1. Introduction
Arising both in nature and in engineering applications, the natural convection model is a coupled system of fluid flow governed by the incompressible NavierStokes equations and heat transfer governed by the energy equation. The natural convection problem has been a hot topic in heat transmission science for a long time, because it has been widely used in many fields of production and life, such as room ventilation, general heating, nuclear reaction systems, fire control, katabatic winds, atmospheric fronts, cooling of electronic equipment, natural ventilation, solar collectors, and so on [
1,
2,
3]. In particular with nanofluids, the literature survey in [
4] evidences the parameters governing the flow and heat behavior of fluids under natural convection and reveals that there are very few generalized correlations between heat transfer and wall heating conditions in enclosures.
Due to its practical significance, a considerable amount of researchers have put forward many efficient numerical methods to obtain the solution to this problem in different geometries [
5,
6,
7,
8,
9,
10]. For example, Boland and Layton [
6,
7] have proposed a Galerkin finite element method for the natural convection problem. Several iterative schemes based on the finite element method for the natural convection equations with different Rayleigh numbers have been studied in [
9]. The coupled NavierStokes/temperature (or Boussinesq) equations [
5] were solved by applying a divergencefree low order stabilized finite element method. A unified analysis approach of a local projection stabilization finite element method for solving natural convection problems was given by [
8]. However, there still remain some important but challenging problems, especially solving the model effectively with the strong coupling between the velocity, pressure, and temperature fields and the saddlepoint problem arising from finite element discretization.
As is known, the Uzawa method [
11] is an efficient iterative algorithm for the saddlepoint system. Since it is simple, efficient, and has minimal computer memory requirements, it has been widely used in computational science and engineering [
12,
13,
14,
15,
16]. In particular, some Uzawa iterative methods were designed for the steady incompressible NavierStokes equations [
17]. Further, the steady magnetohydrodynamic equations [
18] and the steady natural convection equations [
19] were solved by applying some Uzawa iterative algorithms. However, in these works, the weakly divergencefree constraint on the velocity was not enforced.
Recently, a Uzawatype iterative algorithm [
20] was designed for the coupled Stokes equations, where no saddle point system was required to be solved at each iteration step, and the weakly divergencefree velocity approximation was shown. Inspired by [
20], in this article we propose and analyze a Uzawatype iterative algorithm for the natural convection problem and obtain a numerical velocity, which satisfies the weakly divergencefree condition.
2. Preliminaries
Let
$\Omega \subset {\mathbb{R}}^{2}$ be a bounded domain, which has a Lipschitz continuous boundary
$\partial \Omega $ with a regular open subset
$\Gamma $. Consider the following stationary natural convection problem. Seek the velocity
$\mathbf{u}={({u}_{1}(x),{u}_{2}(x))}^{\top}$, the pressure
$p=p(x)$, and the temperature
$T(x)$, such that
where
$\gamma $ is the forcing function,
$\mathbf{n}$ is the outward unit vector, and
$\mathbf{j}={(0,1)}^{\top}$. In addition, the positive parameter
$\kappa $ presents the thermal conductivity,
$Pr$ is the Prandtl number, and
$Ra$ is the Rayleigh number.
Next, in order to write the variational form of (
1)–(4), we introduce the following necessary function spaces:
Here, the space ${L}^{2}(\Omega )$ is endowed with ${L}^{2}$scalar product $(\xb7,\xb7)$ and ${L}^{2}$norm $\parallel \xb7\parallel $. In addition, the space ${H}^{1}(\Omega )$ is used to represent the standard definitions for Sobolev spaces ${W}^{m,p}(\Omega )$, $m,p>0$.
Moreover, we recall the Poincaré inequality [
21] as follows:
where
${C}_{p}$ is the Poincaré constant. Next, we denote two trilinear forms by
which satisfy the following properties [
7,
22,
23]
for all
$\mathbf{u},\mathbf{v},\mathbf{w},\in \mathbf{M}$ and
$T,s\in Z$. Here,
N and
$\overline{N}$ are two fixed positive constants.
