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

Abstract

In this work, a finite element (FE) method is discussed for the 3D steady Navier–Stokes equations by using the finite element pair $X_h\times M_h$. 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 $X_h\times M_h$ of the 3D steady linearized Navier–Stokes equations by using the Stokes, Newton and Oseen iterative methods, where the finite element space pair $X_h\times M_h$ satisfies the discrete inf-sup condition in a 3D domain $\Omega$. 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 $(\sigma ,h)$ 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 $(\sigma ,h)$ 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 $X_h\times M_h$, satisfying the discrete inf-sup condition in a 2D domain $\Omega$, 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 $X_h\times M_h$, satisfying the discrete inf-sup condition in a 2D/3D domain $\Omega$, 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 $\nu \parallel \tilde{\nabla }{u}_h^n{\parallel }_{0,\Omega }\le 2{\parallel F\parallel }_{-1,\Omega }$ and the convergence result is $\nu \parallel \tilde{\nabla }(u_h^n-u_h){\parallel }_{0,\Omega }\le {(3\sigma )}^n{\parallel F\parallel }_{-1,\Omega }$ under the strong stability and convergence condition $0<\sigma \le \frac{1}{4}$. For the Newton iterative finite element method, the stability result is $\nu \parallel \tilde{\nabla }{u}_h^n{\parallel }_{0,\Omega }\le \frac{4}{3}{\parallel F\parallel }_{-1,\Omega }$ and the convergence result is $\nu \parallel \tilde{\nabla }(u_h^n-u_h){\parallel }_{0,\Omega }\le {(\frac{9}{5}\sigma )}^{2^n-1}{\parallel F\parallel }_{-1,\Omega }$ under the strong stability and convergence condition $0<\sigma \le \frac{1}{3}$.

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 $\nu \parallel \tilde{\nabla }{u}_h^n{\parallel }_{0,\Omega }\le \frac{6}{5}{\parallel F\parallel }_{-1,\Omega }$ and the convergence result is $\nu \parallel \tilde{\nabla }(u_h^n-u_h){\parallel }_{0,\Omega }\le {(\frac{11}{5}\sigma )}^n\sigma {\parallel F\parallel }_{-1,\Omega }$ under the weak stability and convergence condition $0<\sigma \le \frac{2}{5}$; for the Newton iterative finite element method, the stability result is $\nu \parallel \tilde{\nabla }{u}_h^n{\parallel }_{0,\Omega }\le \frac{6}{5}{\parallel F\parallel }_{-1,\Omega }$ and the convergence result is $\nu \parallel \tilde{\nabla }(u_h^n-u_h){\parallel }_{0,\Omega }\le {\sigma }^{2^n}{\parallel F\parallel }_{-1,\Omega }$ under the strong stability and convergence condition $0<\sigma \le \frac{5}{11}$. 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 $\nu$ 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 $X_h\times M_h$ 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 $\Omega \subset R^3$. 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 $\Omega$ with the Dirichlet boundary condition:

$$\begin{array}{ccc}\hfill -\tilde{\Delta }u+\tilde{\nabla }p& =& F,\mathrm{in}\Omega ,\hfill \end{array}$$

(1)

$$\begin{array}{ccc}\hfill \tilde{\nabla }·u& =& 0,\mathrm{in}\Omega ,\hfill \end{array}$$

(2)

$$\begin{array}{ccc}\hfill u& =& 0,\mathrm{on}\partial \Omega ,\hfill \end{array}$$

(3)

where $u=(u_1,u_2,u_3)$ represents the velocity, p the pressure with and ${\int }_{\Omega }p(x,y,z)dxdydz=0$, $F=(F_1,F_2,F_3)$ the external volumetric force on the fluid. Additionally, we introduce the following notations: $\tilde{\nabla }=({\nabla }_1,{\nabla }_2,{\nabla }_3)=({\partial }_x,{\partial }_y,{\partial }_z)$, $\tilde{\Delta }=\tilde{\nabla }·\tilde{\nabla }$ and $\tilde{\nabla }u={({\nabla }_i{u}_j)}_{3\times 3}$.

Using the Green formula, we deduce the weak formulation of the 3D steady Stokes Equations (1)–(3): find $(u,p)\in X\times M$ such that for each $(v,q)\in X\times M$ there holds

$$\begin{array}{c}\hfill G((u,p),(v,q))\equiv A(u,v)-d(v,p)-d(u,q)=\tilde{F}(v,q)\equiv {(F,v)}_{\Omega },\end{array}$$

(4)

where $X=H_0^1{(\Omega )}^3$, $M=L_0^2(\Omega )$ and $d(v,q)={(\tilde{\nabla }·v,q)}_{\Omega }$.

In order to consider the existence and the uniqueness of the solution $(u,p)\in X\times M$, 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 $p\in M$, there exists a $\tilde{u}\in X$ such that

$$\begin{array}{c}\hfill d(\tilde{u},p)={\parallel p\parallel }_{0,\Omega }^2,\parallel \tilde{\nabla }\tilde{u}{\parallel }_{0,\Omega }\le {\beta }^{-1}{\parallel p\parallel }_{0,\Omega },\end{array}$$

(5)

or

$$\begin{array}{c}\hfill \underset{v\in X}{sup}\frac{d(v,p)}{\parallel \tilde{\nabla }{v\parallel }_{0,\Omega }}\ge \beta {\parallel p\parallel }_{0,\Omega }\forall p\in M.\end{array}$$

(6)

Next, we need to recall the general Lax–Milgram theorem.

General Lax–Milgram theorem. For the weak formulation, find $(u,p)\in X\times M$ such that for each $(v,q)\in X\times M$, there holds

$$\begin{array}{c}\hfill \tilde{G}((u,p),(v,q))=\tilde{F}(v,q),\end{array}$$

(7)

if there holds the following condition:

$$\begin{array}{ccc}& & |\tilde{F}(v,q)|\le \parallel \tilde{F}{\parallel }_{{(X\times M)}^{\prime }}{(\parallel v\parallel }_X+{\parallel q\parallel }_M),\hfill \end{array}$$

(8)

$$\begin{array}{ccc}& & |\tilde{G}{((u,p),(v,q))|\le \alpha (\parallel u\parallel }_X+{\parallel p\parallel }_M{)(\parallel v\parallel }_X+{\parallel q\parallel }_M),\hfill \end{array}$$

(9)

$$\begin{array}{ccc}& & {\beta (\parallel u\parallel }_X+{\parallel p\parallel }_M)\le \underset{(v,q)\in X\times M}{sup}\frac{\tilde{G}((u,p),(v,q))}{{\parallel v\parallel }_X+{\parallel q\parallel }_M},\hfill \end{array}$$

(10)

$$\begin{array}{ccc}& & {\beta (\parallel v\parallel }_X+{\parallel q\parallel }_M)\le \underset{(u,p)\in X\times M}{sup}\frac{\tilde{G}((u,p),(v,q))}{{\parallel u\parallel }_X+{\parallel p\parallel }_M},\hfill \end{array}$$

(11)

Then, (7) admits a unique solution $(u,p)\in X\times M$, satisfying

$$\begin{array}{ccc}& & {\beta (\parallel u\parallel }_X+{\parallel p\parallel }_M)\le \parallel \tilde{F}{\parallel }_{{(X\times M)}^{\prime }}.\hfill \end{array}$$

(12)

Now, we give the existence, uniqueness and stability of the solution $(u,p)\in X\times M$ for the 3D steady Stokes equations.

Lemma 2.

If $F\in H^{-1}{(\Omega )}^3$, then (4) admits a unique solution $(u,p)\in X\times M$, satisfying the following bound:

$$\begin{array}{c}\hfill \parallel \tilde{\nabla }{u\parallel }_{0,\Omega }\le {\parallel F\parallel }_{-1,\Omega }{,\beta \parallel p\parallel }_{0,\Omega }\le 2{\parallel F\parallel }_{-1,\Omega }.\end{array}$$

(13)

Proof. 

First, we easily prove the following inequalities

$$\begin{array}{c}\hfill |\tilde{F}{(v,q)|=|(F,v)}_{\Omega }{|\le \parallel F\parallel }_{-1,\Omega }{\parallel \tilde{\nabla }v\parallel }_{0,\Omega },\end{array}$$

(14)

$$\begin{array}{c}\hfill G((u,p),(v,q))\le \sqrt{3}(\parallel \tilde{\nabla }{u\parallel }_{0,\Omega }+{\parallel p\parallel }_{0,\Omega })(\parallel \tilde{\nabla }{v\parallel }_{0,\Omega }+{\parallel q\parallel }_{0,\Omega }),\end{array}$$

(15)

for any $(u,p),(v,q)\in X\times M$. Using Lemma 1, for each $(u,p)\in X\times M$, we set $\epsilon ={\beta }^2$ and $(\tilde{v},\tilde{q})=(u-\epsilon \tilde{u},-p)$, where $\tilde{u}$ satisfies

$$\begin{array}{c}\hfill d(\tilde{u},p)={\parallel p\parallel }_{0,\Omega }^2,\parallel \tilde{\nabla }\tilde{u}{\parallel }_{0,\Omega }\le {\beta }^{-1}{\parallel p\parallel }_{0,\Omega }.\end{array}$$

(16)

Thus, we have

$$\begin{array}{ccc}\hfill \underset{(v,q)\in X\times M}{sup}\frac{G((u,p),(v,q))}{\parallel \tilde{\nabla }{v\parallel }_{0,\Omega }+{\parallel q\parallel }_{0,\Omega }}& \ge & \frac{G((u,p),(\tilde{v},\tilde{q}))}{\parallel \tilde{\nabla }\tilde{v}{\parallel }_{0,\Omega }+{\parallel \tilde{q}\parallel }_{0,\Omega }}\hfill \\ & =& \frac{{(\tilde{\nabla }u,\tilde{\nabla }(u-\epsilon \tilde{u}))}_{\Omega }+\epsilon d(\tilde{u},p)}{\parallel \tilde{\nabla }(u-\epsilon \tilde{u}){\parallel }_{0,\Omega }+{\parallel p\parallel }_{0,\Omega }}\hfill \\ & \ge & \frac{\parallel \tilde{\nabla }{u\parallel }_{0,\Omega }^2-\epsilon {\beta }^{-1}{\parallel p\parallel }_{0,\Omega }\parallel \tilde{\nabla }{u\parallel }_{0,\Omega }+\epsilon {\parallel p\parallel }_{0,\Omega }^2}{\parallel \tilde{\nabla }{u\parallel }_{0,\Omega }+(1+\epsilon {\beta }^{-1}){\parallel p\parallel }_{0,\Omega }}\hfill \\ & \ge & \frac{0.5\parallel \tilde{\nabla }{u\parallel }_{0,\Omega }^2+0.5\epsilon {\parallel p\parallel }_{0,\Omega }^2}{\parallel \tilde{\nabla }{u\parallel }_{0,\Omega }+(1+\epsilon {\beta }^{-1}){\parallel p\parallel }_{0,\Omega }}\hfill \\ & \ge & {\beta }_1(\parallel \tilde{\nabla }{u\parallel }_{0,\Omega }+{\parallel p\parallel }_{0,\Omega }).\hfill \end{array}$$

(17)

Since $G((u,p),(v,q))=G((v,q),(u,p))$, there holds

$$\begin{array}{ccc}& & \underset{(u,p)\in X\times M}{sup}\frac{G((u,p),(v,q))}{\parallel \tilde{\nabla }{u\parallel }_{0,\Omega }+{\parallel p\parallel }_{0,\Omega }}=\underset{(u,p)\in X\times M}{sup}\frac{G((v,q),(u,p))}{\parallel \tilde{\nabla }{u\parallel }_{0,\Omega }+{\parallel p\parallel }_{0,\Omega }}\hfill \\ & \ge & {\beta }_1(\parallel \tilde{\nabla }{v\parallel }_{0,\Omega }+{\parallel q\parallel }_{0,\Omega }).\hfill \end{array}$$

(18)

Thus, using the general Lax–Milgram theorem, (4) admits a unique solution $(u,p)\in X\times M$.

Taking $(v,q)=(u,-p)$ in (4), we obtain

$$\begin{array}{c}\hfill \parallel \tilde{\nabla }{u\parallel }_{0,\Omega }\le {\parallel F\parallel }_{-1,\Omega }.\end{array}$$

(19)

Using again Lemma 1 and (4) with $q=0$, we have

$$\begin{array}{c}\hfill {\beta \parallel p\parallel }_{0,\Omega }\le \parallel \tilde{\nabla }{u\parallel }_{0,\Omega }+{\parallel F\parallel }_{-1,\Omega }\le 2{\parallel F\parallel }_{-1,\Omega }.\end{array}$$

(20)

Combining (19) with (20) yields (13). The proof ends. □

Proof. 

We introduce the subspace V of X as follows:

$$\begin{array}{c}\hfill V=\{v\in X;d(v,q)=0,\forall q\in M\}.\end{array}$$

Thus, we deduce from (4) that $u\in V$ satisfies

$$\begin{array}{c}\hfill A(u,v)={(F,v)}_{\Omega },\forall v\in V,\end{array}$$

(21)

where $A(u,v)$ satisfies

$$\begin{array}{c}\hfill |A(u,v)|\le \parallel \tilde{\nabla }{u\parallel }_{0,\Omega }\parallel \tilde{\nabla }{v\parallel }_{0,\Omega },|A(u,u)|\ge \parallel \tilde{\nabla }{u\parallel }_{0,\Omega }^2,\forall u,v\in V.\end{array}$$

(22)

Using the Lax–Miligram theorem, (21) admits a unique solution $u\in V$ such that

$$\begin{array}{c}\hfill \parallel \tilde{\nabla }{u\parallel }_{0,\Omega }\le {\parallel F\parallel }_{-1,\Omega },\end{array}$$

(23)

Now, we introduce a Polar set

$$V^0=\{g\in X^{\prime };<g,v>=0\forall v\in V\},$$

and define two dual operators $Bv\in M$ and $B^{\prime }q\in X^{\prime }$ such that

$$d(v,q)={(Bv,q)}_{\Omega }=<v,B^{\prime }q>\forall (v,q)\in X\times M.$$

Thus, referring to [1,2], we know that the inf-sup condition (6) implies that $B^{\prime }$ is a isomorphic operator from M onto $V^0$. Moreover, we deduce from (21) that $-\tilde{\Delta }u-F\in V^0$. Thus, there exists a unique $p\in M$ such that $-\tilde{\Delta }u-F=B^{\prime }p$ or

$$\begin{array}{c}\hfill A(u,v)-d(v,p)={(F,v)}_{\Omega },\forall v\in X.\end{array}$$

(24)

Due to $u\in V$, there holds $d(u,q)=0$ for each $q\in M$. Thus, we have proved that $(u,p)\in X\times M$ is a unique solution of (4). Using (19) and (20), we show that $(u,p)\in X\times M$ satisfies (13). The proof ends. □

Recalling Temam [1], if $F\in {(L^2(\Omega ))}^3$, there holds the following regularity result of the solution $(u,p)$ for the Stokes equations:

$$\begin{array}{c}\hfill {\parallel u\parallel }_{2,\Omega }+{\parallel p\parallel }_{1,\Omega }\le c_0{\parallel F\parallel }_{0,\Omega }.\end{array}$$

(25)

Next, we consider the 3D steady Navier–Stokes equations with the Dirichlet boundary condition in a bounded domain $\Omega$:

$$\begin{array}{ccc}\hfill -\nu \tilde{\Delta }u+\tilde{\nabla }p+B(u,u)& =& F,\mathrm{in}\Omega ,\hfill \end{array}$$

(26)

$$\begin{array}{ccc}\hfill \tilde{\nabla }·u& =& 0,\mathrm{in}\Omega ,\hfill \end{array}$$

(27)

$$\begin{array}{ccc}\hfill u& =& 0,\mathrm{on}\partial \Omega ,\hfill \end{array}$$

(28)

with ${\int }_{\Omega }p(x,y,z)dxdydz=0$, where $B(u,v)=(u·\tilde{\nabla })v+\frac{1}{2}\tilde{\nabla }·uv$.

