (υ = 1/50), the Newton iteration converges faster than the Stokes iteration. As the viscosity υ decreases (υ = 1/100),the Stokes diverges,and the Newton iteration still converges.

5 Conclusions

In this paper,we obtain some more accurate conditions of stability and convergence for the Stokes and Newton iterations. For solving the stationary Navier-Stokes equations,when 0 < σ = ,the Stokes scheme is efficient. When 0 < σ ≤ ,the Newton scheme is an efficient approach. In our numerical experiment,we find that the Newton scheme has a larger admissible range of data and converges faster than the Stokes iteration,which is consistent with our theoretical results. Unfortunately,because the values of N and are hard to determine,we cannot give a method to verify the new conditions of stability and convergence accurately. How to verify the new conditions precisely will be our future work direction in this field.