Existence and Uniqueness of Generalized and Mixed Finite Element Solutions for Steady Boussinesq Equation

Abstract

Herein, we mainly employ the fixed point theorem and Lax-Milgram theorem in functional analysis to prove the existence and uniqueness of generalized and mixed finite element (MFE) solutions for two-dimensional steady Boussinesq equation. Thus, we can fill in the gap of research for the steady Boussinesq equation since the existing studies for the equation are assumed the existence and uniqueness of generalized solution without providing proof.

1. Introduction

Let $\mathsf{\Omega }\subset {\mathbb{R}}^2$ be a bounded convex domain with suitably smooth boundary $\partial \mathsf{\Omega }$. We consider the following steady Boussinesq equation.

Problem 1.

Find $\mathit{u}={(u_1,u_2)}^T$, p, and T satisfying

$$\begin{array}{ccc}& & \left\{\begin{array}{cc}-\nu \Delta \mathit{u}\left(\mathit{x}\right)+(\mathit{u}\left(\mathit{x}\right)·\nabla )\mathit{u}\left(\mathit{x}\right)+\nabla p\left(\mathit{x}\right)={\gamma }_{0}T\left(\mathit{x}\right)\mathit{j},\hfill & \mathit{x}\in \mathsf{\Omega },\hfill \\ {\displaystyle \mathrm{div}\mathit{u}\left(\mathit{x}\right)=0,}\hfill & \mathit{x}\in \mathsf{\Omega },\hfill \\ {\displaystyle -\Delta T\left(\mathit{x}\right)+{\gamma }_0\mathit{u}\left(\mathit{x}\right)·\nabla T\left(\mathit{x}\right)=0,}\hfill & \mathit{x}\in \mathsf{\Omega },\hfill \\ {\displaystyle \mathit{u}\left(\mathit{x}\right)=\mathit{0},T\left(\mathit{x}\right)=T_0\left(\mathit{x}\right),}\hfill & \mathit{x}\in \partial \mathsf{\Omega }.\hfill \end{array}\right.\hfill \end{array}$$

where $\mathit{u}$ is the unknown velocity vector in the fluid, p represents the unknown pressure in the fluid, T stands for the unknown temperature (or thermal energy) in the fluid, $\nu =\sqrt{Pr/Re}$, $Re$ is the Reynolds number, $Pr$ is the Prandtl number, ${\gamma }_0=1/\sqrt{RePr}$, $\mathit{j}={(0,1)}^T$ is the two-dimensional (2D) vector, and $\mathit{x}=(x_1,x_2)$.

The steady Boussinesq equation is an important nonlinear system of equations with forced dissipation in atmospheric dynamics. It contains not only the velocity and pressure fields, but also the temperature field, and there is even nonlinear coupling between the velocity field with temperature field, so that it is more complex than the steady Navier-Stokes equation. The steady Navier-Stokes equation is only a special case of the steady Boussinesq equation. Since fluid motion is usually viscous, viscous motion must produce heat, so fluid motion must be accompanied by the interconversion of speed, pressure, and temperature. Therefore, the research on this issue has universal significance.

Up to now, there have been few studies on the steady Boussinesq equation (Problem 1), and some numerical methods have been provided (see [1,2,3,4,5,6,7,8,9,10,11]). However, in these numerical methods, the analytical solution for the steady Boussinesq equation (Problem 1) is directly assumed to exist. Herein, we employ the fixed point theorem and Lax-Milgram theorem in the functional analysis to discuss the existence and uniqueness of mixed generalized solution and mixed finite element (MFE) solution for the steady Boussinesq equation (Problem 1), in which some new MFE subspaces satisfying the Babuška-Brezzi (B-B) condition are given.

Generally, the existence and uniqueness of mixed generalized solution and mixed finite element solution for the steady Navier-Stokes equations are analyzed by means of function spacial sequence approximation method (see [12]). Herein, we use the fixed point theorems to analyze the existence and uniqueness of mixed generalized solution and MFE solution for the steady Boussinesq equation, which is a new development over the existing methods, such as that in [12]. The method herein can fill in the gap for the steady Boussinesq equation, which is a useful supplement to the theoretical study of the steady Boussinesq equation.

To this end, in the coming Section 2, we use these theorems to discuss the existence and uniqueness of generalized solution for the steady Boussinesq equation. Then, in Section 3, we use these theorems to discuss the existence and uniqueness of the MFE solution for the equation. Whereafter, we provide some discussions in Section 4. Finally, we summarize the main conclusions in Section 5.

2. The Existence and Uniqueness of Generalized Solution

The Sobolev spaces and norms used in herein are standard (see [13] or [14] or [15]). Let

$$\begin{array}{ccc}& & \mathbb{X}=H_0^1{\left(\mathsf{\Omega }\right)}^2,\mathbb{W}=H^1\left(\mathsf{\Omega }\right),\hfill \\ & & {\displaystyle {\mathbb{W}}_0=H_0^1\left(\mathsf{\Omega }\right),\mathbb{M}=L_0^2\left(\mathsf{\Omega }\right)=\{\phi \in L^2\left(\mathsf{\Omega }\right):{\int }_{\mathsf{\Omega }}\phi \mathrm{d}\mathit{x}=0\},\mathrm{d}\mathit{x}=\mathrm{d}{x}_1\mathrm{d}{x}_2,}\hfill \\ & & {\displaystyle a_1(\mathit{u},\mathit{v},\mathit{w})=\frac{1}{2}({\int }_{\mathsf{\Omega }}{\displaystyle \sum _{i,k=1}^2}{u}_i\frac{\partial v_k}{\partial x_i}{w}_k\mathrm{d}\mathit{x}-{\int }_{\mathsf{\Omega }}{\displaystyle \sum _{i,k=1}^2}{u}_i\frac{\partial w_k}{\partial x_i}{v}_k\mathrm{d}\mathit{x}),\forall \mathit{u},\mathit{v},\mathit{w}\in \mathbb{X},}\hfill \\ & & {\displaystyle a_2(\mathit{u},T,\psi )=\frac{1}{2}\left({\int }_{\mathsf{\Omega }}{\displaystyle \sum _{i=1}^2}{u}_i\frac{\partial T}{\partial x_i}\psi \mathrm{d}\mathit{x}-{\int }_{\mathsf{\Omega }}{\displaystyle \sum _{i=1}^2}{u}_i\frac{\partial \psi }{\partial x_i}T\mathrm{d}\mathit{x}\right),\forall \mathit{u}\in \mathbb{X},T,\psi \in {\mathbb{W}}_0,}\hfill \\ & & b(\phi ,\mathit{v})=(\phi ,\mathrm{div}\mathit{v}),d(T,\psi )=(\nabla T,\nabla \psi ),a(\mathit{u},\mathit{v})=\nu (\nabla \mathit{u},\nabla \mathit{v}),\hfill \end{array}$$

where $(·,·)$ represents the inner product in $L^2\left(\mathsf{\Omega }\right)$ or ${\left[L^2\left(\mathsf{\Omega }\right)\right]}^2$ or ${\left[L^2\left(\mathsf{\Omega }\right)\right]}^{2\times 2}$. Thus, the mixed generalized solution for the steady Boussinesq equation, i.e., Problem 1, can be stated as follows.

Problem 2.