Using the Green formula, we deduce the weak formulation of the 3D steady Navier–Stokes Equations (26)–(28): We find $(u,p)\in X\times M$ such that for each $(v,q)\in X\times M$ there holds

$$\begin{array}{c}\hfill \nu A(u,v)-d(v,p)-d(u,q)+{(B(u,u),v)}_{\Omega }={(F,v)}_{\Omega },\end{array}$$

(29)

where $X=H_0^1{(\Omega )}^3$, $M=L_0^2(\Omega )$.

Here and hereafter, some positive constants N, $\beta$, ${\gamma }_{\Omega }$ and $c_1$ and some inequalities are stated as follows:

$$\begin{array}{ccc}& & {(B(u,v),w)}_{\Omega }=0.5{(u·\tilde{\nabla }v,w)}_{\Omega }-0.5{(u·\tilde{\nabla }w,v)}_{\Omega },\hfill \end{array}$$

(30)

$$\begin{array}{ccc}& & {|(B(u,v),w)}_{\Omega }|\le N\parallel \tilde{\nabla }{u\parallel }_{0,\Omega }\parallel \tilde{\nabla }{v\parallel }_{0,\Omega }{\parallel \tilde{\nabla }w\parallel }_{0,\Omega }\forall u,v,w\in X,\hfill \\ & & {\parallel B(u,v)\parallel }_{0,\Omega }\le N\parallel \tilde{\nabla }{u\parallel }_{0,\Omega }\parallel \tilde{\nabla }{v\parallel }_{0,\Omega }^{\frac{1}{2}}{\parallel v\parallel }_{2,\Omega }^{\frac{1}{2}},\forall u\in X,v\in {(H_0^1(\Omega )\cap H^2(\Omega ))}^3,\hfill \end{array}$$

(31)

$$\begin{array}{ccc}& & \parallel B^{\prime }{(v,w)\parallel }_{0,\Omega }\le N\parallel \tilde{\nabla }{v\parallel }_{0,\Omega }\parallel \tilde{\nabla }{w\parallel }_{0,\Omega }^{\frac{1}{2}}{\parallel w\parallel }_{2,\Omega }^{\frac{1}{2}},\forall v\in X,w\in {(H_0^1(\Omega )\cap H^2(\Omega ))}^3,\hfill \end{array}$$

(32)

$$\begin{array}{ccc}& & {\parallel \varphi \parallel }_{L^2(\Omega )}\le {\gamma }_{\Omega }{\parallel \tilde{\nabla }\varphi \parallel }_{0,\Omega }\forall \varphi \in H_0^1(\Omega ),\hfill \end{array}$$

(33)

$$\begin{array}{ccc}& & {\parallel \varphi \parallel }_{L^{\infty }(\Omega )}\le c_1\parallel \tilde{\nabla }{\varphi \parallel }_{0,\Omega }^{\frac{1}{2}}{\parallel \varphi \parallel }_{2,\Omega }^{\frac{1}{2}}\forall \varphi \in (H^2(\Omega )\cap H_0^1(\Omega )),\hfill \end{array}$$

(34)

$$\begin{array}{ccc}& & {\parallel \varphi \parallel }_{L^6(\Omega )}\le c_1{\parallel \tilde{\nabla }\varphi \parallel }_{0,\Omega }\forall \varphi \in H_0^1(\Omega ),\hfill \end{array}$$

(35)

where $N=(c_1+c_1^{\frac{3}{2}})(1+{\gamma }_{\Omega }^{\frac{1}{2}})$.

In fact, using (33)–(35) and the Green formula, we have

$$\begin{array}{ccc}& & {(B(u,v),w)}_{\Omega }=0.5{(u·\tilde{\nabla }v,w)}_{\Omega }+0.5{(u·\tilde{\nabla }v,w)}_{\Omega }+0.5{(\tilde{\nabla }·uv,w)}_{\Omega }\hfill \\ & =& 0.5{(u·\tilde{\nabla }v,w)}_{\Omega }+0.5\sum _{i,j=1}^3{({\tilde{\nabla }}_i(u_i{v}_j),w_j)}_{\Omega }\hfill \\ & =& 0.5{(u·\tilde{\nabla }v,w)}_{\Omega }-0.5\sum _{i,j=1}^3{(u_i{\tilde{\nabla }}_i{w}_j,v_j)}_{\Omega }\hfill \\ & =& 0.5{(u·\tilde{\nabla }v,w)}_{\Omega }-0.5{(u·\tilde{\nabla }w,v)}_{\Omega }={(u,B^{\prime }(v,w))}_{\Omega },\hfill \\ & & {|(B(u,v),w)}_{\Omega }{|\le 0.5\parallel u\parallel }_{L^3}(\parallel \tilde{\nabla }{v\parallel }_{0,\Omega }{\parallel w\parallel }_{L^6}+\parallel \tilde{\nabla }{w\parallel }_{0,\Omega }{\parallel v\parallel }_{L^6})\hfill \\ & \le & {0.5\parallel u\parallel }_{L^2}^{\frac{1}{2}}{\parallel u\parallel }_{L^6}^{\frac{1}{2}}(\parallel \tilde{\nabla }{v\parallel }_{0,\Omega }{\parallel w\parallel }_{L^6}+\parallel \tilde{\nabla }{w\parallel }_{0,\Omega }{\parallel v\parallel }_{L^6})\hfill \\ & \le & {\gamma }_{\Omega }^{\frac{1}{2}}{c}_1^{\frac{3}{2}}\parallel \tilde{\nabla }{u\parallel }_{0,\Omega }\parallel \tilde{\nabla }{v\parallel }_{0,\Omega }{\parallel \tilde{\nabla }w\parallel }_{0,\Omega }\forall u,v,w\in X,\hfill \\ & & {\parallel B(u,v)\parallel }_{0,\Omega }\le {\parallel u\parallel }_{L^6}\parallel \tilde{\nabla }{v\parallel }_{L^3}+\parallel \tilde{\nabla }{u\parallel }_{0,\Omega }{\parallel v\parallel }_{L^{\infty }}\hfill \\ & \le & {\parallel u\parallel }_{L^6}\parallel \tilde{\nabla }{v\parallel }_{L^2}^{\frac{1}{2}}\parallel \tilde{\nabla }{v\parallel }_{L^6}^{\frac{1}{2}}+c_1\parallel \tilde{\nabla }{u\parallel }_{0,\Omega }\parallel \tilde{\nabla }{v\parallel }_{0,\Omega }^{\frac{1}{2}}{\parallel v\parallel }_{2,\Omega }^{\frac{1}{2}}\hfill \\ & \le & (c_1+c_1^{\frac{3}{2}})\parallel \tilde{\nabla }{u\parallel }_{0,\Omega }\parallel \tilde{\nabla }{v\parallel }_{0,\Omega }^{\frac{1}{2}}{\parallel v\parallel }_{2,\Omega }^{\frac{1}{2}},\hfill \\ & & \parallel B^{\prime }{(v,w)\parallel }_{0,\Omega }\le 0.5(\parallel \tilde{\nabla }{v\parallel }_{0,\Omega }{\parallel w\parallel }_{L^{\infty }}+\parallel \tilde{\nabla }{w\parallel }_{L^3}{\parallel v\parallel }_{L^6})\hfill \\ & \le & 0.5c_1\parallel \tilde{\nabla }{v\parallel }_{0,\Omega }\parallel \tilde{\nabla }{w\parallel }_{L^2}^{\frac{1}{2}}{\parallel w\parallel }_{2,\Omega }^{\frac{1}{2}}+0.5c_1^{\frac{3}{2}}\parallel \tilde{\nabla }{v\parallel }_{0,\Omega }\parallel \tilde{\nabla }{w\parallel }_{0,\Omega }^{\frac{1}{2}}{\parallel w\parallel }_{2,\Omega }^{\frac{1}{2}}\hfill \\ & \le & 0.5(c_1+c_1^{\frac{3}{2}})\parallel \tilde{\nabla }{u\parallel }_{0,\Omega }\parallel \tilde{\nabla }{v\parallel }_{0,\Omega }^{\frac{1}{2}}{\parallel v\parallel }_{2,\Omega }^{\frac{1}{2}},\hfill \end{array}$$

which yield (30)–(32).

Now we discuss the existence, uniqueness and regularity results of the solution $(u,p)$ based on (29).

Theorem 1.

If $F\in H^{-1}{(\Omega )}^3$ and the uniqueness index $\sigma =\frac{{N\parallel F\parallel }_{-1,\Omega }}{{\nu }^2}$ satisfies the uniqueness condition $0<\sigma <1$, then the 3D steady Navier–Stokes equations admit a unique solution $(u,p)$ satisfying the following bound:

$$\begin{array}{c}\hfill \nu \parallel \tilde{\nabla }{u\parallel }_{0,\Omega }\le {\parallel F\parallel }_{-1,\Omega }{,\beta \parallel p\parallel }_{0,\Omega }\le 3{\parallel F\parallel }_{-1,\Omega },\end{array}$$

(36)

and if $F\in L^2{(\Omega )}^3$, then there holds the following regularity result:

$$\begin{array}{c}\hfill {\nu \parallel u\parallel }_{2,\Omega }+{\parallel p\parallel }_{1,\Omega }\le C_0{\parallel F\parallel }_{0,\Omega },\end{array}$$

(37)

where $C_0=2c_0+c_0^2{\gamma }_{\Omega }$.

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 $(u^1,p^1)\in X\times M$ and $(u^2,p^2)\in X\times M$ be the solutions of (29). Then, $(w,\eta )=(u^1-u^2,p^1-p^2)$ satisfies the following relation

$$\begin{array}{c}\hfill \nu A(w,v)-d(v,\eta )-d(w,q)+{(B(w,u^1),v)}_{\Omega }+{(B(u^2,w),v)}_{\Omega }=0,\end{array}$$

(38)

By taking $(v,q)=(w,-\eta )$ in (38) and using (30)–(31) and (36), we obtain

$$\begin{array}{c}\hfill \nu \parallel \tilde{\nabla }{w\parallel }_{0,\Omega }^2={(B(w,u^1),w)}_{\Omega }\le N\parallel \tilde{\nabla }{u}^1{\parallel }_{0,\Omega }\parallel \tilde{\nabla }{w\parallel }_{0,\Omega }^2\le \sigma \nu {\parallel \tilde{\nabla }w\parallel }_{0,\Omega }^2,\end{array}$$

(39)

which, with the uniqueness condition $0<\sigma <1$, yields $w=0$. Using again Lemma 1 and (38) with $w=0$, we deduce $\eta =0$. Thus, the uniqueness of the solution $(u,p)$ of (29) is proved. Moreover, if $F\in {(L^2(\Omega ))}^3$, we deduce from (25), (32) and (36) that

$$\begin{array}{ccc}& & {\nu \parallel u\parallel }_{2,\Omega }+{\parallel p\parallel }_{1,\Omega }\le c_0{\parallel F\parallel }_{0,\Omega }+c_0{\parallel B(u,u)\parallel }_{0,\Omega }\hfill \\ & \le & c_0{\parallel F\parallel }_{0,\Omega }+c_{0}N\parallel \tilde{\nabla }{u\parallel }_{0,\Omega }^{\frac{3}{2}}{\parallel u\parallel }_{2,\Omega }^{\frac{1}{2}}\hfill \\ & \le & \frac{\nu }{2}{\parallel u\parallel }_{2,\Omega }+c_0{\parallel F\parallel }_{0,\Omega }+0.5c_0^2{N}^2{\nu }^{-1}{\parallel \tilde{\nabla }u\parallel }_{0,\Omega }^3\hfill \\ & \le & \frac{\nu }{2}{\parallel u\parallel }_{2,\Omega }+c_0{\parallel F\parallel }_{0,\Omega }+0.5c_0^2(N^2{\nu }^{-4}{\parallel F\parallel }_{-1,\Omega }^2{)\parallel F\parallel }_{-1,\Omega }\hfill \\ & \le & \frac{\nu }{2}{\parallel u\parallel }_{2,\Omega }+c_0{\parallel F\parallel }_{0,\Omega }+0.5c_0^2{\gamma }_{\Omega }{\parallel F\parallel }_{0,\Omega },\hfill \end{array}$$

(40)

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):

$$\begin{array}{ccc}\hfill -\nu \tilde{\Delta }{u}^n+\tilde{\nabla }{p}^n+B(u^{n-1},u^n)& =& F,\mathrm{in}\Omega ,\hfill \end{array}$$

(41)

$$\begin{array}{ccc}\hfill \tilde{\nabla }·u^n& =& 0,\mathrm{in}\Omega ,\hfill \end{array}$$

(42)

$$\begin{array}{ccc}\hfill u^n& =& 0,\mathrm{on}\partial \Omega ,\hfill \end{array}$$

(43)

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 $(u,p)\in X\times M$ such that for each $(v,q)\in X\times M$ there holds

$$\begin{array}{c}\hfill \nu A(u^n,v)-d(v,p^n)-d(u^n,q)+{(B(u^{n-1},u^n),v)}_{\Omega }={(F,v)}_{\Omega },\end{array}$$

(44)

or

$$\begin{array}{c}\hfill G^{n-1}((u^n,p^n),(v,q))={(F,v)}_{\Omega },\end{array}$$

(45)

where

$$\begin{array}{c}\hfill G^{n-1}((u,p),(v,q))=A^{n-1}(u,v)-d(v,p)-d(u,q),\\ \hfill A^{n-1}(u,v)=\nu A(u,v)+{(B(u^{n-1},u),v)}_{\Omega }.\end{array}$$

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

$$\begin{array}{ccc}\hfill A^{n-1}(u,v)& \le & c_2\nu \parallel \tilde{\nabla }{u\parallel }_{0,\Omega }{\parallel \tilde{\nabla }v\parallel }_{0,\Omega },\hfill \end{array}$$

(46)

$$\begin{array}{ccc}\hfill A^{n-1}(u,u)& \ge & \nu \parallel \tilde{\nabla }{u\parallel }_{0,\Omega }^2,\hfill \end{array}$$

(47)

then there exists a ${\beta }_1>0$ such that

$$\begin{array}{ccc}& & G^{n-1}((u,p),(v,q))\le c_2(\sqrt{\nu }\parallel \tilde{\nabla }{u\parallel }_{0,\Omega }+\frac{1}{\sqrt{\nu }}{\parallel p\parallel }_{0,\Omega })(\sqrt{\nu }\parallel \tilde{\nabla }{v\parallel }_{0,\Omega }+\frac{1}{\sqrt{\nu }}{\parallel q\parallel }_{0,\Omega }),\hfill \end{array}$$

(48)

$$\begin{array}{ccc}& & {\beta }_1(\sqrt{\nu }\parallel \tilde{\nabla }{u\parallel }_{0,\Omega }+\frac{1}{\sqrt{\nu }}{\parallel p\parallel }_{0,\Omega })\le \underset{(v,q)\in X\times M}{sup}\frac{G^{n-1}((u,p),(v,q))}{\sqrt{\nu }\parallel \tilde{\nabla }{v\parallel }_{0,\Omega }+\frac{1}{\sqrt{\nu }}{\parallel q\parallel }_{0,\Omega }},\hfill \end{array}$$

(49)

$$\begin{array}{ccc}& & {\beta }_1(\sqrt{\nu }\parallel \tilde{\nabla }{v\parallel }_{0,\Omega }+\frac{1}{\sqrt{\nu }}{\parallel q\parallel }_{0,\Omega })\le \underset{(u,p)\in X\times M}{sup}\frac{G^{n-1}((u,p),(v,q))}{\sqrt{\nu }\parallel \tilde{\nabla }{u\parallel }_{0,\Omega }+\frac{1}{\sqrt{\nu }}{\parallel p\parallel }_{0,\Omega }},\hfill \end{array}$$

(50)

where $c_2\ge \sqrt{3}$.