With the above notations, the weak form of (
1)–(4) reads as: find
$(\mathbf{u},p,T)\in \mathbf{M}\times W\times Z$ such that
The following existence and uniqueness of the solution to (
6) are classical results.
Theorem 1 ([
7,
19])
. There exists at least a solution $(\mathbf{u},p,T)\in \mathbf{M}\times W\times Z$, which satisfies (7)–(9) andwhere ${\parallel \gamma \parallel}_{1}=\underset{s\in Z}{sup}\frac{(\gamma ,s)}{\parallel \nabla s\parallel}.$ Further, if $Pr$, $Ra$, κ, and γ satisfy the uniqueness conditionwhere $\Lambda ={C}_{p}^{2}RaN{\kappa}^{1}{\parallel \gamma \parallel}_{1}$ and $\overline{\Lambda}={C}_{p}^{2}Ra\overline{N}{\kappa}^{2}{\parallel \gamma \parallel}_{1}$, then the solution $(\mathbf{u},p,T)$ of (7)–(9) is unique. Next, we consider a family of quasiuniform and regular triangulations
${K}_{h}=\{K:{\cup}_{K\subset \Omega}\overline{K}=\overline{\Omega}\}$ with mesh size
h, which is a partition of the domain
$\Omega $. Then, we assume that the finite element subspace
${\mathbf{M}}_{h}\times {W}_{h}\times {Z}_{h}\subset \mathbf{M}\times W\times Z$
where
${P}_{i}(K)$,
$i=1,2$ is the set of all polynomials on
K of a degree no more than
i. As is known, the finite element subspaces
${\mathbf{M}}_{h}\times {W}_{h}$ satisfy the following discrete infsup condition [
21]; for each
$q\in {W}_{h}$, there exists
$\mathbf{v}\in {\mathbf{M}}_{h},\mathbf{v}\ne 0$ such that
${inf}_{q\in {W}_{h}}{sup}_{\mathbf{v}\in {\mathbf{M}}_{h}}\frac{(\nabla \xb7\mathbf{v},q)}{\parallel \nabla \mathbf{v}\parallel \parallel q\parallel}\ge \beta $, where the constant
$\beta \in (0,1]$ is proven in [
24].
Moreover, according to the above definition of the finite element subspaces, the finite element approximation for (
7)–(9) is to seek
$({\mathbf{u}}_{h},{p}_{h},{T}_{h})\in {\mathbf{M}}_{h}\times {W}_{h}\times {Z}_{h}$ such that
The following theorem is established for the stability of the finite element discretization.
Theorem 2 ([
6,
9,
25])
. Under the assumptions of Theorem 1, the finite element discretization (10)–(12) has at least a solution $({\mathbf{u}}_{h},{p}_{h},{T}_{h})\in {\mathbf{M}}_{h}\times {W}_{h}\times {Z}_{h}$, such that 3. A UzawaType Iterative Algorithm
In this section, we present a Uzawatype iterative algorithm for solving the considered problem. Before showing the algorithm, we recall the common Uzawa iterative algorithm based on the mixed finite element method as follows Algorithm 1.
According to the above algorithm, we find that
$(\nabla \xb7{\mathbf{u}}_{h}^{n+1},q)\ne 0$, which means that the divergencefree constraint on the velocity is not weakly enforced. In fact, from the finite element approximation (
10)–(12), we have
$(\nabla \xb7{\mathbf{u}}_{h},q)=0$. Although it will result in a saddle problem, it produces weakly divergencefree velocity approximation. Hence, it is interesting to design a Uzawatype iterative algorithm, which does not only retain the benefits of the common Uzawa iterative algorithm but also retains the velocity in a weakly divergencefree condition.
Algorithm 1: Uzawa iterative algorithm [19]. 