Find $(\mathit{u},p,T)\in \mathbb{X}\times \mathbb{M}\times \mathbb{W}$ that satisfies $T_{|\partial \mathsf{\Omega }}=T_0$ and

$$\begin{array}{c}\hfill \left\{\begin{array}{cc}a(\mathit{u},\mathit{v})+a_1(\mathit{u},\mathit{u},\mathit{v})-b(p,\mathit{v})={\gamma }_0(T\mathit{j},\mathit{v}),\hfill & \forall \mathit{v}\in \mathbb{X},\hfill \\ {\displaystyle b(\phi ,\mathit{u})=0,}\hfill & \forall \phi \in \mathbb{M},\hfill \\ {\displaystyle d(T,\psi )+{\gamma }_0{a}_2(\mathit{u},T,\psi )=0,}\hfill & \forall \psi \in {\mathbb{W}}_0.\hfill \end{array}\right.\end{array}$$

The following assumptions can be obtained from Poincaré’s inequality, Sobolev imbedding theorem, Hahn-Banach theorem (see [13]), Cauchy-Schwarz’s inequality, or Green’s formula, which may be referred to in the relevant literature (see [13,14,15]).

(A${}_1$) There exists a positive constant $c_0$ that is dependent on $\mathsf{\Omega }$ satisfying (see Poincaré’s inequality, Sobolev imbedding theorem Theorem 3.5.9 in [13], and Cauchy-Schwarz’s inequality).

(i)

${\parallel \mathit{v}\parallel }_0⩽c_0{\parallel \nabla \mathit{v}\parallel }_0{,\parallel \mathit{v}\parallel }_{0,4}⩽c_0{\parallel \nabla \mathit{v}\parallel }_0,\forall \mathit{v}\in H_0^1{\left(\mathsf{\Omega }\right)}^2$(or $H_0^1\left(\mathsf{\Omega }\right))$,

(ii)

${\parallel \mathit{v}\parallel }_{0,4}⩽c_0{\parallel \mathit{v}\parallel }_1,\forall \mathit{v}\in H^1{\left(\mathsf{\Omega }\right)}^2,$

(iii)

${\parallel \mathit{v}\parallel }_{0,4}⩽2^{\frac{1}{2}}{\parallel \mathit{v}\parallel }_0^{\frac{1}{2}}{\parallel \nabla \mathit{v}\parallel }_0^{\frac{1}{2}},\forall \mathit{v}\in H_0^1{\left(\mathsf{\Omega }\right)}^2$(or $H_0^1\left(\mathsf{\Omega }\right))$,

where ${\parallel \mathit{v}\parallel }_0={\left({\int }_{\mathsf{\Omega }}{\left|\mathit{v}\left(\mathit{x}\right)\right|}^2\mathrm{d}\mathit{x}\right)}^{1/2}$, ${\parallel \mathit{v}\parallel }_{0,4}={\left({\int }_{\mathsf{\Omega }}{\left|\mathit{v}\left(\mathit{x}\right)\right|}^4\mathrm{d}\mathit{x}\right)}^{1/4}$, ${\parallel \nabla \mathit{v}\parallel }_0=$${\left({\int }_{\mathsf{\Omega }}{|\nabla \mathit{v}\left(\mathit{x}\right)|}^2\mathrm{d}\mathit{x}\right)}^{1/2}$ and ${\parallel \mathit{v}\parallel }_1={\left[{\parallel \mathit{v}\parallel }_0^2+{\parallel \nabla \mathit{v}\parallel }_0^2\right]}^{1/2}$.

(A${}_2$) Suppose that for given $T_0\in C^1(\partial \mathsf{\Omega })$ and sufficiently small $\epsilon >0$, there exists a prolongation of $T_0$ to $\mathsf{\Omega }$, which is still denoted by $T_0$, satisfying $T_0\in H^1\left(\mathsf{\Omega }\right)$ and $\parallel T_0{\parallel }_{1,2,\mathsf{\Omega }}⩽\epsilon$.

(A${}_3$) The trilinear forms $a_1(·,·,·)$ and $a_2(·,·,·)$ have the following properties:

$$\begin{array}{c}\hfill \left\{\begin{array}{c}{a}_1(\mathit{u},\mathit{v},\mathit{v})=0,a_1(\mathit{u},\mathit{v},\mathit{w})=-a_1(\mathit{u},\mathit{w},\mathit{v}),\forall \mathit{u},\mathit{w}\in \mathbb{X},\mathit{v}\in H^1{\left(\mathsf{\Omega }\right)}^2;\hfill \\ a_2(\mathit{u},T,T)=0,a_2(\mathit{u},T,\phi )=-a_2(\mathit{u},\phi ,T),\forall \mathit{u}\in \mathbb{X},\forall T\in \mathbb{W},\phi \in {\mathbb{W}}_0;\hfill \end{array}\right.\end{array}$$

(1)

$$\begin{array}{ccc}\hfill & & \left\{\begin{array}{c}|a_1(\mathit{u},\mathit{v},\mathit{w})|⩽N_0{\parallel \nabla \mathit{u}\parallel }_0{\parallel \nabla \mathit{v}\parallel }_0{\parallel \nabla \mathit{w}\parallel }_0,\forall \mathit{u},\mathit{w}\in \mathbb{X},\mathit{v}\in H^1{\left(\mathsf{\Omega }\right)}^2;\hfill \\ |a_2(\mathit{u},T,\phi )|⩽\overline{N}{\parallel \nabla \mathit{u}\parallel }_0{\parallel \nabla T\parallel }_0{\parallel \nabla \phi \parallel }_0,\forall \mathit{u}\in \mathbb{X},T\in \mathbb{W},\phi \in {\mathbb{W}}_0,\hfill \end{array}\right.\hfill \end{array}$$

(2)

where

$$\begin{array}{ccc}& & {\displaystyle N_0=\underset{\mathit{u},\mathit{w}\in \mathbb{X},\mathit{v}\in H^1{\left(\mathsf{\Omega }\right)}^2}{sup}|a_1{(\mathit{u},\mathit{v},\mathit{w})|/(\parallel \nabla \mathit{u}\parallel }_0{\parallel \nabla \mathit{v}\parallel }_0{\parallel \nabla \mathit{w}\parallel }_0),}\hfill \\ \hfill {\displaystyle }& & \overline{N}=\underset{\mathit{u}\in \mathbb{X},T\in \mathbb{W},\phi \in {\mathbb{W}}_0}{sup}|a_2{(\mathit{u},T,\phi )|/(\parallel \nabla \mathit{u}\parallel }_0{\parallel \nabla T\parallel }_0{\parallel \nabla \phi \parallel }_0).\hfill \end{array}$$

(A${}_4$) If we set $A\equiv 2{\nu }^{-1}{\gamma }_0(3c_0+1){\parallel T_0\parallel }_1$ and $B\equiv 2\parallel \nabla T_0{\parallel }_0+{\left(2c_0^2{\gamma }_0\right)}^{-1}\nu A$, then there exist two small positive constants ${\delta }_1$ and ${\delta }_2$ satisfying:

$$\begin{array}{ccc}\hfill & & \begin{array}{c}{\nu }^{-1}{N}_{0}A⩽1-{\delta }_1,0<{\delta }_1⩽1,\hfill \end{array}\hfill \end{array}$$

(3)

$$\begin{array}{ccc}\hfill & & \begin{array}{c}{\delta }_1^{-1}{\nu }^{-1}{c}_0^2{\gamma }_0^{2}B\overline{N}⩽1-{\delta }_2,0<{\delta }_2⩽1.\hfill \end{array}\hfill \end{array}$$

(4)

Obviously, the bilinear functionals $a(·,·)$ and $d(·,·)$ are continuous (i.e., bounded) and positive definite in $\mathbb{X}\times \mathbb{X}$ and ${\mathbb{W}}_0\times {\mathbb{W}}_0$, respectively, and the bilinear functional $b(·,·)$ is continuous (bounded) in $\mathbb{M}\times \mathbb{X}$ and satisfies that following continuous B-B condition:

$$\begin{array}{c}\hfill \begin{array}{c}{\displaystyle \underset{\mathit{v}\in \mathbb{X}}{sup}\frac{b(\phi ,\mathit{v})}{{\parallel \nabla \mathit{v}\parallel }_0}⩾\beta {\parallel \phi \parallel }_0,\forall \phi \in \mathbb{M},}\hfill \end{array}\end{array}$$

(5)

where $\beta >0$ is constant.