Proof. 

First, using (46) and (47), we deduce that $G^{n-1}((u,p),(v,q))$ satisfies (48) and the following inequality

$$\begin{array}{c}\hfill G^{n-1}((u,p),(u,-p))\ge \nu {\parallel \tilde{\nabla }u\parallel }_{0,\Omega }^2,\end{array}$$

(51)

for any $(u,p),(v,q)\in X\times M$.

Using Lemma 1, for each $(u,p)\in X\times M$, we set $\epsilon =c_2^{-2}{\beta }_0^2$ and $(\tilde{v},\tilde{q})=(u-{\nu }^{-1}\epsilon \tilde{u},-p)$, where $\tilde{u}$ satisfies

$$\begin{array}{c}\hfill d(\tilde{u},p)={\parallel p\parallel }_{0,\Omega }^2,\parallel \tilde{\nabla }\tilde{u}{\parallel }_{0,\Omega }\le {\beta }_0^{-1}{\parallel p\parallel }_{0,\Omega }.\end{array}$$

(52)

Thus, we have

$$\begin{array}{ccc}\hfill \underset{(v,q)\in X\times M}{sup}\frac{G^{n-1}((u,p),(v,q))}{\sqrt{\nu }\parallel \tilde{\nabla }{v\parallel }_{0,\Omega }+\frac{1}{\sqrt{\nu }}{\parallel q\parallel }_{0,\Omega }}& \ge & \frac{G^{n-1}((u,p),(\tilde{v},\tilde{q}))}{\sqrt{\nu }\parallel \tilde{v}{\parallel }_{0,\Omega }+\frac{1}{\sqrt{\nu }}{\parallel \tilde{q}\parallel }_{0,\Omega }}\hfill \\ & \ge & \frac{\nu \parallel \tilde{\nabla }{u\parallel }_{0,\Omega }^2-c_2\epsilon \parallel \tilde{\nabla }{u\parallel }_{0,\Omega }{\parallel \tilde{\nabla }\tilde{u}\parallel }_{0,\Omega }+\epsilon {\nu }^{-1}d(\tilde{u},p)}{\sqrt{\nu }\parallel \tilde{\nabla }(u-\epsilon {\nu }^{-1}\tilde{u}){\parallel }_{0,\Omega }+\frac{1}{\sqrt{\nu }}{\parallel \tilde{q}\parallel }_{0,\Omega }}\hfill \\ & \ge & \frac{\nu \parallel \tilde{\nabla }{u\parallel }_{0,\Omega }^2-c_2\epsilon {\beta }_0^{-1}\parallel \tilde{\nabla }{u\parallel }_{0,\Omega }{\parallel p\parallel }_{0,\Omega }+\epsilon {\nu }^{-1}{\parallel p\parallel }_{0,\Omega }^2}{\sqrt{\nu }\parallel \tilde{\nabla }{u\parallel }_{0,\Omega }+(1+\epsilon {\beta }_0^{-1})\frac{1}{\sqrt{\nu }}{\parallel p\parallel }_{0,\Omega }}\hfill \\ & \ge & \frac{0.5\nu \parallel \tilde{\nabla }{u\parallel }_{0,\Omega }^2+0.5\epsilon {\nu }^{-1}{\parallel p\parallel }_{0,\Omega }^2}{\sqrt{\nu }\parallel \tilde{\nabla }{u\parallel }_{0,\Omega }+(1+\epsilon {\beta }_0^{-1})\frac{1}{\sqrt{\nu }}{\parallel p\parallel }_{0,\Omega }}\hfill \\ & \ge & {\beta }_1(\sqrt{\nu }\parallel \tilde{\nabla }{u\parallel }_{0,\Omega }+\frac{1}{\sqrt{\nu }}{\parallel p\parallel }_{0,\Omega }),\hfill \end{array}$$

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 $F\in H^{-1}{(\Omega )}^3$ and $0<\sigma <1$, then (41)–(43) admits a unique solution $(u^n,p^n)\in X\times M$ such that

$$\begin{array}{c}\hfill \nu \parallel \tilde{\nabla }{u}^n{\parallel }_{0,\Omega }\le {\parallel F\parallel }_{-1,\Omega },\beta \parallel p^n{\parallel }_{0,\Omega }\le 3{\parallel F\parallel }_{-1,\Omega }\end{array}$$

(53)

and

$$\begin{array}{ccc}\hfill \nu \parallel \tilde{\nabla }(u^n-u){\parallel }_{0,\Omega }\le {\sigma }^{n+1}{\parallel F\parallel }_{-1,\Omega },\beta {\parallel p^n-p\parallel }_{0,\Omega }& \le & 3{\sigma }^{n+1}{\parallel F\parallel }_{-1,\Omega }.\hfill \end{array}$$

(54)

Furthermore, if $F\in L^2{(\Omega )}^3$, then there holds the following regularity result:

$$\begin{array}{c}\hfill \nu \parallel u^n{\parallel }_{2,\Omega }+\parallel p^n{\parallel }_{1,\Omega }\le C_0{\parallel F\parallel }_{0,\Omega }.\end{array}$$

(55)

Proof. 

For $n=0$ and $u^{-1}=0$, we deduce from (44) that $(u^0,p^0)\in X\times M$ satisfies

$$\begin{array}{c}\hfill \nu A(u^0,v)-d(v,p^0)-d(\nu u^0,q)={(F,v)}_{\Omega },\end{array}$$

(56)

for each $(v,q)\in X\times M$. 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 $(u^0,p^0)\in X\times M$ 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

$$\begin{array}{ccc}& & |A^{n-1}{(u,v)|\le \nu \parallel u\parallel }_X{\parallel v\parallel }_X+N\parallel u^{n-1}{\parallel }_X{\parallel u\parallel }_X{\parallel v\parallel }_X\le {2\nu \parallel u\parallel }_X{\parallel v\parallel }_X,\hfill \end{array}$$

(57)

$$\begin{array}{ccc}& & A^{n-1}(u,u)\ge {\parallel \sqrt{\nu }u\parallel }_X^2,\hfill \end{array}$$

(58)

for any $u,v\in X$. Using Lemma 2 and the general Lax–Miligram theorem, we deduce that (41)–(43) admits a unique solution $(u^n,p^n)\in X\times M$ which satisfies

$$\begin{array}{ccc}& & \nu \parallel u^n{\parallel }_X\le {\parallel F\parallel }_{-1,\Omega }.\hfill \end{array}$$

(59)

Using again Lemma 1, (59), (45) with $q=0$ and the induction assumption for $n-1$, we deduce

$$\begin{array}{ccc}& & \beta \parallel p^n{\parallel }_{0,\Omega }\le \underset{v\in X}{sup}\frac{d(v,p^n)}{{\parallel v\parallel }_X}\le {\parallel F\parallel }_{-1,\Omega }+\nu \parallel u^n{\parallel }_X+N\parallel u^{n-1}{\parallel }_X{\parallel u^n\parallel }_X\hfill \\ & \le & {2\parallel F\parallel }_{-1,\Omega }+N{\nu }^{-2}{\parallel F\parallel }_{-1,\Omega }^2\le 3{\parallel F\parallel }_{-1,\Omega }.\hfill \end{array}$$

(60)

Thus, (59) and (60) yield (53).

Using again (25), (32), (53) and the induction assumptions on $n-1$, we have

$$\begin{array}{ccc}& & \nu \parallel u^n{\parallel }_{2,\Omega }+\parallel p^n{\parallel }_{1,\Omega }\le c_0{\parallel F\parallel }_{0,\Omega }+c_0{\parallel B(u^{n-1},u^n)\parallel }_{0,\Omega }\hfill \\ & \le & c_0{\parallel F\parallel }_{0,\Omega }+0.5c_{0}N(\parallel u^{n-1}{\parallel }_X^{\frac{1}{2}}\parallel u^{n-1}{\parallel }_{2,\Omega }^{\frac{1}{2}}\parallel u^n{\parallel }_X+\parallel u^{n-1}{\parallel }_X\parallel u^n{\parallel }_X^{\frac{1}{2}}\parallel u^n{\parallel }_{2,\Omega }^{\frac{1}{2}})\hfill \\ & \le & c_0{\parallel F\parallel }_{0,\Omega }+0.25\nu \parallel u^n{\parallel }_{2,\Omega }+0.25\nu {\parallel u^{n-1}\parallel }_{2,\Omega }\hfill \\ & +& \frac{1}{8}{\nu }^{-1}{N}^2{c}_0^2(\parallel u^{n-1}{\parallel }_X\parallel u^n{\parallel }_X^2+\parallel u^{n-1}{\parallel }_X^2\parallel u^n{\parallel }_X)\hfill \\ & \le & c_0{\parallel F\parallel }_{0,\Omega }+\frac{1}{4}\nu \parallel u^n{\parallel }_{2,\Omega }+\frac{1}{4}{C}_0{\parallel F\parallel }_{0,\Omega }+\frac{1}{4}{\nu }^{-4}{N}^2{c}_0^2{\parallel F\parallel }_{-1,\Omega }^3,\hfill \end{array}$$

(61)

which, with the uniqueness condition, imply (55).

Next, it follows from (44) and (29) that

$$\begin{array}{ccc}& & \nu A(u^n-u,v)+{(B(u^{n-1}-u,u^n),v)}_{\Omega }+{(B(u,u^n-u),v)}_{\Omega }\hfill \\ & & -d(v,p^n-p)-d(u^n-u,q)=0,\hfill \end{array}$$

(62)

Taking $(v,q)=(u^n-u,-p^n+p)\in X\times M$ in (62) and using (30) and (31) yields

$$\begin{array}{ccc}& & \nu \parallel u^n{-u\parallel }_X\le N\parallel u^{n-1}{-u\parallel }_X{\parallel u^n\parallel }_X\hfill \\ & \le & N{\nu }^{-2}{\parallel F\parallel }_{-1,\Omega }\nu \parallel u^{n-1}{-u\parallel }_X\le \sigma \nu {\parallel u^{n-1}-u\parallel }_X,\hfill \end{array}$$

which, with the induction assumption for $n-1$, yield

$$\begin{array}{ccc}& & \nu \parallel u^n{-u\parallel }_X\le {\sigma }^{n+1}{\parallel F\parallel }_{-1,\Omega }.\hfill \end{array}$$

(63)

Finally, using (31), (62) and Lemma 1, we obtain

$$\begin{array}{ccc}& & \beta \parallel p^n{-p\parallel }_{0,\Omega }\le \nu \parallel u^n{-u\parallel }_X+N\parallel u^{n-1}{-u\parallel }_X\parallel u^n{\parallel }_X+N\parallel u^n{-u\parallel }_X{\parallel u\parallel }_X\hfill \\ & \le & \nu \parallel u^n{-u\parallel }_X+N{\nu }^{-2}{\parallel F\parallel }_{-1,\Omega }\nu (\parallel u^n{-u\parallel }_X+\nu \parallel u^{n-1}-u{\parallel }_X).\hfill \end{array}$$

(64)

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 $X_h\times M_h$.

Let ${\tau }_h=\left\{K\right\}$ be quasi-uniformly regular partition made of tetrahedra with diameters bounded by h of $\Omega$. Define the finite element subspaces $S_h$ and $S_h^b$ of $H^1(\Omega )$ based on $P_1$ and $P_1^b$ elements as follows:

$$S_h=\{v_h\in C(\overline{\Omega })\cap H^1(\Omega );v_h{|}_K\in P_1(K),\forall K\in {\tau }_h\},$$

$$S_h^b=\{v_h\in C(\overline{\Omega })\cap H^1(\Omega );v_h{|}_K\in P_1^b(K),\forall K\in {\tau }_h\},$$

where $P_1^b$ is a bubble element on K and satisfies $P_1^b(K)=P_1(K)\oplus \mathrm{span}\widehat{b}(K)$ and $\widehat{b}(K)$ 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 $X_h\times M_h$ of $X\times M$ as

$$X_h={(S_h^b\cap H_0^1(\Omega ))}^3,M_h=S_h\cap M,$$

$$S_h^b\cap H_0^1(\Omega )=\mathrm{span}\{{\varphi }_1,{\varphi }_2,\cdots ,{\varphi }_m\},M_h=S_h\cap M=\mathrm{span}\{{\psi }_1,{\psi }_2,\cdots ,{\psi }_l\}.$$

Remark 1.

From [39,40,41], the finite element space pair $X_h\times M_h$ satisfies the discrete inf-sup condition.

We easily deduce that the above finite element spaces satisfy the following standard assumption:

• There exist the mappings ${\pi }_h\in \mathcal{L}(X;X_h)$ such that

  $$\begin{array}{c}\hfill \parallel {\pi }_h{v-v\parallel }_{0,\Omega }+h\parallel \nabla ({\pi }_{h}v-v){\parallel }_{0,\Omega }\le c_3{h}^l{\parallel v\parallel }_{H^l},l=1,2,\end{array}$$

  (65)

  for each $v\in X\cap {(H^l(\Omega ))}^d$ with $d=1$ or 3.
• The $L^2$-orthogonal projection operator ${\rho }_h:L^2(\Omega )\to S_h$ satisfies:

  $$\parallel \varphi -{\rho }_h{\varphi \parallel }_{0,\Omega }\le c_3{h}^l{\parallel \varphi \parallel }_{H^l}\forall \varphi \in H^l(\Omega ),l=1,2.$$

  (66)
• The inverse inequality holds:

  $$\parallel \tilde{\nabla }{\varphi }_h{\parallel }_{0,\Omega }\le c_3{h}^{-1}{\parallel {\varphi }_h\parallel }_{0,\Omega }\forall {\varphi }_h\in S_h.$$

  (67)
• There exists a constant ${\beta }_0>0$ such that

  $$\underset{v_h\in X_h}{sup}\frac{d(v_h,q_h)}{\parallel v_h{\parallel }_X}\ge {\beta }_0{\parallel q_h\parallel }_{0,\Omega }{q}_h\in M_h.$$

  (68)

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 $(u_h,p_h)\in X_h\times M_h$ such that for each $(v_h,q_h)\in X_h\times M_h$ there holds

$$\begin{array}{c}\hfill A(u_h,v_h)-d(v_h,p_h)-d(u_h,q_h)={(F,v_h)}_{\Omega }.\end{array}$$

(69)

Lemma 4.