Step 1. Find initial guess $({\mathbf{u}}_{h}^{0},{p}_{h}^{0},{T}_{h}^{0})\in {\mathbf{M}}_{h}\times {W}_{h}\times {Z}_{h}$ by
Step 2. Given a relaxation parameter $\rho >0$, find $({\mathbf{u}}_{h}^{n+1},{p}_{h}^{n+1},{T}_{h}^{n+1})\in {\mathbf{M}}_{h}\times {W}_{h}\times {Z}_{h}$ as solution of

In order to make the velocity of Uzawa algorithm have a weakly divergencefree property, let
g be a gauge variable [
26] and
$\mathbf{d}$ be a variable, such that
$\mathbf{u}=\mathbf{d}+\nabla g$. If
g and
p satisfy an elliptic equation
$Pr\Delta g=p$, then (
1)–(4) can be rewritten as
Furthermore, begin with
${g}^{0}={g}^{1}=0$ and
${\mathbf{d}}^{0}={\mathbf{u}}_{h}^{0}$. Repeat
for
$n=0,1,\dots $Moreover, setting
${\widehat{\mathbf{u}}}^{n+1}={\mathbf{d}}^{n+1}+\nabla {g}^{n}$ in (
13)–(15), we have
where
${\hslash}^{n+1}:={g}^{n+1}{g}^{n}$. So one obtains
and
Now, we are ready to write the Uzawatype finite element iterative algorithm as follows Algorithm 2.
Algorithm 2: Uzawatype iterative algorithm. 
Step 1. Obtain the initial guess $({\mathbf{u}}_{h}^{0},{p}_{h}^{0},{T}_{h}^{0})\in {\mathbf{M}}_{h}\times {W}_{h}\times {Z}_{h}$ from step 1 of Algorithm 1. Step 2. Find $({\widehat{\mathbf{u}}}_{h}^{n+1},{T}_{h}^{n+1})\in {\mathbf{M}}_{h}\times {Z}_{h}$ as the solution of
Step 3. Find ${\hslash}_{h}^{n+1}\in {W}_{h}$ as the solution of
Step 4. Compute ${\mathbf{u}}_{h}^{n+1}$ with ${\mathbf{u}}_{h}^{n+1}={\widehat{\mathbf{u}}}_{h}^{n+1}+\nabla {\hslash}_{h}^{n+1}$. Step 5. Given a relaxation parameter $\rho >0$, find ${p}_{h}^{n+1}\in {W}_{h}$ from the Richardson update
From ( 21) and Step 4 of Algorithm 2, we obtain $(\nabla \xb7{\mathbf{u}}_{h}^{n+1},q)=(\nabla \xb7{\widehat{\mathbf{u}}}_{h}^{n+1},q)(\nabla {\hslash}_{h}^{n+1},\nabla q)=0$. So the velocity obtained by Algorithm 2 satisfies the weakly divergencefree condition. Moreover, we expect to show the iterative errors between the finite element solutions to ( 10)–(12) and the Uzawatype iterative solutions to Algorithm 2. For convenience, assume that ${\mathbf{E}}_{h}^{n}={\mathbf{u}}_{h}{\mathbf{u}}_{h}^{n}$, ${\widehat{\mathbf{E}}}_{h}^{n}={\mathbf{u}}_{h}{\widehat{\mathbf{u}}}_{h}^{n}$, ${\eta}_{h}^{n}={p}_{h}{p}_{h}^{n}$ and ${\theta}_{h}^{n}={T}_{h}{T}_{h}^{n}$. Then, we have ${\widehat{\mathbf{E}}}_{h}^{n}={\mathbf{E}}_{h}^{n}+\nabla {\hslash}_{h}^{n}$.

Firstly, we recall the convergence results of the initial guess. Note that ${\widehat{\mathbf{u}}}_{h}^{0}={\mathbf{d}}^{0}+\nabla {g}^{1}={\mathbf{u}}_{h}^{0}$, which implies ${\mathbf{E}}_{h}^{0}={\widehat{\mathbf{E}}}_{h}^{0}$.