Set

$$\begin{array}{c}\hfill \begin{array}{c}\mathbb{V}=\{\mathit{v}\in \mathbb{X}:b(\phi ,\mathit{v})=0,\forall \phi \in \mathbb{M}\}.\hfill \end{array}\end{array}$$

(6)

It is easily proved that:

$$\begin{array}{c}\hfill \begin{array}{c}\mathbb{V}=\{\mathit{v}\in \mathbb{X}:\mathrm{div}\mathit{v}=0\}.\hfill \end{array}\end{array}$$

(7)

To discuss the existence and uniqueness of solution for Problem 2, we need to first consider the following problem.

Problem 3.

Find $(\mathit{u},T)\in \mathbb{V}\times \mathbb{W}$ that satisfies ${T|}_{\partial \mathsf{\Omega }}=T_0$ and

$$\begin{array}{c}\hfill \left\{\begin{array}{c}a(\mathit{u},\mathit{v})+a_1(\mathit{u},\mathit{u},\mathit{v})={\gamma }_0(T\mathit{j},\mathit{v}),\forall \mathit{v}\in \mathbb{X},\hfill \\ {\displaystyle d(T,\phi )+{\gamma }_0{a}_2(\mathit{u},T,\phi )=0,\forall \phi \in {\mathbb{W}}_0.}\hfill \end{array}\right.\end{array}$$

(8)

In the following, we employ the fixed point theorem and the Lax-Milgram theorem to prove the existence and uniqueness for the generalized solution. We have the following result.

Theorem 1.

If the conditions $\left(A_1\right)$–$\left(A_4\right)$ are satisfied, then Problem 3 possesses a unique solution $(\mathit{u},T)\in \mathbb{V}\times \mathbb{W}$, and there exists a unique $p\in \mathbb{M}$, such that $(\mathit{u},p,T)$ is a unique solution for Problem 2, satisfying:

$$\begin{array}{c}\hfill \begin{array}{c}{\parallel \nabla \mathit{u}\parallel }_0⩽A,{\parallel \nabla T\parallel }_0⩽B.\hfill \end{array}\end{array}$$

(9)

Proof .

(1) Prove that Problem 3 possesses a unique solution.

Let $\tilde{\mathit{u}}\in \mathbb{V}$, which satisfies $\parallel \nabla \tilde{\mathit{u}}{\parallel }_0⩽A$. We consider the following problem:

$$\begin{array}{c}\hfill d(T,\phi )+{\gamma }_0{a}_2(\tilde{\mathit{u}},T,\phi )=0,\forall \phi \in {\mathbb{W}}_0.\end{array}$$

(10)

Let $T=w+T_0$ (where $T_0$ is given by the condition (A${}_2$)) and $w\in {\mathbb{W}}_0$. Inputting $T=w+T_0$ into (10), we obtain:

$$\begin{array}{c}\hfill d(w,\phi )+{\gamma }_0{a}_2(\tilde{\mathit{u}},w,\phi )=-d(T_0,\phi )-{\gamma }_0{a}_2(\tilde{\mathit{u}},T_0,\phi ),\forall \phi \in {\mathbb{W}}_0.\end{array}$$

(11)

If we set $\tilde{d}(w,\phi )=d(w,\phi )+{\gamma }_0{a}_2(\tilde{\mathit{u}},w,\phi )$, by (1), we obtain:

$$\begin{array}{c}\hfill \tilde{d}(w,w)=d(w,w)+{\gamma }_0{a}_2(\tilde{\mathit{u}},w,w)=d(w,w)={\parallel \nabla w\parallel }_0,\end{array}$$

(12)

which implies that $\tilde{d}(·,·)$ is continuous and positive definite in ${\mathbb{W}}_0$. Thus, by the Lax-Milgram theorem (see [13] or [14] or [15]), we can conclude that Equation (11) possesses a unique solution $T\in \mathbb{W}$ satisfying ${T|}_{\partial \mathsf{\Omega }}=T_0$ and ${\parallel \nabla T\parallel }_0⩽2\parallel \nabla T_0{\parallel }_0+{\gamma }_0\overline{N}A{\parallel \nabla T_0\parallel }_0$. Inputting T into the first equation in Problem 3, we obtain:

$$\begin{array}{c}\hfill {\displaystyle a({\mathit{u}}^{*},\mathit{v})+a_1({\mathit{u}}^{*},{\mathit{u}}^{*},\mathit{v})={\gamma }_0(T\mathit{j},\mathit{v}),\forall \mathit{v}\in \mathbb{V}.}\end{array}$$

(13)

It is easily proved by the completeness of Hilbert’s space that Equation (13) possesses a unique solution ${\mathit{u}}^{*}\in \mathbb{V}$. In fact, let ${\mathit{u}}_0^{*}\in \mathbb{V}$, which satisfies $\parallel {\mathit{u}}_0^{*}{\parallel }_1⩽A$, and consider the following iterative recursive process:

$$\begin{array}{c}\hfill {\displaystyle a({\mathit{u}}_m^{*},\mathit{v})+a_1({\mathit{u}}_{m-1}^{*},{\mathit{u}}_m^{*},\mathit{v})={\gamma }_0(T\mathit{j},\mathit{v}),\forall \mathit{v}\in \mathbb{V},m=1,2,\cdots .}\end{array}$$

(14)

Obviously, the above left hand side is continuous in ${\mathit{u}}_m^{*}$ and $\mathit{v}$ and is also positive definite when ${\mathit{u}}_{m-1}^{*}$ is known and the right hand side is continuous in $\mathit{v}$. Therefore, it follows by the Lax-Milgram theorem (see [13] or [14] or [15]) that the system of Equation (14) possesses a unique sequence of solution ${\displaystyle \left\{{\mathit{u}}_m^{*}\right\}}$. Taking $\mathit{v}={\mathit{u}}_m^{*}$ in (14) and using (1), we obtain:

$$\begin{array}{c}\hfill \nu \parallel \nabla {\mathit{u}}_m^{*}{\parallel }_0^2={\gamma }_0|(T\mathit{j},{\mathit{u}}_m^{*})|⩽{\gamma }_0{\parallel T\parallel }_{-1}{\parallel \nabla {\mathit{u}}_m^{*}\parallel }_0.\end{array}$$

(15)

It follows that:

$$\begin{array}{c}\hfill \parallel \nabla {\mathit{u}}_m^{*}{\parallel }_0⩽{\nu }^{-1}{\gamma }_0{\parallel T\parallel }_{-1}⩽{\nu }^{-1}{\gamma }_0(2\parallel \nabla T_0{\parallel }_0+{\gamma }_0\overline{N}A\parallel \nabla T_0{\parallel }_0),\end{array}$$

(16)

which implies that ${\displaystyle \left\{{\mathit{u}}_m^{*}\right\}}$ is bounded. By (14), we obtain:

$$\begin{array}{ccc}& & {\displaystyle a({\mathit{u}}_m^{*}-{\mathit{u}}_{m-1}^{*},\mathit{v})+a_1({\mathit{u}}_{m-1}^{*}-{\mathit{u}}_{m-2}^{*},{\mathit{u}}_m^{*},\mathit{v})+a_1({\mathit{u}}_{m-2}^{*},{\mathit{u}}_m^{*}-{\mathit{u}}_{m-1}^{*},\mathit{v})}\hfill \\ \hfill & & =0,\forall \mathit{v}\in \mathbb{V},m=1,2,\cdots .\hfill \end{array}$$

