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
, satisfying the discrete inf-sup condition in a 2D domain
, 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
, satisfying the discrete inf-sup condition in a 2D/3D domain
, 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
and the convergence result is
under the strong stability and convergence condition
. For the Newton iterative finite element method, the stability result is
and the convergence result is
under the strong stability and convergence condition
.
In this paper, we use the finite element solution
of the 3D steady Stokes, Newton and Oseen iterative equations (the 3D steady linearized Navier–Stokes equations) to approximate the solution
of the 3D steady Navier–Stokes equations. For the Stokes iterative finite element method, the stability result is
and the convergence result is
under the weak stability and convergence condition
; for the Newton iterative finite element method, the stability result is
and the convergence result is
under the strong stability and convergence condition
. Compared with the results of [
37,
38], we obtain better stability and convergence results of the finite element iterative solution
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
of the solution
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
and
on the above equations based on the finite element space pair
are proved in
Section 3. Moreover, the uniform optimal error estimates of the mixed finite element solution
with respect to the exact solution
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
with respect to the exact solution
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
with respect to the exact solution
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
with respect to the exact solution
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 . Then, we consider the 3D steady Navier–Stokes equations and define the iterative solution by the 3D steady Oseen iterative equations (the 3D steady linearized Navier–Stokes equations) and obtain the regularity results of the Oseen iterative solution and the error bound of to .
First, we consider the 3D steady Stokes equations in
with the Dirichlet boundary condition:
where
represents the velocity,
p the pressure with and
,
the external volumetric force on the fluid. Additionally, we introduce the following notations:
,
and
.
Using the Green formula, we deduce the weak formulation of the 3D steady Stokes Equations (
1)–(
3): find
such that for each
there holds
where
,
and
.
In order to consider the existence and the uniqueness of the solution
, we recall the inf-sup condition of
in [
1,
2].
Lemma 1. There exists a positive constant β such that for each , there exists a such thator Next, we need to recall the general Lax–Milgram theorem.
General Lax–Milgram theorem. For the weak formulation, find
such that for each
, there holds
if there holds the following condition:
Then, (
7) admits a unique solution
, satisfying
Now, we give the existence, uniqueness and stability of the solution for the 3D steady Stokes equations.
Lemma 2. If , then (4) admits a unique solution , satisfying the following bound: Proof. First, we easily prove the following inequalities
for any
. Using Lemma 1, for each
, we set
and
, where
satisfies
Since
, there holds
Thus, using the general Lax–Milgram theorem, (
4) admits a unique solution
.
Taking
in (
4), we obtain
Using again Lemma 1 and (
4) with
, we have
Combining (
19) with (
20) yields (
13). The proof ends. □
Proof. We introduce the subspace
V of
X as follows:
Thus, we deduce from (
4) that
satisfies
where
satisfies
Using the Lax–Miligram theorem, (
21) admits a unique solution
such that
Now, we introduce a Polar set
and define two dual operators
and
such that
Thus, referring to [
1,
2], we know that the inf-sup condition (
6) implies that
is a isomorphic operator from
M onto
. Moreover, we deduce from (
21) that
. Thus, there exists a unique
such that
or
Due to
, there holds
for each
. Thus, we have proved that
is a unique solution of (
4). Using (
19) and (
20), we show that
satisfies (
13). The proof ends. □
Recalling Temam [
1], if
, there holds the following regularity result of the solution
for the Stokes equations:
Next, we consider the 3D steady Navier–Stokes equations with the Dirichlet boundary condition in a bounded domain
:
with
, where
.
Using the Green formula, we deduce the weak formulation of the 3D steady Navier–Stokes Equations (
26)–(
28): We find
such that for each
there holds
where
,
.
Here and hereafter, some positive constants
N,
,
and
and some inequalities are stated as follows:
where
.
In fact, using (
33)–(
35) and the Green formula, we have
which yield (
30)–(
32).
Now we discuss the existence, uniqueness and regularity results of the solution
based on (
29).
Theorem 1. If and the uniqueness index satisfies the uniqueness condition , then the 3D steady Navier–Stokes equations admit a unique solution satisfying the following bound:and if , then there holds the following regularity result:where . Proof. For the weak formulation (
29), the existence of the solution
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
and
be the solutions of (
29). Then,
satisfies the following relation
By taking
in (
38) and using (
30)–(
31) and (
36), we obtain
which, with the uniqueness condition
, yields
. Using again Lemma 1 and (
38) with
, we deduce
. Thus, the uniqueness of the solution
of (
29) is proved. Moreover, if
, we deduce from (
25), (
32) and (
36) that
which yield (
37). The proof ends. □
Setting
, we define the iterative solution
by the 3D steady Oseen iterative equations (the 3D steady linearized Navier–Stokes equations):
where
. Using the Green formula, we deduce the weak formulation of the 3D steady Oseen iterative Equations (
41)–(
43): we find
such that for each
there holds
or
where
In order to prove the existence, uniqueness and stability of the solution
based on (
45), we consider the inf-sup condition of the general bilinear form
.