If $F\in H^{-1}{(\Omega )}^3$ and the finite element space pair $X_h\times M_h$ satisfies (68), then (69) admits a unique solution $(u_h,p_h)\in X_h\times M_h$, satisfying the following bound:

$$\begin{array}{c}\hfill \parallel \tilde{\nabla }{u}_h{\parallel }_{0,\Omega }\le {\parallel F\parallel }_{-1,\Omega },{\beta }_0\parallel p_h{\parallel }_{0,\Omega }\le 2{\parallel F\parallel }_{-1,\Omega }.\end{array}$$

(70)

Proof. 

We introduce the subspace $V_h$ of $X_h$ as follows:

$$\begin{array}{c}\hfill V_h=\{v_h\in X_h;d(v_h,q_h)=0,\forall q_h\in M_h\}.\end{array}$$

Thus, we deduce from (69) that $u_h\in V_h$ satisfies

$$\begin{array}{c}\hfill A(u_h,v_h)={(F,v_h)}_{\Omega },\forall v_h\in V_h.\end{array}$$

(71)

Using the Lax–Miligram theorem with $X=V_h$, $\tilde{A}(u,v)=A(u_h,v_h)$ and (22), we show that (69) admits a unique solution $u_h\in V_h$, satisfying

$$\begin{array}{c}\hfill \parallel \tilde{\nabla }{u}_h{\parallel }_{0,\Omega }\le {\parallel F\parallel }_{-1,\Omega }.\end{array}$$

(72)

Now, we introduce a Polar set

$$V_h^0=\{g\in X_h^{\prime };<g,v_h>=0\forall v_h\in V_h\},$$

and define two dual operators $B_h{v}_h\in M_h$ and $B_h^{\prime }{q}_h\in X_h^{\prime }$ such that

$$d(v_h,q_h)={(B_h{v}_h,q_h)}_{\Omega }=<v_h,B_h^{\prime }{q}_h>\forall (v_h,q_h)\in X_h\times M_h.$$

Thus, referring to [1,2], we know that inf-sup condition (68) implies that $B_h^{\prime }$ is a isomorphic operator from $M_h$ onto $V_h^0$. Moreover, we deduce from (71) that $-{\tilde{\Delta }}_h{u}_h-F\in V_h^0$. Thus, there exists a unique $p_h\in M_h$ such that $-{\tilde{\Delta }}_h{u}_h-F=B_h^{\prime }{p}_h$ or

$$\begin{array}{c}\hfill A(u_h,v_h)-d(v_h,p_h)={(F,v_h)}_{\Omega },\forall v_h\in X_h,\end{array}$$

(73)

where the discrete Laplace operator $-{\Delta }_h{u}_h\in X_h$ is defined as

$$(-{\Delta }_h{u}_h,v_h)=A(u_h,v_h),\forall v_h\in X_h.$$

Thus, we have proved that $(u_h,p_h)\in X_h\times M_h$ is a unique solution of (69).

Using again (68) and (69) with $q=0$, we have

$$\begin{array}{c}\hfill {\beta }_0\parallel p_h{\parallel }_{0,\Omega }\le \parallel \tilde{\nabla }{u}_h{\parallel }_{0,\Omega }+{\parallel F\parallel }_{-1,\Omega }\le 2{\parallel F\parallel }_{-1,\Omega }.\end{array}$$

(74)

Combining (74) with (72) yields (70).

The proof ends. □

Due to (68), we can consider the weak formulation of the general Stokes equations: find $(u_h,p_h):X_h\times M_h$ such that for each $(v_h,q_h)\in X_h\times M_h$, there holds

$$\begin{array}{c}\hfill \tilde{A}(u_h,v_h)-d(v_h,p_h)-d(u_h,q_h)={(F,v_h)}_{\Omega }.\end{array}$$

(75)

Lemma 5.

If $F\in X^{\prime }$ and $X_h\times M_h$ satisfies the discrete inf-sup condition (68), the bilinear form $\tilde{A}(u_h,v_h)$ satisfies

$$\begin{array}{ccc}\hfill \tilde{A}(u_h,v_h)& \le & c_4\nu \parallel u_h{\parallel }_X{\parallel v_h\parallel }_X,\hfill \end{array}$$

(76)

$$\begin{array}{ccc}\hfill \tilde{A}(v_h,v_h)& \ge & c_5\nu {\parallel v_h\parallel }_X^2,\hfill \end{array}$$

(77)

then (75) admits a unique solution $(u_h,p_h)\in X_h\times M_h$ such that

$$\begin{array}{ccc}& & c_5\nu \parallel u_h{\parallel }_X\le {\parallel F\parallel }_{-1,\Omega },{\beta }_0\parallel p_h{\parallel }_{0,\Omega }\le (c_4{c}_5^{-1}+1){\parallel F\parallel }_{-1,\Omega },\hfill \end{array}$$

(78)

where $c_4\ge \sqrt{3}$.

Proof. 

First, we deduce from (75) that $u_h\in V_h$ satisfies

$$\begin{array}{c}\hfill \tilde{A}(u_h,v_h)={(F,v_h)}_{\Omega },\forall v_h\in V_h,\end{array}$$

(79)

or

$$\begin{array}{c}\hfill ({\tilde{A}}_h{u}_h,v_h)={(F,v_h)}_{\Omega },\forall v_h\in V_h.\end{array}$$

(80)

Using (76) and (77) and the Lax–Miligram theorem with $X=V_h$ and $\tilde{A}(u,v)=\tilde{A}(u_h,v_h)$, we show that (79) or (80) admits a unique solution $u_h\in V_h$ satisfying

$$\begin{array}{c}\hfill c_5\nu \parallel u_h{\parallel }_X\le {\parallel F\parallel }_{-1,\Omega }.\end{array}$$

(81)

Next, (80) shows ${\tilde{A}}_h{u}_h-F\in V_h^0$. Thus, referring to [1,2], the inf-sup condition (68) implies that $B_h^{\prime }$ is an isomorphic operator from $M_h$ onto $V_h^0$. Thus, there exists a unique $p_h\in M_h$ such that $-{\tilde{A}}_h{u}_h-F=B_h^{\prime }{p}_h$ or

$$\begin{array}{c}\hfill \tilde{A}(u_h,v_h)-d(v_h,p_h)={(F,v_h)}_{\Omega },\forall v_h\in X_h.\end{array}$$

(82)

Thus, we have proved that $(u_h,p_h)\in X_h\times M_h$ is a unique solution of (75).

Using again (68), (76) and (75) with $q_h=0$, we have

$$\begin{array}{c}\hfill {\beta }_0\parallel p_h{\parallel }_{0,\Omega }\le c_4\nu \parallel u_h{\parallel }_X+{\parallel F\parallel }_{-1,\Omega }\le (1+c_4{c}_5^{-1}){\parallel F\parallel }_{-1,\Omega }.\end{array}$$

(83)

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 $(u_h,p_h)\in X_h\times M_h$ such that for each $(v_h,q_h)\in X_h\times M_h$, there holds

$$\begin{array}{c}\hfill \nu A(u_h,v_h)-d(v_h,p_h)-d(u_h,q_h)+{(B(u_h,u_h),v_h)}_{\Omega }={(F,v_h)}_{\Omega }.\end{array}$$

(84)

Lemma 6.

If $F\in X^{\prime }$, the finite element space pair $X_h\times M_h$ satisfies (68) and $0<\sigma <1$, then (84) admits a unique solution $(u_h,p_h)\in X_h\times M_h$ satisfying the following bound:

$$\begin{array}{c}\hfill \nu \parallel \tilde{\nabla }{u}_h{\parallel }_{0,\Omega }\le {\parallel F\parallel }_{-1,\Omega },{\beta }_0\parallel p_h{\parallel }_{0,\Omega }\le 3{\parallel F\parallel }_{-1,\Omega }.\end{array}$$

(85)

Proof. 

We set a bounded convex subset K of $X_h\times M_h$ as

$$\begin{array}{c}\hfill K=\{(v_h,q_h)\in X_h\times M_h;\nu \parallel v_h{\parallel }_{0,\Omega }\le \parallel F{\parallel }_{-1,\Omega },{\beta }_0\parallel q_h{\parallel }_M\le 3\parallel F{\parallel }_{-1,\Omega }\},\end{array}$$

(86)

and define a map $T:K\to X_h\times M_h$ such that for each $(w_h,r_h)\in K$, $(u_h,p_h)=T(w_h,r_h)\in X_h\times M_h$ satisfies

$$\begin{array}{c}\hfill \nu A(u_h,v_h)+{(B(w_h,u_h),v_h)}_{\Omega }-d(v_h,p_h)-d(u_h,q_h)={(F,v_h)}_{\Omega },\forall (v_h,q_h)\in X_h\times M_h.\end{array}$$

(87)

or

$$\begin{array}{c}\hfill \tilde{A}(w_h;u_h,v_h)-d(v_h,p_h)-d(u_h,q_h)={(F,v_h)}_{\Omega },\forall v_h\in V_h.\end{array}$$

(88)

where

$$\begin{array}{c}\hfill \tilde{A}(w_h;u_h,v_h)=\nu A(u_h,v_h)+{(B(w_h,u_h),v_h)}_{\Omega }.\end{array}$$

Due to $(w_h,r_h)\in K$, we deduce from (30) and (31) and the uniqueness condition that $\tilde{A}(w_h;u_h,v_h)$ satisfies

$$\begin{array}{ccc}& & |\tilde{A}(w_h;u_h,v_h)|\le \nu |A(u_h,v_h)|+N\parallel w_h{\parallel }_X\parallel u_h{\parallel }_X\parallel v_h{\parallel }_X\le 2\nu \parallel u_h{\parallel }_X{\parallel v_h\parallel }_X,\hfill \end{array}$$

(89)

$$\begin{array}{ccc}& & \tilde{A}(w_h;u_h,u_h)=\nu {\parallel u_h\parallel }_X^2,\hfill \end{array}$$

(90)

for each $u_h,v_h\in X_h$. Using Lemma 5, we show that (87) or (88) admits a unique solution $(u_h,p_h)$ satisfying

$$\begin{array}{c}\hfill \nu \parallel u_h{\parallel }_X\le {\parallel F\parallel }_{-1,\Omega },{\beta }_0\parallel p_h{\parallel }_{0,\Omega }\le 3{\parallel F\parallel }_{-1,\Omega }.\end{array}$$

(91)

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 $(u_h,p_h)\in K$ such that $T(u_h,p_h)=(u_h,p_h)$ or (84) at least admits a solution $(u_h,p_h)\in K$. Now, we assume that $(u_h^1,p_h^1)\in K$ and $(u_h^2,p_h^2)\in K$ satisfy (84). Then $(w_h,r_h)=(u_h^1-u_h^2,p_h^1-p_h^2)$ satisfies

$$\begin{array}{c}\hfill \nu A(w_h,v_h)+{(B(w_h,u_h^1),v_h)}_{\Omega }+{(B(u_h^2,w_h),v_h)}_{\Omega }-d(v_h,r_h)-d(w_h,q_h)=0,\forall (v_h,q_h)\in X_h\times M_h.\end{array}$$

(92)

for each $(v_h,q_h)\in X_h\times M_h$. Taking $(v_h,q_h)=(w_h,-r_h)$ in (92) and using (30) and (31), we deduce

$$\begin{array}{c}\hfill \nu \parallel w_h{\parallel }_X^2\le N\parallel w_h{\parallel }_{0,\Omega }^2\parallel u_h^1{\parallel }_X\le \sigma \nu {\parallel w_h\parallel }_X^2.\end{array}$$

(93)

Thanks to $0<\sigma <1$, (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 $F\in H^{-1}{(\Omega )}^3$, $0<\sigma <1$, the finite element space pair $X_{h_i}\times M_{h_i}$ is dense in $X\times M$, satisfies (68) and $X_{h_i}\times M_{h_i}\subset X_{h_{i+1}}\times M_{h_{i+1}}$ with ${lim}_{i\to \infty }{h}_i=0$, then the finite element solutions $(u_{h_i},p_{h_i})\in X_{h_i}\times M_{h_i}$ based on (84) satisfying

$$\begin{array}{c}\hfill (u_{h_i},p_{h_i})is weak c\mathrm{on}vergent to(u,p)\mathrm{in} X\times M as i\to \infty ,\end{array}$$

(94)

here $(u,p)\in X\times M$ satisfies (29).

Proof. 

We deduce from Lemma 5 that for each $0<h<1$ (84) admits a unique solution $(u_h,p_h)\in X_h\times M_h$ satisfying (85). Applying the compact theorem in $X\times M$, there exist a sequence $\left\{h_i\right\}$ and $(u,p)$ such that (94) holds. Thanks to $X\subset {(L^2(\Omega ))}^3$ being compact, $u_{h_i}$ is strong convergent to u in ${(L^2(\Omega ))}^3$.