Taking $\mathit{v}={\mathit{u}}_m^{*}-{\mathit{u}}_{m-1}^{*}$ in the above equality, by (16) and (1) we obtain:

$$\begin{array}{ccc}& & {\displaystyle \parallel \nabla ({\mathit{u}}_m^{*}-{\mathit{u}}_{m-1}^{*}){\parallel }_0^2=-{\nu }^{-1}{a}_1({\mathit{u}}_{m-1}^{*}-{\mathit{u}}_{m-2}^{*},{\mathit{u}}_m^{*},{\mathit{u}}_m^{*}-{\mathit{u}}_{m-1}^{*})}\hfill \\ \hfill & & ⩽{\nu }^{-1}{N}_0\parallel \nabla ({\mathit{u}}_{m-1}^{*}-{\mathit{u}}_{m-2}^{*}){\parallel }_0\parallel \nabla {\mathit{u}}_m^{*}{\parallel }_0{\parallel \nabla ({\mathit{u}}_m^{*}-{\mathit{u}}_{m-1}^{*})\parallel }_0\hfill \\ \hfill & & ⩽{\nu }^{-2}{N}_0{\gamma }_0(2\parallel \nabla T_0{\parallel }_0+\overline{N}A\parallel \nabla T_0{\parallel }_0)\parallel \nabla ({\mathit{u}}_{m-1}^{*}-{\mathit{u}}_{m-2}^{*}){\parallel }_0{\parallel \nabla ({\mathit{u}}_m^{*}-{\mathit{u}}_{m-1}^{*})\parallel }_0,\hfill \end{array}$$

where $m=1,2,\cdots .$ If we set that $\tilde{\delta }={\nu }^{-2}{N}_0{\gamma }_0(2\parallel \nabla T_0{\parallel }_0+{\gamma }_0\overline{N}A\parallel \nabla T_0{\parallel }_0)$, then by the conditions (A${}_2$) and (A${}_4$), we obtain $\widehat{\delta }<1$. Thereupon, it follows by the above inequality that:

$$\begin{array}{ccc}& & {\displaystyle \parallel \nabla ({\mathit{u}}_m^{*}-{\mathit{u}}_{m-1}^{*}){\parallel }_0⩽\tilde{\delta }{\parallel \nabla ({\mathit{u}}_{m-1}^{*}-{\mathit{u}}_{m-2}^{*})\parallel }_0}\hfill \\ \hfill & & ⩽\cdots ⩽{\tilde{\delta }}^{m-1}{\parallel \nabla ({\mathit{u}}_1^{*}-{\mathit{u}}_0^{*})\parallel }_0,m=1,2,\cdots .\hfill \end{array}$$

Thus, for any positive integer l, by the above inequality, we obtain:

$$\begin{array}{ccc}& & {\displaystyle \parallel \nabla ({\mathit{u}}_{m+l}^{*}-{\mathit{u}}_m^{*}){\parallel }_0⩽{\displaystyle \sum _{i=1}^l}\parallel \nabla ({\mathit{u}}_{m+i}^{*}-{\mathit{u}}_{m+i-1}^{*}){\parallel }_0⩽{\displaystyle \sum _{i=1}^l}{\tilde{\delta }}^{m+i-1}{\parallel \nabla ({\mathit{u}}_1^{*}-{\mathit{u}}_0^{*})\parallel }_0}\hfill \\ \hfill & & ⩽{\displaystyle \sum _{i=1}^{\infty }}{\tilde{\delta }}^{m+i-1}\parallel \nabla ({\mathit{u}}_1^{*}-{\mathit{u}}_0^{*}){\parallel }_0⩽\frac{{\tilde{\delta }}^m}{1-\tilde{\delta }}{\parallel \nabla ({\mathit{u}}_1^{*}-{\mathit{u}}_0^{*})\parallel }_0\to 0,m\to \infty .\hfill \end{array}$$

It follows that ${\displaystyle \left\{{\mathit{u}}_m^{*}\right\}\subset \mathbb{V}\subset {\left[H_0^1\left(\mathsf{\Omega }\right)\right]}^2}$ is the complete system of fundamental sequence, namely, ${\displaystyle \left\{{\mathit{u}}_m^{*}\right\}\subset \mathbb{V}\subset {\left[H_0^1\left(\mathsf{\Omega }\right)\right]}^2}$ is the Cauchy sequence. By the completeness of Hilbert’s space ${\left[H_0^1\left(\mathsf{\Omega }\right)\right]}^2$, (namely, the Cauchy sequences in Hilbert spaces is convergent, see [15]), we can conclude that there exists a unique ${\displaystyle {\tilde{\mathit{u}}}^{*}\in {\left[H_0^1\left(\mathsf{\Omega }\right)\right]}^2}$ satisfying ${\displaystyle \underset{m\to \infty }{lim}{\mathit{u}}_m^{*}={\tilde{\mathit{u}}}^{*}}$ in ${\left[H_0^1\left(\mathsf{\Omega }\right)\right]}^2$. Taking ${\displaystyle \underset{m\to \infty }{lim}}$ for (14), we obtain:

$$\begin{array}{c}\hfill a({\tilde{\mathit{u}}}^{*},\mathit{v})+a_1({\tilde{\mathit{u}}}^{*},{\tilde{\mathit{u}}}^{*},\mathit{v})={\gamma }_0(T\mathit{j},\mathit{v}),\forall \mathit{v}\in \mathbb{V},\end{array}$$

(17)

which implies that the Equation (13) possesses at least a solution.

In the following, we prove that the solution to the Equation (13) is unique. In fact, if the Equation (13) has another solution ${\mathit{u}}^{*1}$ satisfying:

$$\begin{array}{c}\hfill a({\mathit{u}}^{*1},\mathit{v})+a_1({\mathit{u}}^{*1},{\mathit{u}}^{*1},\mathit{v})={\gamma }_0(T\mathit{j},\mathit{v}),\forall \mathit{v}\in \mathbb{V}.\end{array}$$

Noting that $\parallel \nabla {\mathit{u}}^{*}{\parallel }_0⩽{\nu }^{-1}{\gamma }_0(\parallel \nabla T_0{\parallel }_0+{\gamma }_0\overline{N}A\parallel \nabla T_0{\parallel }_0)$ and taking $\mathit{v}={\mathit{u}}^{*}-{\mathit{u}}^{*1}$, by (14) and the above equation as well as (1), we obtain:

$$\begin{array}{ccc}\hfill & & \nu \parallel {\mathit{u}}^{*}-{\mathit{u}}^{*1}{\parallel }_0^2=-a_1({\mathit{u}}^{*}-{\mathit{u}}^{*1},{\mathit{u}}^{*},{\mathit{u}}^{*}-{\mathit{u}}^{*1})-a_1({\mathit{u}}^{*1},{\mathit{u}}^{*}-{\mathit{u}}^{*1},{\mathit{u}}^{*}-{\mathit{u}}^{*1})\hfill \\ \hfill & & ⩽N_0\parallel {\mathit{u}}^{*}-{\mathit{u}}^{*1}{\parallel }_0^2{\parallel \nabla {\mathit{u}}^{*}\parallel }_0\hfill \\ \hfill & & ⩽{\nu }^{-1}{\gamma }_0{N}_0(2\parallel \nabla T_0{\parallel }_0+{\gamma }_0\overline{N}A\parallel \nabla T_0{\parallel }_0)\parallel {\mathit{u}}^{*}-{\mathit{u}}^{*1}{\parallel }_0^2.\hfill \end{array}$$

Thus, by the conditions (A${}_2$) and (A${}_4$) we obtain $\parallel {\mathit{u}}^{*}-{\mathit{u}}^{*1}{\parallel }_0=0$ so that ${\mathit{u}}^{*}={\mathit{u}}^{*1}$. Therefore, Equation (13) possesses a unique solution ${\mathit{u}}^{*}\in \mathbb{V}$.

It has been shown from the above discussion that, for given $\tilde{\mathit{u}}\in \mathbb{V}$, by (11) and (13), we can uniquely obtain ${\mathit{u}}^{*}\in \mathbb{V}$. The relationship of correspondence is defined as the following mapping ℓ:

$$\begin{array}{c}\hfill u^{*}=\ell \tilde{\mathit{u}}.\end{array}$$

(18)

(i)

Estimate ${\parallel \nabla T\parallel }_0$.

Taking $\phi =w$ in (11) and using the conditions (A${}_1$)–(A${}_4$), we obtain:

$$\begin{array}{c}\hfill \begin{array}{c}{\parallel \nabla w\parallel }_0^2=|d(T_0,w)-{\gamma }_0{a}_2(\tilde{\mathit{u}},w,T_0)|\hfill \\ {\displaystyle ⩽\parallel \nabla T_0{\parallel }_0{\parallel \nabla w\parallel }_0+{\gamma }_0\parallel \tilde{\mathit{u}}{\parallel }_{0,4}{\parallel \nabla w\parallel }_0{\parallel T_0\parallel }_{0,4}}\hfill \\ ⩽(\parallel \nabla T_0{\parallel }_0+{\gamma }_0{c}_0\epsilon \parallel \nabla \tilde{\mathit{u}}{\parallel }_0{)\parallel \nabla w\parallel }_0.\hfill \end{array}\end{array}$$

It follows that:

$$\begin{array}{c}\hfill {\parallel \nabla w\parallel }_0⩽\parallel \nabla T_0{\parallel }_0+{\gamma }_0{c}_0\epsilon {\parallel \nabla \tilde{\mathit{u}}\parallel }_0.\end{array}$$

(19)

Thereupon, we obtain:

$$\begin{array}{ccc}\hfill {\parallel \nabla T\parallel }_0& ⩽& {\parallel \nabla w\parallel }_0+{\parallel \nabla T_0\parallel }_0\hfill \\ \hfill & ⩽& {\displaystyle 2\parallel \nabla T_0{\parallel }_0+{\gamma }_0{c}_0\epsilon \parallel \nabla \tilde{\mathit{u}}{\parallel }_0\equiv \tilde{B}(\parallel \nabla \tilde{\mathit{u}}{\parallel }_0).}\hfill \end{array}$$

(20)

(ii)

Estimate $\parallel \nabla {\mathit{u}}^{*}{\parallel }_0$.

Taking $\mathit{v}={\mathit{u}}^{*}$ in (13) and using the condition (A${}_1$), we obtain:

$$\begin{array}{c}\hfill \nu \parallel \nabla {\mathit{u}}^{*}{\parallel }_0^2=|{\gamma }_0(T\mathit{j},{\mathit{u}}^{*})|⩽{\gamma }_0{\parallel T\parallel }_0\parallel {\mathit{u}}^{*}{\parallel }_0⩽{\gamma }_0{c}_0{\parallel T\parallel }_0{\parallel \nabla {\mathit{u}}^{*}\parallel }_0.\end{array}$$

Thus, by the condition (A${}_1$) and $T=w+T_0$, we obtain:

$$\begin{array}{c}\hfill \begin{array}{c}\parallel \nabla {\mathit{u}}^{*}{\parallel }_0⩽{\nu }^{-1}{\gamma }_0{c}_0{\parallel T\parallel }_0\parallel ⩽{\gamma }_0{\nu }^{-1}{c}_0{(\parallel w\parallel }_0+\parallel T_0{\parallel }_0)\hfill \\ ⩽{\gamma }_0{\nu }^{-1}{c}_0(c_0{\parallel \nabla w\parallel }_0+\parallel T_0{\parallel }_0)\hfill \\ ⩽{\gamma }_0{\nu }^{-1}{c}_0(c_0{\parallel \nabla T\parallel }_0+(1+c_0)\parallel T_0{\parallel }_1).\hfill \end{array}\end{array}$$

Inputting (20) into the above inequality, we obtain:

$$\begin{array}{ccc}\hfill & & {\displaystyle \parallel \nabla {\mathit{u}}^{*}{\parallel }_0⩽{\nu }^{-1}{\gamma }_0{c}_0((3c_0+1)\parallel T_0{\parallel }_1+{\gamma }_0\epsilon c_0^2\parallel \nabla \tilde{\mathit{u}}{\parallel }_0)}\hfill \\ \hfill & & {\displaystyle ={\nu }^{-1}{\gamma }_0^2\epsilon c_0^3\parallel \nabla \tilde{\mathit{u}}{\parallel }_0+{\nu }^{-1}{\gamma }_0{c}_0(3c_0+1)\parallel T_0{\parallel }_1\equiv \tilde{A}(\parallel \nabla \tilde{\mathit{u}}{\parallel }_0).}\hfill \end{array}$$

(21)

(iii)

Prove that the operator ℓ is continuous.

Suppose that when $\tilde{\mathit{u}}$ is equal to ${\tilde{\mathit{u}}}^1$ and ${\tilde{\mathit{u}}}^2\in \mathbb{V}$ in the Equation (10), the Equations (11) and (13) have, respectively, solutions $({\mathit{u}}^{*1},$$T_1)\in \mathbb{V}\times \mathbb{W}$ and $({\mathit{u}}^{*2},T_2)\in \mathbb{V}\times \mathbb{W}$ satisfying $T_1{{|}_{\partial \mathsf{\Omega }}=T_2|}_{\partial \mathsf{\Omega }}=T_0$. Thus, by (11), we obtain:

$$d(T_1-T_2,\phi )+{\gamma }_0{a}_2({\tilde{\mathit{u}}}^1-{\tilde{\mathit{u}}}^2,T_1,\phi )+{\gamma }_0{a}_2({\tilde{\mathit{u}}}^2,T_1-T_2,\phi )=0,\phi \in {\mathbb{W}}_0.$$

(22)

Owing $(T_1-T_2){|}_{\partial \mathsf{\Omega }}=0$, namely, $(T_1-T_2)\in {\mathbb{W}}_0$, taking $\phi =T_1-T_2$ in (22) and using the condition (A${}_3$), we obtain:

$$\begin{array}{ccc}\hfill \parallel \nabla (T_1-T_2){\parallel }_0^2& =& {\displaystyle |{\gamma }_0{a}_2({\tilde{\mathit{u}}}^1-{\tilde{\mathit{u}}}^2,T_1,T_1-T_2)|}\hfill \\ \hfill & ⩽& {\displaystyle {\gamma }_0\overline{N}\parallel \nabla ({\tilde{\mathit{u}}}^1-{\tilde{\mathit{u}}}^2){\parallel }_0\parallel \nabla T_1{\parallel }_0{\parallel \nabla (T_1-T_2)\parallel }_0.}\hfill \end{array}$$

It follows that:

$$\begin{array}{c}\hfill \parallel \nabla (T_1-T_2){\parallel }_0⩽{\gamma }_0\overline{N}\parallel \nabla ({\tilde{\mathit{u}}}^1-{\tilde{\mathit{u}}}^2){\parallel }_0{\parallel \nabla T_1\parallel }_0.\end{array}$$

(23)

By (13), we obtain:

$$\begin{array}{ccc}\hfill & & {\displaystyle a({\mathit{u}}^{*1}-{\mathit{u}}^{*2},\mathit{v})+a_1({\mathit{u}}^{*1}-{\mathit{u}}^{*2},{\mathit{u}}^{*1},\mathit{v})+a_1({\mathit{u}}^{*2},{\mathit{u}}^{*1}-{\mathit{u}}^{*2},\mathit{v})}\hfill \\ \hfill & & {\displaystyle ={\gamma }_0((T_1-T_2)\mathit{j},\mathit{v}),\forall \mathit{v}\in \mathbb{V}.}\hfill \end{array}$$