Lemma 3. If the bilinear form satisfiesthen there exists a such thatwhere . Proof. First, using (
46) and (
47), we deduce that
satisfies (
48) and the following inequality
for any
.
Using Lemma 1, for each
, we set
and
, where
satisfies
Thus, we have
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 .
Theorem 2. If and , then (41)–(43) admits a unique solution such thatand Furthermore, if , then there holds the following regularity result: Proof. For
and
, we deduce from (
44) that
satisfies
for each
. Using Lemma 2, we show the existence and uniqueness of the solution
satisfying (
53) of (
56). Moreover, using (
25), (
31) and Lemma 1, we easily show that
satisfies (
54) and (
55). Thus, we show that Theorem 2 holds for
.
Now, assuming that the conclusions of Theorem 2 hold for
, we want to prove that Theorem 2 holds for
n. Using (
31) and the induction assumption for
, we deduce
for any
. Using Lemma 2 and the general Lax–Miligram theorem, we deduce that (
41)–(
43) admits a unique solution
which satisfies
Using again Lemma 1, (
59), (
45) with
and the induction assumption for
, we deduce
Thus, (
59) and (
60) yield (
53).
Using again (
25), (
32), (
53) and the induction assumptions on
, we have
which, with the uniqueness condition, imply (
55).
Next, it follows from (
44) and (
29) that
Taking
in (
62) and using (
30) and (
31) yields
which, with the induction assumption for
, yield
Finally, using (
31), (
62) and Lemma 1, we obtain
Combining (
63) and (
64) and using the induction assumption for
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 , and on the above equations based on the finite element space pair .
Let
be quasi-uniformly regular partition made of tetrahedra with diameters bounded by
h of
. Define the finite element subspaces
and
of
based on
and
elements as follows:
where
is a bubble element on
K and satisfies
and
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
of
as
Remark 1. From [39,40,41], the finite element space pair satisfies the discrete inf-sup condition. We easily deduce that the above finite element spaces satisfy the following standard assumption:
There exist the mappings such thatfor each with or 3. The -orthogonal projection operator satisfies: The inverse inequality holds: There exists a constant such that
FE method of the 3D Stokes equations.
Referring to the weak formulation (
4), we design the FE method of the 3D steady Stokes equations as follows: find
such that for each
there holds
Lemma 4. If and the finite element space pair satisfies (68), then (69) admits a unique solution , satisfying the following bound: Proof. We introduce the subspace
of
as follows:
Thus, we deduce from (
69) that
satisfies
Using the Lax–Miligram theorem with
,
and (
22), we show that (
69) admits a unique solution
, satisfying
Now, we introduce a Polar set
and define two dual operators
and
such that
Thus, referring to [
1,
2], we know that inf-sup condition (
68) implies that
is a isomorphic operator from
onto
. Moreover, we deduce from (
71) that
. Thus, there exists a unique
such that
or
where the discrete Laplace operator
is defined as
Thus, we have proved that
is a unique solution of (
69).
Using again (
68) and (
69) with
, we have
Combining (
74) with (
72) yields (
70).
The proof ends. □
Due to (
68), we can consider the weak formulation of the general Stokes equations: find
such that for each
, there holds
Lemma 5. If and satisfies the discrete inf-sup condition (68), the bilinear form satisfiesthen (75) admits a unique solution such thatwhere . Proof. First, we deduce from (
75) that
satisfies
or
Using (
76) and (
77) and the Lax–Miligram theorem with
and
, we show that (
79) or (
80) admits a unique solution
satisfying
Next, (
80) shows
. Thus, referring to [
1,
2], the inf-sup condition (
68) implies that
is an isomorphic operator from
onto
. Thus, there exists a unique
such that
or
Thus, we have proved that
is a unique solution of (
75).
Using again (
68), (
76) and (
75) with
, we have
Combining (
83) with (
81) yields (
78).
The proof ends. □
FE method of the 3D Navier-Stokes equations.
Referring to the weak formulation (
29), we design the FE method of the 3D steady Navier–Stokes equations as follows: find
such that for each
, there holds
Lemma 6. If , the finite element space pair satisfies (68) and , then (84) admits a unique solution satisfying the following bound: Proof. We set a bounded convex subset
K of
as
and define a map
such that for each
,
satisfies
or
where
Due to
, we deduce from (
30) and (
31) and the uniqueness condition that
satisfies
for each
. Using Lemma 5, we show that (
87) or (
88) admits a unique solution
satisfying
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
such that
or (
84) at least admits a solution
. Now, we assume that
and
satisfy (
84). Then
satisfies
for each
. Taking
in (
92) and using (
30) and (
31), we deduce
Thanks to
, (
93) yields
or
. Next, using (
68) and (
93) with
, we deduce
. The proof ends. □
Lemma 7. If , , the finite element space pair is dense in , satisfies (68) and with , then the finite element solutions based on (84) satisfyinghere satisfies (29). Proof. We deduce from Lemma 5 that for each
(
84) admits a unique solution
satisfying (
85). Applying the compact theorem in
, there exist a sequence
and
such that (
94) holds. Thanks to
being compact,
is strong convergent to
u in
.