Thus, for a fixed $i_0$ and $i\ge i_0$, there hold

$$\begin{array}{ccc}& & {(B(u_{h_i},u_{h_i}),v_{h_{i_0}})}_{\Omega }-{(B(u,u),v_{h_{i_0}})}_{\Omega }\hfill \\ & & ={(B(u_{h_i}-u,u_{h_i}),v_{h_{i_0}})}_{\Omega }+{(B(u,u_{h_i}-u),v_{h_{i_0}})}_{\Omega }\hfill \\ & & \le 0.5\parallel u_{h_i}{-u\parallel }_{L^3(\Omega )}(\parallel \tilde{\nabla }{u}_{h_i}{\parallel }_{0,\Omega }\parallel v_{h_{i_0}}{\parallel }_{L^6(\Omega )}+\parallel u_{h_i}{\parallel }_{L^6(\Omega )}\parallel \tilde{\nabla }{v}_{h_{i_0}}{\parallel }_{0,\Omega })\hfill \\ & & +{(\parallel u\parallel }_{L^6(\Omega )}\parallel \tilde{\nabla }{v}_{h_{i_0}}{\parallel }_{0,\Omega }+\parallel \tilde{\nabla }{u\parallel }_{0,\Omega }\parallel v_{h_{i_0}}{\parallel }_{L^6(\Omega )})\parallel u_{h_i}{-u\parallel }_{L^3(\Omega )}\hfill \\ & & \le c_1^{\frac{3}{2}}\parallel u_{h_i}{-u\parallel }_{L^2(\Omega )}^{\frac{1}{2}}\parallel \tilde{\nabla }(u_{h_i}-u){\parallel }_{0,\Omega }^{\frac{1}{2}}\parallel \tilde{\nabla }{u}_{h_i}{\parallel }_{0,\Omega }{\parallel \tilde{\nabla }{v}_{h_{i_0}}\parallel }_{0,\Omega }\hfill \\ & & +2c_1^{\frac{3}{2}}\parallel u_{h_i}{-u\parallel }_{L^2(\Omega )}^{\frac{1}{2}}\parallel \tilde{\nabla }(u_{h_i}-u){\parallel }_{0,\Omega }^{\frac{1}{2}}\parallel \tilde{\nabla }{u\parallel }_{0,\Omega }{\parallel \tilde{\nabla }{v}_{h_{i_0}}\parallel }_{0,\Omega }.\hfill \end{array}$$

(95)

Thus, using (94) and (95), we deduce

$$\begin{array}{ccc}& & \underset{i\to \infty }{lim}\nu A(u_{h_i},v_{h_{i_0}})=\nu A(u,v_{h_{i_0}}),\hfill \end{array}$$

(96)

$$\begin{array}{ccc}& & \underset{i\to \infty }{lim}d(v_{h_{i_0}},p_{h_i})=d(v_{h_{i_0}},p),\hfill \end{array}$$

(97)

$$\begin{array}{ccc}& & \underset{i\to \infty }{lim}d(u_{h_i},q_{h_{i_0}})=d(u,q_{h_{i_0}}),\hfill \end{array}$$

(98)

$$\begin{array}{ccc}& & \underset{i\to \infty }{lim}{(B(u_{h_i},u_{h_i}),v_{h_{i_0}})}_{\Omega }={(B(u,u),v_{h_{i_0}})}_{\Omega }.\hfill \end{array}$$

(99)

Setting $(u_h,p_h)=(u_{h_i},p_{h_i})\in X_{h_i}\times M_{h_i}$ and $(v_h,q_h)=(v_{h_{i_0}},q_{h_{i_0}})\in X_{h_{i_0}}\times M_{h_{i_0}}$ in (84), we obtain

$$\begin{array}{c}\hfill \nu A(u_{h_i},v_{h_{i_0}})-d(v_{h_{i_0}},p_{h_i})-d(u_{h_i},q_{h_{i_0}})+{(B(u_{h_i},u_{h_i}),v_{h_{i_0}})}_{\Omega }={(F,v_{h_{i_0}})}_{\Omega }.\end{array}$$

(100)

Setting $i\to \infty$ in (100) and using (96)–(99), we obtain

$$\begin{array}{c}\hfill \nu A(u,v_{h_{i_0}})-d(v_{h_{i_0}},p)-d(u,q_{h_{i_0}})+{(B(u,u),v_{h_{i_0}})}_{\Omega }={(F,v_{h_{i_0}})}_{\Omega }.\end{array}$$

(101)

Since $X_{h_{i_0}}\times M_{h_{i_0}}$ is dense in $X\times M$, setting $i_0\to \infty$ in (101), we deduce that $(u,p)\in X\times M$ 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 $(u_h,p_h)\in X_h\times M_h$ of the Stokes equations, there hold the following error estimates.

Lemma 8.

If $F\in L^2(\Omega )$ and $X_h$ satisfy (65), then the FE solution $(u_h,p_h)$ of the Stokes equations satisfies the following error estimates:

$$\begin{array}{c}\hfill \parallel u-u_h{\parallel }_{0,\Omega }+h\parallel u-u_h{\parallel }_X\le C{h}^2{\parallel F\parallel }_{0,\Omega },{\beta }_0\parallel p-p_h{\parallel }_{0,\Omega }\le Ch{\parallel F\parallel }_{0,\Omega }.\end{array}$$

(102)

Proof. 

First, we deduce from (4) and (69) that

$$\begin{array}{ccc}& & A(u-u_h,v_h)-d(v_h,p-p_h)-d(u-u_h,q_h)=0\forall (v_h,q_h)\in X_h\times M_h.\hfill \end{array}$$

(103)

Using (103) with $q_h=0$ and (68), we deduce

$$\begin{array}{ccc}& & {\beta }_0\parallel {\rho }_{h}p-p_h{\parallel }_{0,\Omega }\le \parallel u-u_h{\parallel }_X+\sqrt{3}{\parallel p-{\rho }_{h}p\parallel }_{0,\Omega }.\hfill \end{array}$$

(104)

Next, taking $(v_h,q_h)=({\pi }_{h}u-u_h,-{\rho }_{h}p+p_h)$ in (103), using (104), (65) and (66) and (25), we deduce

$$\begin{array}{ccc}& & \frac{1}{2}\parallel u-u_h{\parallel }_X^2+\frac{1}{2}\parallel {\pi }_{h}u-u_h{\parallel }_X^2=\frac{1}{2}{\parallel {\pi }_{h}u-u\parallel }_X^2+d({\pi }_{h}u-u_h,p-{\rho }_{h}p)+d(u-{\pi }_{h}u,{\rho }_{h}p-p_h)\hfill \\ & \le & \frac{1}{2}\parallel {\pi }_h{u-u\parallel }_X^2+\sqrt{3}(\parallel {\pi }_{h}u-u_h{\parallel }_X\parallel p-{\rho }_h{p\parallel }_{0,\Omega }+\parallel u-{\pi }_h{u\parallel }_X\parallel {\rho }_{h}p-p_h{\parallel }_{0,\Omega })\hfill \\ & \le & \frac{1}{2}\parallel {\pi }_h{u-u\parallel }_X^2+\sqrt{3}\parallel {\pi }_{h}u-u_h{\parallel }_X{\parallel p-{\rho }_{h}p\parallel }_{0,\Omega }\hfill \\ & +& \sqrt{3}{\beta }_0^{-1}\parallel u-{\pi }_h{u\parallel }_X(\parallel u-u_h{\parallel }_X+\sqrt{3}\parallel {\rho }_{h}p-p{\parallel }_{0,\Omega })\hfill \\ & \le & \frac{1}{2}\parallel {\pi }_{h}u-u_h{\parallel }_X^2+\frac{1}{4}\parallel u-u_h{\parallel }_X^2+c(\parallel p-{\rho }_h{p\parallel }_{0,\Omega }^2+\parallel u-{\pi }_{h}u{\parallel }_X^2)\hfill \\ & \le & \frac{1}{2}\parallel {\pi }_{h}u-u_h{\parallel }_X^2+\frac{1}{4}\parallel u-u_h{\parallel }_X^2+c{h}^2{(\parallel u\parallel }_{2,\Omega }^2+{\parallel p\parallel }_{1,\Omega }^2)\hfill \\ & \le & \frac{1}{2}\parallel {\pi }_{h}u-u_h{\parallel }_X^2+\frac{1}{4}\parallel u-u_h{\parallel }_X^2+c{h}^2{\parallel F\parallel }_{0,\Omega }^2.\hfill \end{array}$$

(105)

Combining (104) and (105) and (66), we deduce

$$\begin{array}{c}\hfill \parallel u-u_h{\parallel }_X+{\beta }_0\parallel p-p_h{\parallel }_{0,\Omega }\le Ch{\parallel F\parallel }_{0,\Omega }.\end{array}$$

(106)

In order to estimate the $L^2(\Omega )$ bound of the error estimate $u-u_h$, we consider the dual equation of the 3D Stokes equations with the Dirichlet boundary condition

$$\begin{array}{ccc}\hfill -\tilde{\Delta }\varphi +\tilde{\nabla }\psi & =& u-u_h,\mathrm{in}\Omega ,\hfill \end{array}$$

(107)

$$\begin{array}{ccc}\hfill \tilde{\nabla }·\varphi & =& 0,\mathrm{in}\Omega ,\hfill \end{array}$$

(108)

$$\begin{array}{ccc}\hfill \varphi & =& 0,\mathrm{on}\partial \Omega .\hfill \end{array}$$

(109)

Using the Green formula, we deduce the weak formulation of dual Equations (107) and (109): find $\varphi \in X$ such that for each $v\in X$ there holds

$$\begin{array}{c}\hfill A(\varphi ,v)-d(v,\psi )-d(\varphi ,q)={(u-u_h,v)}_{\Omega }.\end{array}$$

(110)

Using again (25), we have

$$\begin{array}{c}\hfill {\parallel \varphi \parallel }_{2,\Omega }+{\parallel \psi \parallel }_{1,\Omega }\le c_0{\parallel u-u_h\parallel }_{0,\Omega }.\end{array}$$

(111)

Taking $(v,q)=(u-u_h,p-p_h)$ in (110), using (103) and (65) and (66) and (111), we deduce

$$\begin{array}{ccc}& & \parallel u-u_h{\parallel }_{0,\Omega }^2=A(u-u_h,\varphi )-d(u-u_h,\psi )-d(\varphi ,p-p_h)\hfill \\ & =& A(u-u_h,\varphi -{\pi }_h\varphi )-d(u-u_h,\psi -{\rho }_h\psi )-d(\varphi -{\pi }_h\varphi ,p-p_h)\hfill \\ & \le & \parallel u-u_h{\parallel }_X\parallel \varphi -{\pi }_h{\varphi \parallel }_X+\sqrt{3}(\parallel u-u_h{\parallel }_X\parallel \psi -{\rho }_h{\psi \parallel }_{0,\Omega }+\parallel \varphi -{\pi }_h{\varphi \parallel }_X\parallel p-p_h{\parallel }_{0,\Omega })\hfill \\ & \le & ch\parallel u-u_h{\parallel }_X{\parallel \varphi \parallel }_{2,\Omega }+ch\parallel u-u_h{\parallel }_X{\parallel \psi \parallel }_{1,\Omega }+{ch\parallel \varphi \parallel }_{2,\Omega }{\parallel p-p_h\parallel }_{0,\Omega }\hfill \\ & \le & ch(\parallel u-u_h{\parallel }_X+\parallel p-p_h{\parallel }_{0,\Omega }{)(\parallel \varphi \parallel }_{2,\Omega }+{\parallel \psi \parallel }_{1,\Omega })\hfill \\ & \le & ch(\parallel u-u_h{\parallel }_X+\parallel p-p_h{\parallel }_{0,\Omega })\parallel u-u_h{\parallel }_{0,\Omega }.\hfill \end{array}$$

(112)

Combining (112) with (106), we obtain (102).

The proof ends. □

Thanks to (68), we can define the Stokes projection $(R_h,Q_h):V\times M\to X_h\times M_h$ such that for each $(u,p)\in V\times M$, $(R_h,Q_h)=(R_h(u,p),Q_h(u,p))\in X_h\times M_h$ satisfies

$$\begin{array}{c}\hfill {(\nabla R_h,\nabla v_h)}_{\Omega }-d(v_h,Q_h)-d(R_h,q_h)={(\nabla u,\nabla v_h)}_{\Omega }-d(v_h,p)-d(u,q_h),\end{array}$$

(113)

for each $(v_h,q_h)\in X_h\times M_h$. From Lemma 4 and Lemma 8, we have

$$\begin{array}{ccc}& & \parallel R_h{\parallel }_X\le \sqrt{3}{(\parallel u\parallel }_X+{\parallel p\parallel }_{0,\Omega }),{\beta }_0\parallel Q_h{\parallel }_{0,\Omega }\le (1+\sqrt{3}){(\parallel u\parallel }_X+{\parallel p\parallel }_{0,\Omega })\forall (u,p)\in V\times M,\hfill \\ & & \parallel R_h{(u,p)-u\parallel }_{0,\Omega }+h\parallel R_h{(u,p)-u\parallel }_X+h{\beta }_0{\parallel Q_h(u,p)-p\parallel }_{0,\Omega }\hfill \end{array}$$

(114)

$$\begin{array}{ccc}& \le & c_6{h}^2{(\parallel u\parallel }_{2,\Omega }+{\parallel p\parallel }_{1,\Omega })\forall (u,p)\in (V\cap {(H^2(\Omega ))}^3)\times (M\cap H^1(\Omega )).\hfill \end{array}$$

(115)

For the FE solution $(u_h,p_h)\in X_h\times M_h$ of the Navier–Stokes equations, there hold the following error estimates.

Lemma 9.

If $F\in L^2(\Omega )$ and $X_h\times M_h$ satisfies (65)–(68), $0<\sigma <1$ and $\sigma h^{\frac{1}{2}}\le 1-\sigma$, then the FE solution $(u_h,p_h)$ satisfies the following error estimates:

$$\begin{array}{c}\hfill \nu \parallel u-u_h{\parallel }_X\le 2c_6{C}_0{h\parallel F\parallel }_{0,\Omega },{\beta }_0\parallel p-p_h{\parallel }_{0,\Omega }\le 4c_6{C}_0{h\parallel F\parallel }_{0,\Omega },\nu \parallel u-u_h{\parallel }_{0,\Omega }\le \frac{C}{1-\sigma }{h}^2{\parallel F\parallel }_{0,\Omega }.\end{array}$$

(116)

Proof. 

First, we deduce from (4), (69) and (113) that

$$\begin{array}{ccc}& & \nu A(R_h-u_h,v_h)-d(v_h,Q_h-p_h)-d(R_h-u_h,q_h)+{(B(u-u_h,u),v_h)}_{\Omega }+{(B(u_h,u-u_h),v_h)}_{\Omega }=0,\hfill \end{array}$$

(117)

or

$$\begin{array}{ccc}& & \nu A(R_h-u_h,v_h)-d(v_h,Q_h-p_h)-d(R_h-u_h,q_h)+{(B(R_h-u_h,u),v_h)}_{\Omega }+{(B(u_h,R_h-u_h),v_h)}_{\Omega }\hfill \\ & =& {(B(R_h-u,u),v_h)}_{\Omega }+{(B(u_h,R_h-u),v_h)}_{\Omega }\forall (v_h,q_h)\in X_h\times M_h,\hfill \end{array}$$

(118)

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 $0<\sigma <1$, we deduce

$$\begin{array}{ccc}& & {\beta }_0\parallel Q_h-p_h{\parallel }_{0,\Omega }\le \nu \parallel R_h-u_h{\parallel }_X+N\parallel R_h-u_h{\parallel }_X{(\parallel u\parallel }_X+\parallel u_h{\parallel }_X)\hfill \\ & +& N\parallel R_h{-u\parallel }_{0,\Omega }^{\frac{1}{2}}\parallel R_h{-u\parallel }_X^{\frac{1}{2}}{(\parallel u\parallel }_X+\parallel u_h{\parallel }_X)\hfill \\ & \le & 3\nu \parallel R_h-u_h{\parallel }_X+2\sigma c_6{h}^{\frac{3}{2}}{(\nu \parallel u\parallel }_{2,\Omega }+{\parallel p\parallel }_{1,\Omega })\hfill \\ & \le & 3\nu \parallel R_h-u_h{\parallel }_X+2\sigma c_6{h}^{\frac{3}{2}}{C}_0{\parallel F\parallel }_{0,\Omega }.\hfill \end{array}$$

(119)

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

$$\begin{array}{ccc}& & \nu (1-\sigma )\parallel R_h-u_h{\parallel }_X^2=-{(B(u-R_h,u),R_h-u_h)}_{\Omega }-{(B(u_h,u-R_h),R_h-u_h)}_{\Omega }\hfill \\ & \le & 0.5\parallel u-R_h{\parallel }_{L^3}(\parallel \tilde{\nabla }{u\parallel }_{0,\Omega }\parallel R_h-u_h{\parallel }_{L^6}+\parallel \tilde{\nabla }(R_{h}u-u_h){\parallel }_{0,\Omega }{\parallel u\parallel }_{L^6})\hfill \\ & +& \parallel u-R_h{\parallel }_{L^3}(\parallel u_h{\parallel }_{L^6}\parallel \tilde{\nabla }(R_h-u_h){\parallel }_{0,\Omega }+\parallel \tilde{\nabla }{u}_h{\parallel }_{0,\Omega }\parallel R_{h}u-u_h{\parallel }_{L^6})\hfill \\ & \le & 2c_1^{\frac{3}{2}}\parallel u-R_h{\parallel }_{0,\Omega }^{\frac{1}{2}}\parallel u-R_h{\parallel }_X^{\frac{1}{2}}{(\parallel u\parallel }_X+\parallel u_h{\parallel }_X)\parallel R_h-u_h{\parallel }_X\hfill \\ & \le & 2c_1^{\frac{3}{2}}{c}_6{h}^{\frac{3}{2}}{\nu }^{-1}{(\nu \parallel u\parallel }_{2,\Omega }+{\parallel p\parallel }_{1,\Omega }{)(\parallel u\parallel }_x+\parallel u_h{\parallel }_X)\parallel R_h-u_h{\parallel }_X\hfill \\ & \le & N{c}_6{h}^{\frac{3}{2}}{\nu }^{-2}{(\nu \parallel u\parallel }_{2,\Omega }+{\parallel p\parallel }_{1,\Omega }{)(\nu \parallel u\parallel }_x+\nu \parallel u_h{\parallel }_X)\parallel R_h-u_h{\parallel }_X\hfill \\ & \le & 2\sigma c_6{h}^{\frac{3}{2}}{C}_0{\parallel F\parallel }_{0,\Omega }{\parallel R_h-u_h\parallel }_X.\hfill \end{array}$$

(120)

Combining (119) and (120) and (65) and (66), we deduce

$$\begin{array}{c}\hfill \nu \parallel R_h-u_h{\parallel }_X\le 2c_6{C}_0{h}^{\frac{3}{2}}\frac{\sigma }{1-\sigma }{\parallel F\parallel }_{0,\Omega },{\beta }_0\parallel Q_h-p_h{\parallel }_{0,\Omega }\le 5c_6{C}_0{h}^{\frac{3}{2}}\frac{\sigma }{1-\sigma }{\parallel F\parallel }_{0,\Omega }.\end{array}$$

(121)

Combining (121) with (115) and using the stability condition $\sigma h^{\frac{1}{2}}\le 1-\sigma$, we obtain

$$\begin{array}{c}\hfill \nu \parallel u-u_h{\parallel }_X\le {Ch\parallel F\parallel }_{0,\Omega },{\beta }_0\parallel p-p_h{\parallel }_{0,\Omega }\le Ch{\parallel F\parallel }_{0,\Omega }.\end{array}$$

(122)

In order to estimate the $L^2(\Omega )$ bound of the error estimate $u-u_h$, we consider the dual equation of the 3D Navier–Stokes equations with the Dirichlet boundary condition

$$\begin{array}{ccc}\hfill -\nu \tilde{\Delta }\varphi +\tilde{\nabla }\psi +B^{\prime }(u,\varphi )-B(u_h,\varphi )& =& u-u_h,\mathrm{in}\Omega ,\hfill \end{array}$$

(123)

$$\begin{array}{ccc}\hfill \tilde{\nabla }·\varphi & =& 0,\mathrm{in}\Omega ,\hfill \end{array}$$

(124)

$$\begin{array}{ccc}\hfill \varphi & =& 0,\mathrm{on}\partial \Omega ,\hfill \end{array}$$

(125)

where ${(B(v,u),\varphi )}_{\Omega }={(v,B^{\prime }(u,\varphi ))}_{\Omega }$.

Using the Green formula, we deduce the weak formulation of the dual Equations (123) and (125): Find $(\varphi ,\psi )\in X\times M$ such that for each $(v,q)\in X\times M$ there holds

$$\begin{array}{c}\hfill \nu A(\varphi ,v)-d(v,\psi )-d(\varphi ,q)+{(B(v,u),\varphi )}_{\Omega }+{(B(u_h,v),\varphi )}_{\Omega }={(u-u_h,v)}_{\Omega }.\end{array}$$

(126)

Setting

$$\tilde{A}(\varphi ,v)=\nu A(\varphi ,v)+{(B(v,u),\varphi )}_{\Omega }+{(B(u_h,v),\varphi )}_{\Omega },$$

we deduce from (30) and (31) that

$$\begin{array}{ccc}& & |\tilde{A}{(\varphi ,v)|\le \nu \parallel \varphi \parallel }_X{\parallel v\parallel }_X+{N\parallel v\parallel }_X{\parallel \varphi \parallel }_X{(\parallel u\parallel }_X+\parallel u_h{\parallel }_X{)\le 3\nu \parallel \varphi \parallel }_X{\parallel v\parallel }_X,\hfill \end{array}$$

(127)

$$\begin{array}{ccc}& & \tilde{A}(\varphi ,\varphi )=\nu A(\varphi ,\varphi )+{(B(\varphi ,u),\varphi )}_{\Omega }\ge (1-\sigma )\nu {\parallel \varphi \parallel }_X^2.\hfill \end{array}$$

(128)

Using Lemma 3 with $A^{n-1}(u,v)=\tilde{A}(\varphi ,v)$, we show that (126) admits a unique solution $(\varphi ,\psi )\in X\times M$. Using (126)–(128) and Lemma 1, we have

$$\begin{array}{ccc}& & {\nu (1-\sigma )\parallel \varphi \parallel }_X\le {\parallel u-u_h\parallel }_{-1,\Omega },\hfill \end{array}$$

(129)

$$\begin{array}{ccc}& & {\beta \parallel \psi \parallel }_{0,\Omega }\le \parallel u-u_h{\parallel }_{-1,\Omega }+{3\nu \parallel \varphi \parallel }_X\le (1+\frac{3}{1-\sigma }){\parallel u-u_h\parallel }_{-1,\Omega }.\hfill \end{array}$$

(130)

Using again (25), (30)–(32) and (126), we have

$$\begin{array}{ccc}& & {\nu \parallel \varphi \parallel }_{2,\Omega }+{\parallel \psi \parallel }_{1,\Omega }\le c_0\parallel u-u_h{\parallel }_{0,\Omega }+c_0\parallel B^{\prime }{(u,\varphi )\parallel }_{0,\Omega }+c_0{\parallel B(u_h,\varphi )\parallel }_{0,\Omega }\hfill \\ & \le & c_0\parallel u-u_h{\parallel }_{0,\Omega }+2c_0{N(\parallel u\parallel }_X+\parallel u_h{\parallel }_X{)\parallel \varphi \parallel }_X^{\frac{1}{2}}{\parallel \varphi \parallel }_{2,\Omega }^{\frac{1}{2}}\hfill \\ & \le & \frac{1}{2}{\nu \parallel \varphi \parallel }_{2,\Omega }+c_0\parallel u-u_h{\parallel }_{0,\Omega }+4c_0^2{N}^2{\nu }^{-1}{(\parallel u\parallel }_X^2+\parallel u_h{\parallel }_X^2{)\parallel \varphi \parallel }_X\hfill \\ & \le & \frac{1}{2}{\nu \parallel \varphi \parallel }_{2,\Omega }+c_0\parallel u-u_h{\parallel }_{0,\Omega }+4c_0^2{N}^2{\nu }^{-4}{\parallel F\parallel }_{-1,\Omega }^2\nu {\parallel \varphi \parallel }_X\hfill \\ & \le & \frac{1}{2}{\nu \parallel \varphi \parallel }_{2,\Omega }+c_0\parallel u-u_h{\parallel }_{0,\Omega }+4c_0^2{\gamma }_{\Omega }\frac{1}{1-\sigma }{\parallel u-u_h\parallel }_{0,\Omega },\hfill \end{array}$$

(131)

which yields

$$\begin{array}{ccc}& & {\nu \parallel \varphi \parallel }_{2,\Omega }+{\parallel \psi \parallel }_{1,\Omega }\le 2c_0\parallel u-u_h{\parallel }_{0,\Omega }+8c_0^2{\gamma }_{\Omega }{(1-\sigma )}^{-1}{\parallel u-u_h\parallel }_{0,\Omega }.\hfill \end{array}$$

(132)

Taking $(v,q)=\nu (u-u_h,-p+p_h)$ in (126), using (118), (132) and (65) and (66), we deduce

$$\begin{array}{ccc}& & \nu \parallel u-u_h{\parallel }_{0,\Omega }^2={\nu }^{2}A(u-u_h,\varphi )-\nu d(u-u_h,\psi )-\nu d(\varphi ,p-p_h)\hfill \\ & +& \nu {(B(u-u_h,u),\varphi )}_{\Omega }+\nu {(B(u_h,u-u_h),\varphi )}_{\Omega }\hfill \\ & =& {\nu }^{2}A(u-u_h,\varphi -{\pi }_h\varphi )-\nu d(u-u_h,\psi -{\rho }_h\psi )-\nu d(\varphi -{\pi }_h\varphi ,p-p_h)\hfill \\ & +& \nu {(B(u-u_h,u),\varphi -{\pi }_h\varphi )}_{\Omega }+\nu {(B(u_h,u-u_h),\varphi -{\pi }_h\varphi )}_{\Omega }\hfill \\ & \le & {\nu }^2\parallel u-u_h{\parallel }_X\parallel \varphi -{\pi }_h{\varphi \parallel }_X+\sqrt{3}(\nu \parallel u-u_h{\parallel }_X\parallel \psi -{\rho }_h{\psi \parallel }_{0,\Omega }+\nu \parallel \varphi -{\pi }_h{\varphi \parallel }_X\parallel p-p_h{\parallel }_{0,\Omega })\hfill \\ & +& N\parallel u-u_h{\parallel }_X{(\parallel u\parallel }_X+\parallel u_h{\parallel }_X)\nu \parallel \varphi -{\pi }_h{\varphi \parallel }_X\hfill \\ & \le & \sqrt{3}{c}_{3}h(\nu \parallel u-u_h{\parallel }_X+\parallel p-p_h{\parallel }_{0,\Omega }{)(\nu \parallel \varphi \parallel }_{2,\Omega }+{\parallel \psi \parallel }_{1,\Omega })\hfill \\ & +& N\nu \parallel u-u_h{\parallel }_X{(\parallel u\parallel }_X+\parallel u_h{\parallel }_X)c_{3}h{\parallel \varphi \parallel }_{2,\Omega }\hfill \\ & \le & \frac{C}{1-\sigma }h(\nu \parallel u-u_h{\parallel }_X+\parallel p-p_h{\parallel }_{0,\Omega })\parallel u-u_h{\parallel }_{0,\Omega }.\hfill \end{array}$$

(133)

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 $(u_h^n,p_h^n)\in X_h\times M_h$ such that for each $(v_h,q_h)\in X_h\times M_h$ there holds

$$\begin{array}{c}\hfill \nu A(u_h^n,v_h)-d(v_h,p_h^n)-d(u_h^n,q_h)+{(B(u_h^{n-1},u_h^n),v_h)}_{\Omega }={(F,v_h)}_{\Omega },\end{array}$$

(134)

or

$$\begin{array}{c}\hfill A^{n-1}(u_h^n,v_h)-d(v_h,p_h^n)-d(u_h^n,q_h)={(F,v_h)}_{\Omega },\end{array}$$

(135)

where

$$\begin{array}{c}\hfill A^{n-1}(u_h,v_h)=\nu A(u_h,v_h)+{(B(u_h^{n-1},u_h),v_h)}_{\Omega }.\end{array}$$

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 $\nu \parallel u_h^{n-1}{\parallel }_X\le {\parallel F\parallel }_{-1,\Omega }$ and $0<\sigma <1$, then the bilinear form $A^{n-1}(u_h,v_h)$ satisfies

$$\begin{array}{ccc}\hfill A^{n-1}(u_h,v_h)& \le & 2\nu \parallel u_h{\parallel }_X{\parallel v_h\parallel }_X,\hfill \end{array}$$

(136)

$$\begin{array}{ccc}\hfill A^{n-1}(u_h,u_h)& =& \nu \parallel u_h{\parallel }_X^2,\hfill \end{array}$$

(137)

for each $u_h,v_h\in X_h$.

Proof. 

Using (30) and (31), we have

$$\begin{array}{ccc}\hfill A^{n-1}(u_h,v_h)& \le & \nu \parallel u_h{\parallel }_X\parallel v_h{\parallel }_X+N\parallel u_h^{n-1}{\parallel }_X\parallel u_h{\parallel }_X\parallel v_h{\parallel }_X\le 2\nu \parallel u_h{\parallel }_X{\parallel v_h\parallel }_X,\hfill \\ \hfill A^{n-1}(u_h,u_h)& =& \nu \parallel u_h{\parallel }_X^2+{(B(u_h^{n-1},u_h),u_h)}_{\Omega }=\nu {\parallel u_h\parallel }_X^2,\hfill \end{array}$$

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 $F\in H^{-1}{(\Omega )}^3$ and $0<\sigma <1$, $X_h\times M_h$ satisfies (68), then (134) or (135) admits a unique solution $(u_h^n,p_h^n)\in X_h\times M_h$ such that

$$\begin{array}{c}\hfill \nu \parallel u_h^n{\parallel }_X\le {\parallel F\parallel }_{-1,\Omega },{\beta }_0\parallel p_h^n{\parallel }_{0,\Omega }\le 3{\parallel F\parallel }_{-1,\Omega },\end{array}$$

(138)

and

$$\begin{array}{ccc}\hfill \nu \parallel u_h^n-u_h{\parallel }_X\le {\sigma }^{n+1}{\parallel F\parallel }_{-1,\Omega },{\beta }_0{\parallel p_h^n-p_h\parallel }_{0,\Omega }& \le & 3{\sigma }^{n+1}{\parallel F\parallel }_{-1,\Omega }.\hfill \end{array}$$

(139)

Proof. 

For $n=0$ and $u_h^{-1}=0$, we deduce from (134) that $(u_h^0,p_h^0)\in X_h\times M_h$ satisfies

$$\begin{array}{c}\hfill \nu A(u_h^0,v_h)-d(v_h,p_h^0)-d(\nu u_h^0,q_h)={(F,v_h)}_{\Omega },\end{array}$$

(140)

for each $(v_h,q_h)\in X_h\times M_h$.

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 $(u_h^0,p_h^0)\in X_h\times M_h$ satisfies

$$\begin{array}{c}\hfill \nu A(u_h^0-u_h,v_h)-d(v_h,p_h^0-p_h)-d(\nu u_h^0-\nu u_h,q_h)=-{(B(u_h,u_h),v_h)}_{\Omega }.\end{array}$$

(141)

Using again Lemma 4, Lemma 6 and (141), we have

$$\begin{array}{c}\hfill \nu \parallel u_h^0-u_h{\parallel }_X\le N\parallel u_h{\parallel }_X^2\le {\sigma \parallel F\parallel }_{-1,\Omega },{\beta }_0\parallel p_h^0-p_h\parallel \le 2N\parallel u_h{\parallel }_X^2\le 2\sigma {\parallel F\parallel }_{-1,\Omega }.\end{array}$$

(142)

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 $(u_h^n,p_h^n)\in X\times M$ which satisfies

$$\begin{array}{ccc}& & \nu \parallel u_h^n{\parallel }_X\le {\parallel F\parallel }_{-1,\Omega },{\beta }_0\parallel p_h^n{\parallel }_{0,\Omega }\le 3{\parallel F\parallel }_{-1,\Omega }.\hfill \end{array}$$

(143)

Next, it follows from (134) and (84) that

$$\begin{array}{ccc}& & \nu A(u_h^n-u_h,v_h)+{(B(u_h^{n-1}-u_h,u_h^n),v_h)}_{\Omega }+{(B(u_h,u_h^n-u_h),v_h)}_{\Omega }\hfill \\ & & -d(v_h,p_h^n-p_h)-d(u_h^n-u_h,q_h)=0.\hfill \end{array}$$

(144)

Taking $(v_h,q_h)=(u_h^n-u_h,-p_h^n+p_h)\in X_h\times M_h$ in (144) and using (30) and (31) yields

$$\begin{array}{ccc}& & \nu \parallel u_h^n-u_h{\parallel }_X\le N\parallel u_h^{n-1}-u_h{\parallel }_X{\parallel u_h^n\parallel }_X\hfill \\ & \le & N{\nu }^{-2}{\parallel F\parallel }_{-1,\Omega }\nu \parallel u_h^{n-1}-u_h{\parallel }_X\le \sigma \nu {\parallel u_h^{n-1}-u_h\parallel }_X,\hfill \end{array}$$

which, with the induction assumption for $n-1$, yields

$$\begin{array}{ccc}& & \nu \parallel u_h^n-u_h{\parallel }_X\le {\sigma }^{n+1}{\parallel F\parallel }_{-1,\Omega }.\hfill \end{array}$$

(145)

Finally, using (31), (145) and Lemma 6, we obtain

$$\begin{array}{ccc}& & {\beta }_0\parallel p_h^n-p_h{\parallel }_{0,\Omega }\le \nu \parallel u_h^n-u_h{\parallel }_X+N\parallel u_h^{n-1}-u_h{\parallel }_X\parallel u_h^n{\parallel }_X+N\parallel u_h^n-u_h{\parallel }_X{\parallel u_h\parallel }_X\hfill \\ & \le & \nu \parallel u_h^n-u_h{\parallel }_X+N{\nu }^{-2}{\parallel F\parallel }_{-1,\Omega }\nu (\parallel u_h^n-u_h{\parallel }_X+\nu \parallel u_h^{n-1}-u_h{\parallel }_X)\hfill \\ & \le & 2\nu \parallel u_h^n-u_h{\parallel }_X+\sigma \nu \parallel u_h^{n-1}-u_h{\parallel }_X\le 3{\sigma }^{n+1}{\parallel F\parallel }_{-1,\Omega }.\hfill \end{array}$$

(146)

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 $F\in L^2(\Omega )$ and $X_h\times M_h$ satisfies (65)–(68), $0<\sigma <1$ and $\sigma h^{\frac{1}{2}}\le 1-\sigma$, then the Oseen iterative FE solution $(u_h^n,p_h^n)$ satisfies the following error estimates:

$$\begin{array}{c}\hfill \nu \parallel u-u_h^n{\parallel }_X\le {\sigma }^{n+1}{\parallel F\parallel }_{-1,\Omega }+Ch{\parallel F\parallel }_{0,\Omega },\end{array}$$

(147)

$$\begin{array}{c}\hfill {\beta }_0\parallel p-p_h^n{\parallel }_{0,\Omega }\le 3{\sigma }^{n+1}{\parallel F\parallel }_{-1,\Omega }+Ch{\parallel F\parallel }_{0,\Omega },\end{array}$$

(148)

$$\begin{array}{c}\hfill \nu \parallel u-u_h^n{\parallel }_{0,\Omega }\le {\sigma }^{n+1}{\gamma }_{\Omega }{\parallel F\parallel }_{-1,\Omega }+\frac{C}{1-\sigma }{h}^2{\parallel F\parallel }_{0,\Omega }.\end{array}$$

(149)

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 $(u_h^n,p_h^n)\in X_h\times M_h$ such that for each $(v_h,q_h)\in X_h\times M_h$ there holds

$$\begin{array}{ccc}& & \nu A(u_h^n,v_h)-d(v_h,p_h^n)-d(u_h^n,q_h)+{(B(u_h^{n-1},u_h^n),v_h)}_{\Omega }\hfill \\ & +& {(B(u_h^n,u_h^{n-1}),v_h)}_{\Omega }-{(B(u_h^{n-1},u_h^{n-1}),v_h)}_{\Omega }={(F,v_h)}_{\Omega },\hfill \end{array}$$

(150)

or

$$\begin{array}{c}\hfill A^{n-1}(u_h^n,v_h)-d(v_h,p_h^n)-d(u_h^n,q_h)=\tilde{F}(v_h),\end{array}$$

(151)

where

$$\begin{array}{ccc}& & \tilde{F}(v_h)={(F,v_h)}_{\Omega }+{(B(u_h^{n-1},u_h^{n-1}),v_h)}_{\Omega },\hfill \\ & & A^{n-1}(u_h,v_h)=\nu A(u_h,v_h)+{(B(u_h^{n-1},u_h),v_h)}_{\Omega }+{(B(u_h,u_h^{n-1}),v_h)}_{\Omega }.\hfill \end{array}$$

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 $\tilde{F}(v_h)$ and the continuous and elliptic condition of the bilinear form $A^{n-1}(u_h,v_h)$.

Lemma 12.

If $\nu \parallel u_h^{n-1}{\parallel }_X\le \frac{6}{5}{\parallel F\parallel }_{-1,\Omega }$ and $0<\sigma \le \frac{5}{11}$, then the bilinear form $A^{n-1}(u_h,v_h)$ satisfies

$$\begin{array}{ccc}& & |\tilde{F}(v_h){|\le 2\parallel F\parallel }_{-1,\Omega }{\parallel v_h\parallel }_X,\hfill \end{array}$$

(152)

$$\begin{array}{ccc}& & A^{n-1}(u_h,v_h)\le 3\nu \parallel u_h{\parallel }_X{\parallel v_h\parallel }_X,\hfill \end{array}$$

(153)

$$\begin{array}{ccc}& & A^{n-1}(u_h,u_h)\ge \sigma \nu {\parallel u_h\parallel }_X^2,\hfill \end{array}$$

(154)

for each $u_h,v_h\in X_h$.

Proof. 

Using (30) and (31), we have

$$\begin{array}{ccc}& & |\tilde{F}(v_h){|\le \parallel F\parallel }_{-1,\Omega }\parallel v_h{\parallel }_X+N\parallel u_h^{n-1}{\parallel }_X^2\parallel v_h{\parallel }_X\le {\parallel F\parallel }_{-1,\Omega }\parallel v_h{\parallel }_X(1+\sigma \frac{36}{25})\le {2\parallel F\parallel }_{-1,\Omega }{\parallel v_h\parallel }_X,\hfill \\ & & A^{n-1}(u_h,v_h)\le \nu \parallel u_h{\parallel }_X\parallel v_h{\parallel }_X+2N\parallel u_h^{n-1}{\parallel }_X\parallel u_h{\parallel }_X\parallel v_h{\parallel }_X\le \frac{23}{11}\nu \parallel u_h{\parallel }_X{\parallel v_h\parallel }_X,\hfill \\ & & A^{n-1}(u_h,u_h)=\nu \parallel u_h{\parallel }_X^2+{(B(u_h,u_h^{n-1}),u_h)}_{\Omega }\ge \sigma \nu {\parallel u_h\parallel }_X^2,\hfill \end{array}$$

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 $F\in H^{-1}{(\Omega )}^3$ and $0<\sigma \le \frac{5}{11}$, $X_h\times M_h$ satisfies (68), then (150) or (151) admits a unique solution $(u_h^n,p_h^n)\in X_h\times M_h$ such that

$$\begin{array}{ccc}& & \nu \parallel u_h^n{\parallel }_X\le \frac{6}{5}{\parallel F\parallel }_{-1,\Omega },\hfill \end{array}$$

(155)

$$\begin{array}{ccc}& & {\beta }_0\parallel p_h^n{\parallel }_{0,\Omega }\le 3{\parallel F\parallel }_{-1,\Omega },\hfill \end{array}$$

(156)

and

$$\begin{array}{ccc}& & \nu \parallel u_h^n-u_h{\parallel }_X\le {\sigma }^{2^n}{\parallel F\parallel }_{-1,\Omega },\hfill \end{array}$$

(157)

$$\begin{array}{ccc}& & {\beta }_0\parallel p_h^n-p_h{\parallel }_{0,\Omega }\le 3{\sigma }^{2^n}{\parallel F\parallel }_{-1,\Omega }.\hfill \end{array}$$

(158)

Proof. 

For $n=0$ and $u_h^{-1}=0$, we deduce from (150) that $(u_h^0,p_h^0)\in X_h\times M_h$ satisfies

$$\begin{array}{c}\hfill \nu A(u_h^0,v_h)-d(v_h,p_h^0)-d(\nu u_h^0,q_h)={(F,v_h)}_{\Omega },\end{array}$$

(159)

for each $(v_h,q_h)\in X_h\times M_h$.

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 $(u_h^0,p_h^0)\in X_h\times M_h$ satisfies

$$\begin{array}{c}\hfill \nu A(u_h^0-u_h,v_h)-d(v_h,p_h^0-p_h)-d(\nu u_h^0-\nu u_h,q_h)=-{(B(u_h,u_h),v_h)}_{\Omega }.\end{array}$$

(160)

Using again Lemma 4, Lemma 6 and (160), we have

$$\begin{array}{ccc}& & \nu \parallel u_h^0-u_h{\parallel }_X\le N\parallel u_h{\parallel }_X^2\le \sigma {\parallel F\parallel }_{-1,\Omega },\hfill \end{array}$$

(161)

$$\begin{array}{ccc}& & {\beta }_0\parallel p_h^0-p_h\parallel \le \nu \parallel u_h^0-u_h{\parallel }_X+N\parallel u_h{\parallel }_X^2\le 2\sigma {\parallel F\parallel }_{-1,\Omega },\hfill \end{array}$$

(162)

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 $(e^n,{\eta }^n)=(u_h^n-u_h,p_h^n-p_h)$ satisfy

$$\begin{array}{ccc}& & \nu A(u_h^n,v_h)-d(v_h,p_h^n)-d(u_h^n,q_h)+{(B(u_h^n,u_h^n),v_h)}_{\Omega }-{(B(u_h^n-u_h^{n-1},u_h^n-u_h^{n-1}),v_h)}_{\Omega }=0,\hfill \end{array}$$

(163)

$$\begin{array}{ccc}& & \nu A(u_h^n,u_h^n)-{(B(u_h^n-u_h^{n-1},u_h^n-u_h^{n-1}),u_h^n)}_{\Omega }={(F,u_h^n)}_{\Omega },\hfill \end{array}$$

(164)

$$\begin{array}{c}\nu A(e^n,v_h)-d(v_h,{\eta }^n)-d(e^n,q_h)+{(B(e^n,u_h^{n-1}),v_h)}_{\Omega }\\ +{(B(u_h^{n-1},e^n),v_h)}_{\Omega }-{(B(e^{n-1},e^{n-1}),v_h)}_{\Omega }=0,\end{array}$$

(165)

which yield

$$\begin{array}{c}(1-\sigma )\nu \parallel e^1{\parallel }_X\le \nu (1-N{\nu }^{-1}\parallel u_h^0{\parallel }_X)\parallel e^1{\parallel }_X\hfill \\ \le N\parallel e^0{\parallel }_X^2\le N{\nu }^{-2}{\sigma }^2{\parallel F\parallel }_{-1,\Omega }^2\le {\sigma }^3{\parallel F\parallel }_{-1,\Omega }\le (1-\sigma )\frac{5}{6}{\sigma }^2{\parallel F\parallel }_{-1,\Omega },\end{array}$$

(166)

$$\begin{array}{c}\nu \parallel u_h^1-u_h^0{\parallel }_X\le \nu (\parallel e^1{\parallel }_X+\parallel e^0{\parallel }_X)\le (1+\frac{5}{6}\sigma ){\sigma \parallel F\parallel }_{-1,\Omega }{\parallel F\parallel }_{-1,\Omega },\hfill \end{array}$$

(167)

$$\begin{array}{c}\nu \parallel u_h^1{\parallel }_X\le N\parallel u_h^1-u_h^0{\parallel }_X^2+{\parallel F\parallel }_{-1,\Omega }\hfill \\ \le {\parallel F\parallel }_{-1,\Omega }+N{\nu }^{-2}{(1+\frac{5}{6}\sigma )}^2{\sigma }^2{\parallel F\parallel }_{-1,\Omega }^2\le [1+{(1+\frac{5}{6}\sigma )}^2{\sigma }^3]{\parallel F\parallel }_{-1,\Omega }\le \frac{6}{5}{\parallel F\parallel }_{-1,\Omega },\end{array}$$

(168)

$$\begin{array}{ccc}& & {\beta }_0\parallel p_h^1{\parallel }_{0,\Omega }\le \nu \parallel u_h^1{\parallel }_X+{\parallel F\parallel }_{-1,\Omega }+N\parallel u_h^0{\parallel }_X(\parallel u_h^1{\parallel }_X+\parallel u_h^1-u_h^0{\parallel }_X{)\le 3\parallel F\parallel }_{-1,\Omega },\hfill \end{array}$$

(169)

$$\begin{array}{ccc}& & {\beta }_0\parallel {\eta }^1{\parallel }_{0,\Omega }\le \nu \parallel e^1{\parallel }_X+N(2\parallel u_h^0{\parallel }_X\parallel e^1{\parallel }_X+\parallel e^0{\parallel }_X^2)\le 3{\sigma }^2{\parallel F\parallel }_{-1,\Omega },\hfill \end{array}$$

(170)

for $n=1$ and

$$\begin{array}{ccc}& & \sigma \nu \parallel e^n{\parallel }_X\le \nu (1-N{\nu }^{-1}\parallel u_h^{n-1}{\parallel }_X)\parallel e^n{\parallel }_X\hfill \end{array}$$

$$\begin{array}{ccc}& \le & N\parallel e^{n-1}{|}_X^2\le N{\nu }^{-2}{({\sigma }^{2^{n-1}})}^2{\parallel F\parallel }_{-1,\Omega }^2\le \sigma {\sigma }^{2^n}{\parallel F\parallel }_{-1,\Omega },\hfill \end{array}$$

(171)

$$\begin{array}{ccc}& & \nu \parallel u_h^n-u_h^{n-1}{\parallel }_X\le \nu \parallel e^n-e^{n-1}{\parallel }_X\le ({\sigma }^{2^n}+{\sigma }^{2^{n-1}}){\parallel F\parallel }_{-1,\Omega },\hfill \\ & & \nu \parallel u_h^n{\parallel }_X\le N\parallel u_h^n-u_h^{n-1}{\parallel }_X^2+{\parallel F\parallel }_{-1,\Omega }\hfill \end{array}$$

(172)

$$\begin{array}{ccc}& \le & {\parallel F\parallel }_{-1,\Omega }+\sigma {({\sigma }^{2^n}+{\sigma }^{2^{n-1}})}^2{\parallel F\parallel }_{-1,\Omega }\le \frac{6}{5}{\parallel F\parallel }_{-1,\Omega },\hfill \end{array}$$