(24)

Taking $\mathit{v}={\mathit{u}}^{*1}-{\mathit{u}}^{*2}$ in (24) and using the condition (A${}_3$), we obtain:

$$\begin{array}{c}\nu \parallel \nabla ({\mathit{u}}^{*1}-{\mathit{u}}^{*2}){\parallel }_0^2\hfill \\ =|{\gamma }_0((T_1-T_2)\mathit{j},{\mathit{u}}^{*1}-{\mathit{u}}^{*2})-a_1({\mathit{u}}^{*1}-{\mathit{u}}^{*2},{\mathit{u}}^{*1},{\mathit{u}}^{*1}-{\mathit{u}}^{*2})|\hfill \\ ⩽N_0\parallel \nabla ({\mathit{u}}^{*1}-{\mathit{u}}^{*2}){\parallel }_0^2\parallel \nabla {\mathit{u}}^{*1}{\parallel }_0+c_0^2{\gamma }_0\parallel \nabla (T_1-T_2){\parallel }_0{\parallel \nabla ({\mathit{u}}^{*1}-{\mathit{u}}^{*2})\parallel }_0.\hfill \end{array}$$

It follows that:

$$\begin{array}{ccc}\hfill {\displaystyle \parallel \nabla ({\mathit{u}}^{*1}-{\mathit{u}}^{*2}){\parallel }_0}& ⩽& {\displaystyle {\nu }^{-1}{N}_0\parallel \nabla ({\mathit{u}}^{*1}-{\mathit{u}}^{*2}){\parallel }_0{\parallel \nabla {\mathit{u}}^{*1}\parallel }_0}\hfill \\ \hfill & & {\displaystyle +{\nu }^{-1}{c}_0^2{\gamma }_0{\parallel \nabla (T_1-T_2)\parallel }_0.}\hfill \end{array}$$

(25)

Inputting (23) into (25), we obtain:

$$\begin{array}{ccc}\hfill {\displaystyle \parallel \nabla ({\mathit{u}}^{*1}-{\mathit{u}}^{*2}){\parallel }_0}& ⩽& {\displaystyle {\nu }^{-1}{N}_0\parallel \nabla ({\mathit{u}}^{*1}-{\mathit{u}}^{*2}){\parallel }_0{\parallel \nabla {\mathit{u}}^{*1}\parallel }_0}\hfill \\ \hfill & & {\displaystyle +{\nu }^{-1}{c}_0^2{\gamma }_0^2\overline{N}\parallel \nabla T_1{\parallel }_0{\parallel \nabla ({\tilde{\mathit{u}}}^1-{\tilde{\mathit{u}}}^2)\parallel }_0.}\hfill \end{array}$$

(26)

It follows by (20) and (21) that, if both $\parallel \nabla {\tilde{\mathit{u}}}^1{\parallel }_0$ and $\parallel \nabla {\tilde{\mathit{u}}}^2{\parallel }_0$ are bounded, then both $\parallel \nabla {\mathit{u}}^{*i}{\parallel }_0⩽\tilde{A}(\parallel \nabla {\tilde{\mathit{u}}}^i{\parallel }_0)$ and $\parallel \nabla T_i{\parallel }_0⩽\tilde{B}(\parallel \nabla {\tilde{\mathit{u}}}^i{\parallel }_0)$$(i=1,2)$ are also bounded. Thus, when

$$\begin{array}{c}\hfill {\nu }^{-1}{N}_0\tilde{A}⩽1-{\delta }_1,0<{\delta }_1⩽1,\end{array}$$

(27)

by (26), we obtain:

$$\begin{array}{c}\hfill \parallel \nabla ({\mathit{u}}^{*1}-{\mathit{u}}^{*2}){\parallel }_0⩽{\nu }^{-1}{\delta }_1^{-1}{c}_0^2{\gamma }_0^2\overline{N}\tilde{B}{\parallel \nabla ({\tilde{\mathit{u}}}^1-{\tilde{\mathit{u}}}^2)\parallel }_0,\end{array}$$

(28)

so ${\mathit{u}}^{*}$ is continuously dependent on $\tilde{\mathit{u}}$.

(iv)

Consider the following fixed point equation

$$\begin{array}{c}\hfill s\ell \mathit{u}=\mathit{u},0⩽s⩽1.\end{array}$$

(29)

When $s=0$, taking $\mathit{u}=\mathit{0}$, Equation (29) holds. Therefore, it is only necessary to consider the case of $0<s⩽1$. Inputting $s^{-1}\mathit{u}=\ell \mathit{u}$ into (21), we obtain:

$$\begin{array}{c}\hfill s^{-1}{\parallel \nabla \mathit{u}\parallel }_0⩽{\nu }^{-1}{\gamma }_0^2\epsilon c_0^3{\parallel \nabla \mathit{u}\parallel }_0+{\nu }^{-1}{\gamma }_0{c}_0(3c_0+1){\parallel T_0\parallel }_1.\end{array}$$

(30)

Noting that $2s{(2-s)}^{-1}⩽2$ and $0<s⩽1$, and taking

$$\begin{array}{c}\hfill \epsilon ={\left(2c_0^3{\gamma }_0^2\right)}^{-1}\nu ,\end{array}$$

(31)

by (20), we obtain:

$$\begin{array}{ccc}\hfill {\displaystyle {\parallel \nabla \mathit{u}\parallel }_0}& ⩽& {\displaystyle 2s{(2-s)}^{-1}{c}_0{\gamma }_0{\nu }^{-1}(3c_0+1){\parallel T_0\parallel }_1}\hfill \\ \hfill & ⩽& {\displaystyle 2c_0{\gamma }_0{\nu }^{-1}(3c_0+1){\parallel T_0\parallel }_1\equiv A.}\hfill \end{array}$$

(32)

It follows that ${\parallel \nabla \mathit{u}\parallel }_0$ is also bounded. Inputting (32) into (20) and using (31), we obtain:

$$\begin{array}{c}\hfill {\parallel \nabla T\parallel }_0⩽2\parallel \nabla T_0{\parallel }_0+\frac{{\gamma }_0{c}_{0}A}{2c_0^3{\gamma }_0{\nu }^{-1}}⩽2{\parallel T_0\parallel }_1+{\left(2{\gamma }_0{c}_0^2\right)}^{-1}\nu A=B.\end{array}$$

(33)

Thus, when the condition (A${}_4$) holds, from (28), we conclude that ℓ is the compression operator. By the Banach compression fixed point theorem (see [13]), we can conclude that Problem 3 possesses a unique solution $(\mathit{u},T)\in \mathbb{V}\times \mathbb{W}$ satisfying ${T|}_{\partial \mathsf{\Omega }}=T_0$.

(2) Prove that Problem 2 possesses at least a solution.

Obviously, the solution $(\mathit{u},T)$ for Problem 3 satisfies the second and third equations for Problem 2. For the obtained $(\mathit{u},T)$, let:

$$\begin{array}{c}\hfill f\left(\mathit{v}\right)=-{\gamma }_0(T\mathit{j},\mathit{v})+\nu a(\mathit{u},\mathit{v})+a_1(\mathit{u},\mathit{u},\mathit{v}),\forall \mathit{v}\in \mathbb{X}.\end{array}$$

(34)

Then, $f\left(\mathit{v}\right)$ is a bounded linear functional, and when $\mathit{v}\in \mathbb{V}$, $f\left(\mathit{v}\right)=0$. Therefore, there exists some $p\in \mathbb{M}$ satisfying:

$$\begin{array}{c}\hfill f\left(\mathit{v}\right)=b(p,\mathit{v})=(p,\mathrm{div}\mathit{v}),\forall \mathit{v}\in \mathbb{X},\end{array}$$

(35)

which implies that $(\mathit{u},p,T)\in \mathbb{X}\times \mathbb{M}\times \mathbb{W}$ is a solution for Problem 2 satisfying ${T|}_{\partial \mathsf{\Omega }}=T_0$, namely, Problem 2 possesses at least a solution $(\mathit{u},p,T)\in \mathbb{X}\times \mathbb{M}\times \mathbb{W}$ satisfying ${T|}_{\partial \mathsf{\Omega }}=T_0$.

(3) Prove that $(\mathit{u},p,T)\in \mathbb{X}\times \mathbb{M}\times \mathbb{W}$ is a unique solution for Problem 2 satisfying ${T|}_{\partial \mathsf{\Omega }}=T_0$.

If $({\mathit{u}}^1,p_1,T_1)$ is anther solution for Problem 2 that satisfies $T_1{|}_{\partial \mathsf{\Omega }}=T_0$, then by Problem 2, we obtain:

$$\begin{array}{ccc}\hfill & & {\displaystyle a(\mathit{u}-{\mathit{u}}^1,\mathit{v})+a_1(\mathit{u}-{\mathit{u}}^1,\mathit{u},\mathit{v})+a_1({\mathit{u}}^1,\mathit{u}-{\mathit{u}}^1,\mathit{v})+b(p-p_1,\mathit{v})}\hfill \\ \hfill & & {\displaystyle ={\gamma }_0((T-T_1)\mathit{j},\mathit{v}),\forall \mathit{v}\in \mathbb{X},}\hfill \end{array}$$

(36)

$$\begin{array}{ccc}\hfill & & {\displaystyle b(\phi ,\mathit{u}-{\mathit{u}}^1)=0,\forall \phi \in \mathbb{M},}\hfill \end{array}$$

(37)

$$\begin{array}{ccc}\hfill & & {\displaystyle d(T-T_1,\psi )+{\gamma }_0{a}_2(\mathit{u}-{\mathit{u}}^1,T,\psi )+{\gamma }_0{a}_2({\mathit{u}}^1,T-T_1,\psi )}\hfill \\ \hfill & & {\displaystyle =0,\forall \psi \in {\mathbb{W}}_0.}\hfill \end{array}$$

(38)

Taking $\mathit{v}=\mathit{u}-{\mathit{u}}^1$ in (36) and using the condition (A${}_1$) and (37), we obtain:

$$\begin{array}{ccc}& & {\displaystyle \nu \parallel \nabla (\mathit{u}-{\mathit{u}}^1){\parallel }_0^2=|a_1(\mathit{u}-{\mathit{u}}^1,\mathit{u},\mathit{u}-{\mathit{u}}^1)-{\gamma }_0((T-T_1)\mathit{j},\mathit{u}-{\mathit{u}}^1)|}\hfill \\ & & {\displaystyle ⩽N_{0}A\parallel \nabla (\mathit{u}-{\mathit{u}}^1){\parallel }_0^2+{\gamma }_0{c}_0^2\parallel \nabla (T-T_1){\parallel }_0{\parallel \nabla (\mathit{u}-{\mathit{u}}^1)\parallel }_0,}\hfill \end{array}$$

Thus, when ${\nu }^{-1}{N}_{0}A⩽1-{\delta }_1$ and $0<{\delta }_1⩽1$, we obtain:

$$\begin{array}{c}\hfill \parallel \nabla (\mathit{u}-{\mathit{u}}^1){\parallel }_0⩽{\delta }_1^{-1}{\gamma }_0{\nu }^{-1}{c}_0^2{\parallel \nabla (T-T_1)\parallel }_0.\end{array}$$

(39)

Taking $\psi =T-T_1$ in (38), by the condition (A${}_1$), we obtain:

$$\begin{array}{ccc}\hfill \parallel \nabla (T-T_1){\parallel }_0^2& =& {\displaystyle |-{\gamma }_0{a}_2(\mathit{u}-{\mathit{u}}^1,T,T-T_1)|}\hfill \\ \hfill & ⩽& {\displaystyle {\gamma }_{0}B\overline{N}\parallel \nabla (\mathit{u}-{\mathit{u}}^1){\parallel }_0{\parallel \nabla (T-T_1)\parallel }_0.}\hfill \end{array}$$

It follows that:

$$\begin{array}{ccc}\hfill & & \parallel \nabla (T-T_1){\parallel }_0⩽{\delta }_1^{-1}{\gamma }_0^{2}B\overline{N}{\nu }^{-1}{c}_0^2{\parallel \nabla (T-T_1)\parallel }_0.\hfill \end{array}$$

(40)

Thus, when ${\delta }_1^{-1}{\gamma }_0^{2}B\overline{N}{\nu }^{-1}{c}_0^2⩽1-{\delta }_2$ and $0<{\delta }_2⩽1$, we obtain $\parallel \nabla (T-T_1){\parallel }_0=0$. It follow that $T=T_1$. Further, from (39) we obtain $\parallel \nabla (\mathit{u}-{\mathit{u}}^1){\parallel }_0=0$. It follows that: $\mathit{u}={\mathit{u}}^1$. Therefore, by (36), we obtain:

$$\begin{array}{ccc}\hfill & & b(p-p_1,\mathit{v})=0,\forall \mathit{v}\in \mathbb{X}.\hfill \end{array}$$

(41)

Hence, by the continuous B-B condition, we obtain:

$$\begin{array}{ccc}\hfill & & \parallel p-p_1{\parallel }_0⩽\underset{\mathit{v}\in \mathbb{X}}{sup}\frac{b(p-p_1,\mathit{v})}{{\parallel \nabla \mathit{v}\parallel }_0}=0.\hfill \end{array}$$

(42)

It follows that $p=p_1$. This proves that Problem 2 possesses a unique solution. Theorem 1 is proved. □

Remark 1.

In the past, the existence and uniqueness of generalized solutions of Navier-Stokes equation were proved by using spatial sequence approximation (see [16]). Herein, the existence and uniqueness of the mixed generalized solutions of the steady Boussinesq equation are proved by the Banach compression fixed point theorem, which provides another approach for proving the existence and uniqueness of generalized solutions of complex hydrodynamics problems.

3. The Existence and Uniqueness of the MFE Solution

To establish the MFE format for the steady Boussinesq equation, one needs the following assumption.

(A${}_5$) Supposed that ${\mathsf{\Omega }}_h\subset \mathsf{\Omega }$ is the bounded polygonal domain that satisfies mes$(\mathsf{\Omega }-{\mathsf{\Omega }}_h)=0$, ${\Im }_h$ is the quasi-uniform triangle or quadrilateral subdivision on ${\overline{\mathsf{\Omega }}}_h$, ${\mathbb{X}}_h\subset \mathbb{X}$ is at least formed by piecewise mth-degree polynomial vectors ($m>0$ is integer), ${\mathbb{M}}_h\subset \mathbb{M}$ is formed by the piecewise $(m-1)$th-degree polynomials, ${\mathbb{W}}_h\subset \mathbb{W}$ is formed by the piecewise mth-degree polynomials, ${\mathbb{W}}_{0h}={\mathbb{W}}_h\cap H_0^1\left(\mathsf{\Omega }\right)$, and $({\mathbb{X}}_h,{\mathbb{M}}_h)$ satisfies the following discrete B-B condition:

$$\begin{array}{c}\hfill \underset{v_h\in {\mathbb{X}}_h}{sup}\frac{b({\phi }_h,v_h)}{\parallel \nabla {\mathit{v}}_h{\parallel }_0}⩾\beta {\parallel {\phi }_h\parallel }_0,\forall {\phi }_h\in {\mathbb{M}}_h,\end{array}$$

(43)

where $\beta$ is a positive constant independent of h.