Thus, for a fixed
and
, there hold
Thus, using (
94) and (
95), we deduce
Setting
and
in (
84), we obtain
Setting
in (
100) and using (
96)–(
99), we obtain
Since
is dense in
, setting
in (
101), we deduce that
satisfies (
29).
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
, we define the Oseen iterative FE solution
of the 3D steady Navier–Stokes equations: find
such that for each
there holds
or
where
In order to prove the existence, uniqueness and stability of the solution
based on (
134) or (
135), we consider the continuous and elliptic condition of the bilinear form
.
Lemma 10. If and , then the bilinear form satisfiesfor each . Proof. Using (
30) and (
31), we have
which are (
136) and (
137). The proof ends. □
Furthermore, we obtain the existence, uniqueness, stability and convergence results of the solution
based on (
134) or (
135).
Lemma 11. If and , satisfies (68), then (134) or (135) admits a unique solution such thatand Proof. For
and
, we deduce from (
134) that
satisfies
for each
.
Using Lemma 4, we show the existence and uniqueness of the solution
satisfying (
138). Moreover, using (
140) and (
84), we easily show that
satisfies
Using again Lemma 4, Lemma 6 and (
141), we have
Thus, we show that Lemma 11 holds for .
Now, assuming that the conclusions of Lemma 11 hold for
, we want to prove that Lemma 11 holds for
n. Using the induction assumption for
, Lemma 10 and Lemma 5, we deduce that (
135) admits a unique solution
which satisfies
Next, it follows from (
134) and (
84) that
Taking
in (
144) and using (
30) and (
31) yields
which, with the induction assumption for
, yields
Finally, using (
31), (
145) and Lemma 6, we obtain
Combining (
146) and (
145) and using the induction assumption for
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 with respect to the exact solution of the 3D steady Navier–Stokes equations.
Theorem 3. If and satisfies (65)–(68), and , then the Oseen iterative FE solution satisfies the following error estimates: 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
, we define the Newton iterative FE solution
of the 3D steady Navier–Stokes equations: find
such that for each
there holds
or
where
In order to prove the existence, uniqueness and stability of the solution
based on (
150) or (
151), we consider the continuous of the linear form
and the continuous and elliptic condition of the bilinear form
.
Lemma 12. If and , then the bilinear form satisfiesfor each . Proof. Using (
30) and (
31), we have
which are (
152)–(
154). The proof ends. □
Furthermore, we obtain the existence, uniqueness, stability and convergence results of the solution
based on (
150) or (
151).
Lemma 13. If and , satisfies (68), then (150) or (151) admits a unique solution such thatand Proof. For
and
, we deduce from (
150) that
satisfies
for each
.
Using Lemma 4, we show the existence and uniqueness of the solution
satisfying (
155) and (
156). Moreover, using (
159) and (
84), we easily show that
satisfies
Using again Lemma 4, Lemma 6 and (
160), we have
which yield (
157) and (
158) for
. Thus, we show that Lemma 13 holds for
.
Now, assuming that the conclusions of Lemma 13 hold for , we want to prove that Lemma 13 holds for n.
Next, we deduce from (
150), (
161), (
30) and (
31) and the induction assumption for
that
and
satisfy
which yield
for
and
for
. 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 with respect to the exact solution of the 3D steady Navier–Stokes equations.
Theorem 4. If and satisfies (65)–(68) and , then the Newton iterative FE solution satisfies the following error estimates: 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
, we define the Stokes iterative FE solution
of the 3D steady Navier–Stokes equations: find
such that for each
there holds
or
where
In order to prove the existence, uniqueness and stability of the solution
based on (
179) or (
180), we consider the continuous condition of the linear form
and the continuous and elliptic conditions of the bilinear form
.
Lemma 14. If and , then the bilinear form satisfiesfor each . 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
based on (
179) or (
180). □
Lemma 15. If and , satisfy (68), then (179) or (180) admits a unique solution such thatand Proof. For
and
, we deduce from (
179) that
satisfies
for each
.
Using Lemma 4, we show the existence and uniqueness of the solution
satisfying (
184) and (
185). Moreover, using (
188) and (
84), we easily show that
satisfies
Using again Lemma 4, Lemma 6 and (
189), we have
Thus, we show that Lemma 15 holds for .
Now, assuming that the conclusions of Lemma 15 hold for , we want to prove that Lemma 15 holds for n.
Next, we deduce from (
179), (
190), (
30) and (
31) and the induction assumption for
that
and
satisfy
which yield
for
and
for
. 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 with respect to the exact solution of the 3D steady Navier–Stokes equations.
Theorem 5. If and satisfies (65)–(68) and , then the Stokes FE solution satisfies the following error estimates: