1. Introduction
The double-diffusive natural convection model, which does not only incorporate the velocity vector field as well as the pressure field, but also contains the temperature field and the concentration field, has been widely used in scientific, engineering and industrial applications such as nuclear design, cooling of electronic equipment, aircraft cabins, insulation with double pane windows, and so on. For greater understanding of the physical background, authors can refer to [
1,
2,
3]. In recent years, the impact of nanofluid on free convection heat transfer was investigated by researchers in [
4]. The free convective flow of a Nano-Encapsulated Phase Change Material (NEPCM) suspension in an eccentric annulus was investigated numerically in [
5]. The authors obtained that the volume fraction of the NEPCM particles and Stefan number effect the thermal and hydrodynamic characteristics of the suspension. The effect of the arrangement of the tubes in a multi-tube heat exchanger during the solidification process was considered in [
6], which focused on the natural convection effect in phase change material in this research.
Let be a open bounded domain with a Lipschitz continuous boundary and is a subset of , denotes the velocity field, p is the fluid pressure, T is the temperature, C is the concentration, is the gravitational acceleration vector, is the forcing function, . Moreover, represents the outer normal vector, is the viscosity, is the Darcy number, is the heat diffusivity, is the mass diffusivity, and are the thermal and solutal expansion coefficients.
The governing equations of this double-diffusive natural convection model are presented as follows [
7].
Many numerical studies were made concerning the double-diffusive natural convection model. A projection-based stabilized finite element method for steady-state natural convection problem was considered in [
8]. A stabilized finite element error analysis for the Darcy–Brinkman model of double-diffusive convection in a porous medium was discussed in [
9]. An efficient two-step algorithm for the steady-state natural convection problem was presented in [
10]. The melting process of a nano-enhanced phase change material in a square cavity was investigated in [
11]. In numerical test, the author used the Galerkin finite element method to solve the dimensionless partial differential equations. Based on the idea of curvature stabilization, Çıbık et al. [
12] discussed a family of second order time stepping methods for the Darcy–Brinkman equations. A decoupled finite element method called the modified characteristics method was considered in [
13]. Rajabi et al. performed the detailed uncertainty propagation analysis and variance-based global sensitivity analysis on the widely adopted double-diffuse convection benchmark problem of a square porous cavity with horizontal temperature and concentration gradients in [
14]. In [
15], the mixed convection heat transfer of AL2O3 nanofuid in a horizontal channel subjected with two heat sources was considered. In [
16], the curvature-based stabilization method was considered for double-diffusive natural convection flows in the presence of a magnetic field and unconditionally stable and optimally accurate second order approximations were obtained. There are several works devoted to the efficient numerical methods for the treatment of nonlinear problems. For example, several iterative methods for the 2D steady penalty Navier–Stokes equations were presented and discussed in [
17]. He et al. [
18] discussed a combination of two-level methods and iterative methods for solving the 2D/3D steady Navier–Stokes equations. Some iterative finite element methods for steady Navier–Stokes equations with different viscosities were discussed in [
19]. Furthermore, the authors refer to the Oseen method [
20], the Newton method [
21] and the Euler implicit-explicit methods [
22]. Recently, Huang et al. [
23] have considered and analyzed the Oseen, Newton and Stokes iterative methods for the 2D steady Navier–Stokes equations. He et al. [
24] considered and analyzed three iterative methods for the 3D steady MHD equations.
The main work in this paper is to design, analyze, and compare three iteration methods to solve nonlinear equations based on the finite element discretization. Then, we will show the performance of these numerical methods in both theoretical analysis and numerical experiments. By setting , we obtain the conclusion that the three iterative methods are stable and convergent as . Iterative method I and II are valid as and only iterative method I runs well as .
In this paper, by developing some techniques and using some ideas in [
7], we prove the existence and uniqueness with a different method, then we obtain a different uniqueness condition. Furthermore, we propose and analyze iterative methods I and III. In addition to this, we use iterative method II to computer a smaller viscosity than them in numerical experiments. Compared with He et al. [
24], although the iterative methods are the same, the considered problems are different.
The paper is organized as follows. In
Section 2, we describe the considered problem and some mathematical preliminaries. In the next section, we prove the existence and uniqueness of the weak solution to the considered equations. Then, we analyze stability and iterative error estimates of three iterative methods in
Section 4. In
Section 5, we show some numerical experiments to verify the correctness of theoretical results. In the last section, conclusions are presented.
2. Preliminaries
In this section, we present some basic notations and properties for the stationary double-diffusive natural convection problem. First, for
and
we use standard notations for Sobolev space
and Lebegue space
. In particular,
norm and its inner product are denoted by
and
Moreover, for
f, an element in the dual space of
, its norm is defined by
Next, we introduce the functional spaces associated with the velocity, the pressure, the temperature, and the concentration:
Then, we define the following particular subspace of the velocity space
Moreover, define several continuous bilinear forms
and
on
and
, respectively,
Further, denote three skew-symmetric trilinear forms:
Please note that the bilinear form
is continuous on
and satisfies the inf-sup condition [
25]: there exists a positive constant
such that
The trilinear forms [
18] satisfy
and
where
are three constants defined as
and
Furthermore, we recall the Poincaré inequality [
25]
where
is a positive constant depending on
.
The variational form of the model (
1) is presented as follows: find
such that for all
3. Existence and Uniqueness
This section gives the existence and uniqueness of (
5), which is crucial to consider the discrete scheme.
Theorem 1. There exists at least a solution pair which satisfies (
5)
and Proof.
First, for
, it is easy to see that
and
are continuous, elliptic bilinear forms of
and
, respectively. Hence, according to the Lax–Milgram theorem, there exists a unique solution
to the second equation of (
5), and a unique solution
to the third equation of (
5). The theorem will be proved if we can show that there is at least a solution
in the first equation of (
5).
Secondly,
is a continuous and elliptic bilinear form on
. Using (
2) and (
4) we obtain
where
. Then, we define a mapping
by
where
Clearly,
is a solution of the first equation of (
5) with
, if it is a solution of
Using the Leray-Schauder Principle [
26],
has at least one solution
, if (a) A is completely continuous; (b) there exists
such that for every
and
with
satisfies the bound
.
Assume
and subtract the equations obtained from (
7) with
and
. Then, set
and choose
to yield
Moreover, in order to estimate
and
, we substitute
and
in the second equation of (
5) and subtract the ensuing equations to obtain
Taking
and using (
3) we obtain
Analogously, we have
Further, combining (
9) and (
10), we obtain the bound of (
8) as follows
Hence,
A is completely continuous.
Now, we prove (b). If
then
and
. Assume
and
satisfies
. Then, from (
7), we have
Using (
2) and (
4), we arrive at
Thirdly, setting
in the second equation of (
5), we have
Thus, applying (
3) leads to
Similarly, taking
in the third equation of (
5), we obtain
Moreover, choosing
in the first equation of (
5) and using (
4), we arrive at
which combines with the above two equations to give
The proof is completed. □
Theorem 2. Assume that is a solution pair of (5). If ν, , C and T satisfy the following uniqueness conditionthen is unique solution pair of (5). Proof. Suppose
is also a solution pair of (
5) and
then
for all
.
Now, choosing
and
we obtain
Hence, applying (
4), (
9), (
10), Theorem 1 and the uniqueness condition, we have
a contradiction. Hence,
. □
4. Several Iterative Methods Based on the Finite Element Discretization
In this section, we propose three iterative methods for the double-diffusive natural convection model. Then the stability and convergence of these iterative methods are considered. First, let
denote the mesh size which is a real positive parameter and
be a uniform partition of
into non-overlapping triangles. Next, given a
, we consider the finite element spaces
,
and
where
represents the space of the order polynomial on the set
,
. Please note that the Taylor-Hood element
satisfies the following discret inf-sup condition
where the constant
is independent of
h.
With the above notations, the finite element scheme for the natural convection problem is defined as follows: find
such that
for all
The following stability and convergence results of the numerical solutions to (
12) are showed.
Theorem 3. ([
7,
8,
26,
27])
Let . Under the assumption of Theorem 2, the numerical solution pair to (12) satisfies and Moreover, the following error estimate holds where c is a positive constant depending on h.
In the following part of this section, we propose and analyse three iterative methods.
Iterative method I. Find
such that
for all
.
Iterative method II. Find
such that
for all
Iterative method III. Find
such that
for all
For the above three iterative methods, the initial guess
is defined by solving the following equations
for all
Now, we will establish the stability and iterative error estimates of the presented iterative methods for the double-diffusive natural convection model. For the sake of simplicity, let
Theorem 4. Under the assumptions of Theorem 3, defined by iterative method I satisfiesfor all . Furthermore, the following iterative error bounds holdfor all . Proof. First, the induction method is used to consider the stability of iterative method I. Setting
in (
16) leads to
which shows that (
17) holds for
.
Next, assuming that it holds for
, we prove that it is valid for
. Taking
in (
13) with
and applying (
2), (
3) and (
4) yield
Hence, we finish the induction method.
Moreover, we consider the iterative error estimates of iterative method I. Making use of (
12) and (
13) yields the error equations
Setting
,
in the second and the third equation of (
20) and using (
3), (
17), and Theorem 3, we obtain
Then, taking
in the first equation of (
20) and using (
2), (
4), (
17), (
21) and Theorem 3, we arrive at
Hence, using uniqueness condition, we have
Furthermore, subtracting (
16) from (
12), we obtain
Applying (
4), the Theorem 2 and the Theorem 3, we obtain
which combines with (
21) and (
22), we arrive at
for all
Finally, applying the discrete inf-sup condition, from the first equation of (
20) with
, the error estimate of the pressure can be stated as follows.
for all
. □
Theorem 5. Under the assumptions of Theorem 3, suppose that the following condition (the strong uniqueness condition)holds. Then generated by iterative method II satisfiesfor all . Furthermore, the following iterative error bounds holdfor all . Proof. Combining with (
19) and (
23), it is found that (
25) and (
26) hold for
. Supposing that (
25) and (
26) hold for
, we shall prove that they are valid for
.
Subtracting (
14) from (
12), we obtain the error equations
Setting
in (
27) with
and applying (
2), (
3), (
4) and the assumptions on
we have
and
Moreover, imply the strong uniqueness condition (
24) on (
29), we obtain
Hence, making use of (
30), we rewrite (
28) as
Combining the first equation of (
27) with
and
and the discrete inf-sup condition, we have
Furthermore, subtracting (
16) from (
14) with
that
Then, taking
in the second equation of (
33), we observe that
and
Moreover, setting
in the first equation of (
33), we obtain
Combining (
14) with
and using (
34), we obtain
In view of the strong uniqueness condition (
24), we arrive at
Next, taking
in (
14) with
, and using (
2), (
3) and (
26), we obtain
Similarly, we obtain
Finally, it has
The proof is completed. □
Theorem 6. Under the assumptions of Theorem 3, suppose that the following condition (the stronger uniqueness condition),holds. Then defined by the iterative method III
satisfiesfor all . Furthermore, the following iterative error bounds holdfor all . Proof. From (
19) and (
23), it is obvious that (
36) and (
37) hold for
. Supposing that (
36) and (
37) hold for
, we shall prove that they are valid for
.
Setting
in (
15) with
and using (
2), (
3), (
4) and (
36), we obtain that
Hence, (
36) is valid for
. Consequently, subtracting (
15) from (
12) yields
Now, choosing
, in the second equation of (
38) and using (
3), (
36), (
37) and Theorem 3, we can deduce that
Similarly, one has
Moreover, taking
in the first equation of (
38) and using (
2), (
4), (
36), (
37) and the Theorem 3, we find that
Finally, combining the first equation of (
38) with
and the discrete inf-sup condition, the error estimate for the pressure can be stated as follows
□
6. Conclusions
In conclusion, for solving stationary double-diffusive natural convection equations, three iterative methods have their own advantages under different viscosity numbers. In the case of , all methods can export data. Moreover, in the case of , iterative method I and II can run well. Finally, in the case of , only iterative method I can export data.
From the perspective of physical applications, these finite element iterative methods can be used to simulate different double-diffusive natural convection models, such as the aluminum oxide nanofluid natural convection heat transfer, the natural convection flow of a suspension containing nano-encapsulated. Furthermore, some different boundary conditions of these models with some different calculation areas should be considered, such as the T-geometry enclosure porous cavity, L-geometry cavity, and porous cavity.