Lemma 1 ([
19])
. Let $({\mathbf{u}}_{h}^{0},{p}_{h}^{0},{T}_{h}^{0})\in {\mathbf{M}}^{h}\times {W}^{h}\times {Z}^{h}$ be the solution of Step 1 of Algorithm 1. Then, under the assumptions of Theorem 2, we have the following results Secondly, we show that the solution sequence generated by Algorithm 2 is bounded.
Theorem 3. Let $\{{\mathbf{u}}_{h}^{n},{p}_{h}^{n},{T}_{h}^{n}\}$ be the solution sequence of Algorithm 2. Then, under the assumptions of Theorem 2, if the relaxation parameter satisfies $\rho \in (0,2(1\overline{\Lambda}P{r}^{1}\Lambda ))$, the sequences $\{\parallel \nabla {\mathbf{u}}_{h}^{n}\parallel \}$, $\{\parallel \nabla {\widehat{\mathbf{u}}}_{h}^{n}\parallel \}$, $\{\parallel {p}_{h}^{n}\parallel \}$ and $\{\parallel \nabla {T}_{h}^{n}\parallel \}$ are uniformly bounded with respect to $h$.
Proof. Subtracting (
19) from (12), we have
Setting
$s={\theta}_{h}^{n+1}$ obtains
According to (
6) and Theorem 2, we arrive at
Then, subtracting (20) from (
10), we have
Choosing
$\mathbf{v}={\widehat{\mathbf{E}}}_{h}^{n+1}$ in (24) and combining the ensuing equation with (
21) lead to
Next, according to (
22), we have
which, by using (
5), (
6), (
23), Theorem 2, and the Proposition identity
$(u,v)=\frac{1}{2}{(\parallel u+v\parallel}^{2}{\parallel u\parallel}^{2}{\parallel v\parallel}^{2})$, we have
Then, using (
21) and (
22), we obtain
which leads to
where we have applied the fact that
$\parallel \nabla \xb7\mathbf{v}\parallel \le \parallel \nabla \mathbf{v}\parallel $ in [
24].
Moreover, substituting (
26) into (
25) and using the Young inequality, we obtain
where
$\varsigma >0$ is a parameter to be determined later on.
Furthermore, we solve a quadratic algebraic equation
to obtain a positive root
$\varsigma ={\varsigma}^{*}$, which makes
$(2PrPr\rho \varsigma (\Lambda +Pr\overline{\Lambda}))={\varsigma}^{1}(\Lambda +Pr\overline{\Lambda})$ hold. In fact, we have
where
$\Delta :=(2PrPr\rho +2(\Lambda +Pr\overline{\Lambda}))(2PrPr\rho 2(\Lambda +Pr\overline{\Lambda}))$.
Thus, the inequality (
27) is rewritten as
which, along with (
23), implies that
Finally, applying (
26) into (
22), we obtain
which combines with
${\widehat{\mathbf{E}}}_{h}^{n+1}={\mathbf{E}}_{h}^{n+1}+\nabla {\hslash}_{h}^{n+1}$; then, we have
Finally, combining (
29) with (
28), we obtain
Hence, using (
28), (
30), and Lemma 1, we finish the proof of the theorem. □
Thirdly, we are going to develop the convergence analysis for Algorithm 2.
Theorem 4. Under the assumptions of Theorem 3, the following estimates holdwhere $D\in (0,\frac{1}{2})$ and $H\in (\frac{3}{4},1)$ are two constants independent of n and h. Proof. By Theorem 3, there exists a positive constant
${D}_{2}$, independent of
n and
h, such that
Then, rewrite (
24) to obtain
Applying the infsup condition, (
5), (
6), (
23), and Theorem 2 to the above equation, we obtain
which combines with (
31) to obtain
Next, using the inequality
${(a+b)}^{2}\le 2{a}^{2}+2{b}^{2}$, we have
Hence, one obtains
where
${D}_{3}:=\frac{{\beta}^{2}}{2{(Pr+N{D}_{2})}^{2}}$ and
${D}_{4}:=\frac{{(\Lambda +Pr\overline{\Lambda})}^{2}}{{(Pr+N{D}_{2})}^{2}}$. Obviously, if we let
${C}_{\rho ,\varsigma}:=Pr\rho (2PrPr\rho \varsigma (\Lambda +Pr\overline{\Lambda}))$, then (
27) becomes
where
$\delta \in (0,{C}_{\rho ,\varsigma})$ is a parameter to be determined. From (
32) and (
33), we obtain
Then, we will choose parameters
$\varsigma $ and
$\delta $ such that
and
$1\delta {D}_{3}>0$, which leads to
In fact, one finds that
which, along with the definition of
${C}_{\rho ,\varsigma}$, yields
and
where the notation
$\Delta $ is defined in the proof of Theorem 3. Note that we have used condition
$0<\rho <2(1\overline{\Lambda}P{r}^{1}\Lambda )$. Here, we select
Substituting this parameter into (
36), we arrive at
$a{\delta}^{2}b\delta +c=0$, where
$a={D}_{3}$,
$b=1+{D}_{4}+{s}_{1}a$,
$c={s}_{1}\frac{P{r}^{2}{\rho}^{2}{(\Lambda +Pr\overline{\Lambda})}^{2}}{{s}_{1}}$, and
${s}_{1}=P{r}^{2}\rho (1\frac{1}{2}\rho )$. Obviously,
$b>1+{s}_{1}a$,
$c<{s}_{1}$; so, we deduce that
Then, the Equation (
36) has a real root
${\delta}^{*}=\frac{b\sqrt{{b}^{2}4ac}}{2a}$.
With the parameter
$\epsilon $ and
$\delta $ given by
${\epsilon}^{+}$ and
${\delta}^{*}$, it follows from (
34) that
where
$\overline{D}={s}_{1}{\delta}^{*}$ and
$H=1{\delta}^{*}{D}_{3}$.
Note that
$\overline{D}>0$ and
$H>0$. Now, we will prove them. Consider the quadratic function
$f(\delta )=a{\delta}^{2}b\delta +c$. Because
$a>0$,
${s}_{1}>0$,
$b>1+{s}_{1}a$ and
$c<{s}_{1}$, we obtain
${lim}_{\delta \to \infty}f(\delta )=\infty $ and
Thus, the smallest root
${\delta}^{*}$ of
$f(\delta )$ must belong to
$(\infty ,{s}_{1})$. So, the inequality
$\overline{D}>0$ holds. Noticing that
${C}_{\rho ,{\varsigma}^{+}}{\delta}^{*}={s}_{1}{\delta}^{*}>0$, it follows readily from (
35) that
$H>0$.
Finally, note that
$0<\overline{D}<{s}_{1}=P{r}^{2}\rho \frac{1}{2}P{r}^{2}{\rho}^{2}\le \frac{P{r}^{2}}{2}$. If, we choose the
$\overline{D}=P{r}^{2}D$ and
$0<D<\frac{1}{2}$, the inequality (
37) is rewritten as
According to the definition of ${D}_{3}$ and $\beta \le 1$, we arrive at ${D}_{3}\le \frac{1}{2P{r}^{2}}$. Noticing that ${\delta}^{*}<{s}_{1}<\frac{P{r}^{2}}{2}$, we easily find that $1>H=1{\delta}^{*}{D}_{3}>\frac{3}{4}$.
Next, using (
38) and (
29), we obtain
Finally, using the above estimates with (
23), we finish the proof. □
4. Numerical Study
We will represent some numerical tests to claim the accuracy and performance of the proposed algorithm for the steady natural convection problem in this section. We used the public finite element software FreeFem++ [
27] and applied
${P}_{2}{P}_{1}{P}_{2}$ element to approximate the velocity, temperature, and pressure, respectively.