(173)

$$\begin{array}{ccc}& & {\beta }_0\parallel p_h^n{\parallel }_{0,\Omega }\le \nu \parallel u_h^n{\parallel }_X+{\parallel F\parallel }_{-1,\Omega }+N\parallel u_h^{n-1}{\parallel }_X(\parallel u_h^n{\parallel }_X+\parallel u_h^n-u_h^{n-1}{\parallel }_X{)\le 3\parallel F\parallel }_{-1,\Omega },\hfill \end{array}$$

(174)

$$\begin{array}{ccc}& & {\beta }_0\parallel {\eta }^n{\parallel }_{0,\Omega }\le \nu \parallel e^n{\parallel }_X+2N\parallel u_h^{n-1}{\parallel }_X\parallel e^n{\parallel }_X+N\parallel e^{n-1}{\parallel }_X^2\le 3{\sigma }^{2^n}{\parallel F\parallel }_{-1,\Omega },\hfill \end{array}$$

(175)

for $n\ge 2$. 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 $F\in L^2(\Omega )$ and $X_h\times M_h$ satisfies (65)–(68) and $0<\sigma \le \frac{5}{11}$, then the Newton iterative FE solution $(u_h^n,p_h^n)$ satisfies the following error estimates:

$$\begin{array}{c}\hfill \nu \parallel u-u_h^n{\parallel }_X\le {\sigma }^{2^n}{\parallel F\parallel }_{-1,\Omega }+Ch{\parallel F\parallel }_{0,\Omega },\end{array}$$

(176)

$$\begin{array}{c}\hfill {\beta }_0\parallel p-p_h^n{\parallel }_{0,\Omega }\le 3{\sigma }^{2^n}{\parallel F\parallel }_{-1,\Omega }+Ch{\parallel F\parallel }_{0,\Omega },\end{array}$$

(177)

$$\begin{array}{c}\hfill \nu \parallel u-u_h^n{\parallel }_{0,\Omega }\le {\sigma }^{2^n}{\gamma }_{\Omega }{\parallel F\parallel }_{-1,\Omega }+C{h}^2{\parallel F\parallel }_{0,\Omega }.\end{array}$$

(178)

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 $(u_h^n,p_h^n)\in X_h\times M_h$ such that for each $(v_h,q_h)\in X_h\times M_h$ there holds

$$\begin{array}{c}\hfill \nu A(u_h^n,v_h)-d(v_h,p_h^n)-d(u_h^n,q_h)+{(B(u_h^{n-1},u_h^{n-1}),v_h)}_{\Omega }={(F,v_h)}_{\Omega },\end{array}$$

(179)

or

$$\begin{array}{c}\hfill A^{n-1}(u_h^n,v_h)-d(v_h,p_h^n)-d(u_h^n,q_h)=\tilde{F}(v_h),\end{array}$$

(180)

where

$$\begin{array}{c}\hfill \tilde{F}(v_h)={(F,v_h)}_{\Omega }-{(B(u_h^{n-1},u_h^{n-1}),v_h)}_{\Omega },A^{n-1}(u_h,v_h)=\nu A(u_h,v_h).\end{array}$$

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 $\tilde{F}(v_h)$ and the continuous and elliptic conditions of the bilinear form $A^{n-1}(u_h,v_h)$.

Lemma 14.

If $\nu \parallel u_h^{n-1}{\parallel }_X\le \frac{6}{5}{\parallel F\parallel }_{-1,\Omega }$ and $0<\sigma \le \frac{2}{5}$, then the bilinear form $A^{n-1}(u_h,v_h)$ satisfies

$$\begin{array}{ccc}& & |\tilde{F}(v_h){|\le 2\parallel F\parallel }_{-1,\Omega }{\parallel v_h\parallel }_X,\hfill \end{array}$$

(181)

$$\begin{array}{ccc}& & A^{n-1}(u_h,v_h)\le \nu \parallel u_h{\parallel }_X{\parallel v_h\parallel }_X,\hfill \end{array}$$

(182)

$$\begin{array}{ccc}& & A^{n-1}(u_h,u_h)=\nu {\parallel u_h\parallel }_X^2,\hfill \end{array}$$

(183)

for each $u_h,v_h\in X_h$.

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 $F\in H^{-1}{(\Omega )}^3$ and $0<\sigma \le \frac{2}{5}$, $X_h\times M_h$ satisfy (68), then (179) or (180) admits a unique solution $(u_h^n,p_h^n)\in X_h\times M_h$ such that

$$\begin{array}{ccc}& & \nu \parallel u_h^n{\parallel }_X\le \frac{6}{5}{\parallel F\parallel }_{-1,\Omega },{\beta }_0\parallel p_h^n{\parallel }_{0,\Omega }\le 3{\parallel F\parallel }_{-1,\Omega },\hfill \end{array}$$

(184)

$$\begin{array}{ccc}& & \nu \parallel u_h^n-u_h^{n-1}{\parallel }_X\le {(\frac{12}{5}\sigma )}^{n-1}\sigma {\parallel F\parallel }_{-1,\Omega }(n\ge 1),\hfill \end{array}$$

(185)

and

$$\begin{array}{ccc}& & \nu \parallel u_h^n-u_h{\parallel }_X\le {(\frac{11}{5}\sigma )}^n\sigma {\parallel F\parallel }_{-1,\Omega },\hfill \end{array}$$

(186)

$$\begin{array}{ccc}& & {\beta }_0\parallel p_h^n-p_h{\parallel }_{0,\Omega }\le 3{(\frac{11}{5}\sigma )}^n\sigma {\parallel F\parallel }_{-1,\Omega }.\hfill \end{array}$$

(187)

Proof. 

For $n=0$ and $u_h^{-1}=0$, we deduce from (179) that $(u_h^0,p_h^0)\in X_h\times M_h$ satisfies

$$\begin{array}{c}\hfill \nu A(u_h^0,v_h)-d(v_h,p_h^0)-d(\nu u_h^0,q_h)={(F,v_h)}_{\Omega },\end{array}$$

(188)

for each $(v_h,q_h)\in X_h\times M_h$.

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 $(u_h^0,p_h^0)\in X_h\times M_h$ satisfies

$$\begin{array}{c}\hfill \nu A(u_h^0-u_h,v_h)-d(v_h,p_h^0-p_h)-d(\nu u_h^0-\nu u_h,q_h)=-{(B(u_h,u_h),v_h)}_{\Omega }.\end{array}$$

(189)

Using again Lemma 4, Lemma 6 and (189), we have

$$\begin{array}{ccc}& & \nu \parallel u_h^0-u_h{\parallel }_X\le N\parallel u_h{\parallel }_X^2\le \sigma {\parallel F\parallel }_{-1,\Omega },\hfill \end{array}$$

(190)

$$\begin{array}{ccc}& & {\beta }_0\parallel p_h^0-p_h\parallel \le \nu \parallel u_h^0-u_h{\parallel }_X+N\parallel u_h{\parallel }_X^2\le 2\sigma {\parallel F\parallel }_{-1,\Omega }.\hfill \end{array}$$

(191)

Thus, we show that Lemma 15 holds for $n=0$.

Now, assuming that the conclusions of Lemma 15 hold for $0,\cdots ,n-1$, we want to prove that Lemma 15 holds for n.

Next, we deduce from (179), (190), (30) and (31) and the induction assumption for $0,\cdots ,n-1$ that $(u_h^n,p_h^n)$ and $(e^n,{\eta }^n)=(u_h^n-u_h,p_h^n-p_h)$ satisfy

$$\begin{array}{ccc}& & \nu A(u_h^n-u_h^{n-1},v_h)-d(v_h,p_h^n-p_h^{n-1})-d(u_h^n-u_h^{n-1},q_h)\hfill \end{array}$$

$$\begin{array}{ccc}& +& {(B(u_h^{n-1}-u_h^{n-2},u_h^{n-1}),v_h)}_{\Omega }+{(B(u_h^{n-2},u_h^{n-1}-u_h^{n-2}),v_h)}_{\Omega }=0,\hfill \end{array}$$

(192)

$$\begin{array}{ccc}& & \nu A(u_h^n,u_h^n)-{(B(u_h^{n-1},u_h^n-u_h^{n-1}),u_h^n)}_{\Omega }={(F,u_h^n)}_{\Omega },\hfill \end{array}$$

(193)

$$\begin{array}{ccc}& & \nu A(e^n,v_h)-d(v_h,{\eta }^n)-d(e^n,q_h)+{(B(e^{n-1},u_h^{n-1}),v_h)}_{\Omega }+{(B(u_h,e^{n-1}),v_h)}_{\Omega }=0,\hfill \end{array}$$

(194)

which yield

$$\begin{array}{ccc}& & \nu \parallel e^1{\parallel }_X\le N(\parallel u_h^0{\parallel }_X+\parallel u_h{\parallel }_X)\parallel e^0{\parallel }_X\hfill \end{array}$$

$$\begin{array}{ccc}& \le & 2N{\nu }^{-2}{\parallel F\parallel }_{-1,\Omega }\parallel e^0{\parallel }_X\le 2{\sigma }^2{\parallel F\parallel }_{-1,\Omega },\hfill \end{array}$$

(195)

$$\begin{array}{ccc}& & \nu \parallel u_h^1-u_h^0{\parallel }_X\le N\parallel u_h^0{\parallel }_X^2\le \sigma {\parallel F\parallel }_{-1,\Omega },\hfill \\ & & \nu \parallel u_h^1{\parallel }_X\le N\parallel u_h^1-u_h^0{\parallel }_X\parallel u_h^0{\parallel }_X+{\parallel F\parallel }_{-1,\Omega }\le {\parallel F\parallel }_{-1,\Omega }+N{\nu }^{-2}\sigma {\parallel F\parallel }_{-1,\Omega }^2\hfill \end{array}$$

(196)

$$\begin{array}{ccc}& \le & (1+{\sigma }^2){\parallel F\parallel }_{-1,\Omega }\le \frac{6}{5}{\parallel F\parallel }_{-1,\Omega },\hfill \end{array}$$

(197)

$$\begin{array}{ccc}& & {\beta }_0\parallel p_h^1{\parallel }_{0,\Omega }\le \nu \parallel u_h^1{\parallel }_X+{\parallel F\parallel }_{-1,\Omega }+N\parallel u_h^0{\parallel }_X^2\le 3{\parallel F\parallel }_{-1,\Omega },\hfill \end{array}$$

(198)

$$\begin{array}{ccc}& & {\beta }_0\parallel {\eta }^1{\parallel }_{0,\Omega }\le \nu \parallel e^1{\parallel }_X+N(\parallel u_h^0{\parallel }_X+\parallel u_h{\parallel }_X)\parallel e^0{\parallel }_X\le 3{\sigma }^2{\parallel F\parallel }_{-1,\Omega },\hfill \end{array}$$

(199)

for $n=1$ and

$$\begin{array}{ccc}& & \nu \parallel u_h^n-u_h^{n-1}{\parallel }_X\le N(\parallel u_h^{n-1}{\parallel }_X+\parallel u_h^{n-2}{\parallel }_X)\parallel u_h^{n-1}-u_h^{n-2}{\parallel }_X\hfill \end{array}$$

$$\begin{array}{ccc}& \le & \frac{12}{5}\sigma \parallel u_h^{n-1}-u_h^{n-2}{\parallel }_X\le {(\frac{12}{5}\sigma )}^{n-1}\sigma {\parallel F\parallel }_{-1,\Omega },\hfill \\ & & \nu \parallel u^n{\parallel }_X\le N\parallel u_h^{n-1}{\parallel }_X\parallel u_h^n-u_h^{n-1}{\parallel }_X+{\parallel F\parallel }_{-1,\Omega }\hfill \end{array}$$

(200)

$$\begin{array}{ccc}& \le & {\parallel F\parallel }_{-1,\Omega }+\frac{6}{5}\sigma {(\frac{12}{5}\sigma )}^{n-1}{\sigma \parallel F\parallel }_{-1,\Omega }\le \frac{6}{5}{\parallel F\parallel }_{-1,\Omega },\hfill \end{array}$$

(201)

$$\begin{array}{ccc}& & {\beta }_0\parallel p_h^n{\parallel }_{0,\Omega }\le \nu \parallel u_h^n{\parallel }_X+{\parallel F\parallel }_{-1,\Omega }+N\parallel u_h^{n-1}{\parallel }_X^2\le 3{\parallel F\parallel }_{-1,\Omega },\hfill \end{array}$$

(202)

$$\begin{array}{ccc}& & \nu \parallel e^n{\parallel }_X\le N(\parallel u_h{\parallel }_X+\parallel u_h^{n-1}{\parallel }_X)\parallel e^{n-1}{\parallel }_X\le \frac{11}{5}\sigma \parallel e^{n-1}{\parallel }_X\le {(\frac{11}{5}\sigma )}^n\sigma {\parallel F\parallel }_{-1,\Omega },\hfill \end{array}$$

(203)

$$\begin{array}{ccc}& & {\beta }_0\parallel {\eta }^n{\parallel }_{0,\Omega }\le \nu \parallel e^n{\parallel }_X+2N(\parallel u_h^{n-1}{\parallel }_X+\parallel u_h{\parallel }_X)\parallel e^{n-1}{\parallel }_X\le 3{(\frac{11}{5}\sigma )}^n\sigma {\parallel F\parallel }_{-1,\Omega },\hfill \end{array}$$

(204)

for $n\ge 2$. 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 $F\in L^2(\Omega )$ and $X_h\times M_h$ satisfies (65)–(68) and $0<\sigma \le \frac{2}{5}$, then the Stokes FE solution $(u_h^n,p_h^n)$ satisfies the following error estimates:

$$\begin{array}{c}\hfill \nu \parallel u-u_h^n{\parallel }_X\le {(\frac{11}{5}\sigma )}^n{\sigma \parallel F\parallel }_{-1,\Omega }+Ch{\parallel F\parallel }_{0,\Omega },\end{array}$$

(205)

$$\begin{array}{c}\hfill {\beta }_0\parallel p-p_h^n{\parallel }_{0,\Omega }\le 3{(\frac{11}{5}\sigma )}^n{\sigma \parallel F\parallel }_{-1,\Omega }+Ch{\parallel F\parallel }_{0,\Omega },\end{array}$$

(206)

$$\begin{array}{c}\hfill \nu \parallel u-u_h^n{\parallel }_{0,\Omega }\le {(\frac{11}{5}\sigma )}^n\sigma {\gamma }_{\Omega }{\parallel F\parallel }_{-1,\Omega }+C{h}^2{\parallel F\parallel }_{0,\Omega }.\end{array}$$

(207)

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 $(\nu ,1-\sigma )$; 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 $0<\sigma \le \frac{2}{5}$.

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 $(\nu ,1-\sigma )$; 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 $\frac{2}{5}<\sigma \le \frac{5}{11}$.

Remark 4.

From Theorem 3, we find that the Oseen iterative FE method is $H^1$-uniform stable and convergent with respect to $(\nu ,1-\sigma )$ and the Oseen iterative FE method is $L^2$-uniform stable and second-order convergent with respect to ν in the case of $\frac{5}{11}<\sigma <1$.