In the first numerical test, let the domain
$\Omega =[0,1]\times [0,1]$, and the righthand side of (
1)–(4) is selected such that the exact solutions are given by
Here, we set the parameters
$Ra=Pr=\kappa =1$ and use the stopping rule
Figure 1 displays the iteration errors of the velocity, temperature in
${H}^{1}$seminorm, and the pressure in
${L}^{2}$norm for different iterative steps
n solved by Algorithm 2. Here, we set the relaxation parameter
$\rho =1.6$ and choose five different mesh sizes
h. From
Figure 1, we observe that the proposed algorithm worked well and kept the convergence when iteration step
n became large.
In the above test, we fixed the relaxation parameter and varied the mesh size. Now, we consider different relaxation parameters with the mesh size
$h=\frac{1}{32}$.
Figure 2 expresses different iterative steps of the log errors with different values
$\rho $. From
Figure 2, we observe that
${\mathbf{u}}_{h}^{n}$,
${p}_{h}^{n}$, and
${T}_{h}^{n}$ converged faster when
$\rho $ was larger. However, we have an interesting observation that it became slow when
$\rho $ was too large (e.g.,
$\rho =1.7$ or 1.9). It is not surprising since from Theorem 3 and 4 the relaxation parameter
$\rho $ had a limited interval, and the value
$\rho =1.7$ or 1.9 may have been out of its interval.
Hence, we should reveal the convergence on the relaxation parameter
$\rho $ by showing the values with respect to
n and
$\rho $ under the mesh size
$h=\frac{1}{32}$. From
Table 1, we find that Algorithms 1 and 2 converged faster when we chose larger
$\rho $. However, if the
$\rho $ chosen was very large, then these algorithms either need more iterative steps or diverge. In addition, Algorithms 1 and 2 achieved the tolerance error when
$\rho =1.6$ with the least iterative steps
$n=44$ and
$n=42$, respectively.
Based on the previous section, Algorithm 2 produced the divergencefree velocity approximation. Hence, in
Table 2 we list the value of
$\parallel \nabla \xb7{\mathbf{u}}_{h}^{n}\parallel $. From this table, Algorithms 1 and 2 obtain good numerical results when
$Ra=10$. However, when the value of
$Ra$ increased, then Algorithm 1 could not achieve the tolerance error and converge. Meanwhile, Algorithm 2 still ran well.
In the second numerical test, we considered the hot cylinder problem solving the proposed algorithm with different Rayleigh numbers. The boundary conditions are given in [
28,
29], i.e.,
$\frac{\partial T}{\partial n}=1$ on inner wall,
$T=0$ on the other wall, and zero Dirichlet condition on velocity were imposed. Set
$Pr=0.7,\kappa =1$,
$\gamma =0$, and
$h=\frac{1}{80}$.
Figure 3 and
Figure 4 express the numerical streamlines, isobars, and isotherms for different radii of inner circle
${r}_{in}$ based on
$Ra=100$ and
$Ra=250$ with
$\rho =1.6$. We observe that it shapes two vortices when
${r}_{in}=0.2$ and four vortices when
${r}_{in}=0.8$, which were found to be in good agreement with those reported in [
28,
29]. Therefore, the given method captured this classical model well.
In
Table 3 and
Table 4, we show the CPU time and the maximum value of velocity at
$x=0.5$ and
$y=0.5$ by Algorithms 1 and 2 with
$\rho =1.6$ and Wang’s algorithm [
29] for
${r}_{in}=0.2$ and
${r}_{in}=0.8$, respectively. From
Table 3 and
Table 4, we find that the proposed algorithm took the least computational time among these algorithms to obtain almost the same maximum value of velocity. In particular, Algorithm 1 did not work when
$Ra=250$. Therefore, the proposed algorithm solved this model well.