In the following, we provide some examples of MFE subspaces that satisfy the condition (A${}_5$). To this end, suppose that ${\mathbb{P}}_l\left(K\right)$ and ${\mathbb{Q}}_l\left(K\right)$ are spaces formed by lth- and bilth-degree polynomials on K, respectively.

Example 1.

The MFE subspaces for first-order MFE format in the triangle partition ${\Im }_h$ are defined as follows:

$$\begin{array}{c}\hfill \left\{\begin{array}{c}{\mathbb{W}}_h=\{{\psi }_h\in \mathbb{W}\cap C^0\left(\overline{\mathsf{\Omega }}\right):{\psi }_h{|}_K\in {\mathbb{P}}_1\left(K\right),\forall K\in {\Im }_h\},\hfill \\ {\mathbb{M}}_h=\{{\phi }_h\in \mathbb{M}:{\phi }_h{|}_K\in {\mathbb{P}}_0\left(K\right),\forall K\in {\Im }_h\},\hfill \\ {\mathbb{X}}_h=\{{\mathit{v}}_h\in \mathbb{X}\cap C^0{\left(\overline{\mathsf{\Omega }}\right)}^2:{\mathit{v}}_h{|}_K\in {\mathbb{P}}_K,\forall K\in {\Im }_h\},\hfill \end{array}\right.\end{array}$$

where ${\mathbb{P}}_K={\mathbb{P}}_1{\left(K\right)}^2\oplus \mathrm{span}\left\{{\displaystyle \prod _{j=1,j\ne i}^3}{\lambda }_j{\mathit{n}}_i:i=1,2,3\right\}$, ${\mathit{n}}_i$ are the outward unit normal vector on opposite side or face of vertex $A_i$ of element K, and ${\lambda }_i$ are the areal coordinates corresponding to $A_i$ ($i=1,$$2,$ 3). The degrees of freedom of ${\mathbb{X}}_h$ are taken as the values of $\mathit{u}$ at the vertexes of element K and the flow rate through the three sides $F_i$$(i=1,2,3)$ of K, the degrees of freedom of ${\mathbb{M}}_h$ are taken as the average ${\displaystyle {\int }_{K}p\mathrm{d}\mathit{x}/\mathrm{mes}\left(K\right)}$ of p over the element K, and the degrees of freedom of ${\mathbb{W}}_h$ are taken as the values of T at the vertexes of element K. Noting that the ratio for the number $N_p$ of vertexes, the number $N_K$ of elements, and the number $N_S$ of sides in the triangle partition ${\Im }_h$ is $1:2:3$, we can calculate out that the number of total degrees of freedom for first-order MFE format in triangle partition ${\Im }_h$ is $3N_p+N_K+N_s\approx 8N_p$.

Example 2.

The MFE subspaces for the second-order MFE format in the triangle partition ${\Im }_h$ are defined as follows:

$$\begin{array}{ccc}& & \left\{\begin{array}{c}{\mathbb{W}}_h=\{{\psi }_h\in \mathbb{W}\cap C^0\left(\overline{\mathsf{\Omega }}\right):{\psi }_h{|}_K\in {\mathbb{P}}_2\left(K\right),\forall K\in {\Im }_h\},\hfill \\ {\mathbb{M}}_h=\{{\phi }_h\in \mathbb{M}\cap C^0\left(\overline{\mathsf{\Omega }}\right):{\phi }_h{|}_K\in {\mathbb{P}}_1\left(K\right),\forall K\in {\Im }_h\},\hfill \\ {\mathbb{X}}_h=\{{\mathit{v}}_h\in \mathbb{X}\cap C^0{\left(\overline{\mathsf{\Omega }}\right)}^2:{\mathit{v}}_h{|}_K\in {\mathbb{P}}_K,\forall K\in {\Im }_h\},\hfill \end{array}\right.\hfill \end{array}$$

where ${\mathbb{P}}_K={\mathbb{P}}_2{\left(K\right)}^2\oplus {\left(\mathrm{span}\left\{{\lambda }_1{\lambda }_2{\lambda }_3\right\}\right)}^2$. The degrees of freedom of ${\mathbb{X}}_h$, ${\mathbb{W}}_h$, and ${\mathbb{M}}_h$ can be, respectively, taken as the values of $\mathit{u}$, T, and p at the vertexes K and the values of $\mathit{u}$ and T at midpoints of three sides on K, as well as the integral value ${\displaystyle {\int }_K\mathit{u}\mathrm{d}\mathit{x}}$. Thus, the number of total degrees of freedom in the format is $4N_p+2N_K+3N_s\approx 17N_p$.

Example 3.

The MFE subspaces for the third-order MFE format in the triangle partition ${\Im }_h$ are defined as follows:

$$\begin{array}{ccc}& & \left\{\begin{array}{c}{\mathbb{W}}_h=\{{\psi }_h\in \mathbb{W}\cap C^0\left(\overline{\mathsf{\Omega }}\right):{\psi }_h{|}_K\in {\mathbb{P}}_3\left(K\right),\forall K\in {\Im }_h\},\hfill \\ {\mathbb{M}}_h=\{{\phi }_h\in \mathbb{M}\cap C^0\left(\overline{\mathsf{\Omega }}\right):{\phi }_h{|}_K\in {\mathbb{P}}_2\left(K\right),\forall K\in {\Im }_h\},\hfill \\ {\mathbb{X}}_h=\{{\mathit{v}}_h\in \mathbb{X}\cap C^0{\left(\overline{\mathsf{\Omega }}\right)}^2:{\mathit{v}}_h{|}_K\in {\mathbb{P}}_K,\forall K\in {\Im }_h\},\hfill \end{array}\right.\hfill \end{array}$$

where ${\mathbb{P}}_K={\mathbb{P}}_3{\left(K\right)}^2\oplus {\left(\mathrm{span}\{{\lambda }_1{\lambda }_2{\lambda }_3{\lambda }_i:i=1,2,3\}\right)}^2$. The degrees of freedom of ${\mathbb{X}}_h$, ${\mathbb{W}}_h$, and ${\mathbb{M}}_h$ can be, respectively, taken as the values of $\mathit{u}$, T, and p at the vertexes K, the values of $\mathit{u}$ and T at the trisection points on three sides of K, and the values of p at midpoints of three sides on K as well as the integral values ${\displaystyle {\int }_K{\lambda }_i\mathit{u}\mathrm{d}\mathit{x}}$$(i=1,$$2,$ 3). Thus, the number of total degrees of freedom in the format is $4N_p+6N_K+7N_s=37N_p$.

Example 4.

The MFE subspaces for the first-order MFE format in the quadrilateral partition ${\Im }_h$ are defined as follows:

$$\begin{array}{ccc}& & \left\{\begin{array}{c}{\mathbb{W}}_h=\{{\psi }_h\in \mathbb{W}\cap C^0\left(\overline{\mathsf{\Omega }}\right):{\psi }_h{|}_K\in {\mathbb{Q}}_1\left(K\right),\forall K\in {\Im }_h\},\hfill \\ {\mathbb{M}}_h=\{{\phi }_h\in \mathbb{M}:{\phi }_h{|}_K\in {\mathbb{P}}_0\left(K\right),\forall K\in {\Im }_h\},\hfill \\ {\mathbb{X}}_h=\{{\mathit{v}}_h\in \mathbb{X}\cap C^0{\left(\overline{\mathsf{\Omega }}\right)}^2:{\mathit{v}}_h{|}_K\in {\mathbb{Q}}_K,\forall K\in {\Im }_h\},\hfill \end{array}\right.\hfill \end{array}$$

where ${\mathbb{Q}}_K={\mathbb{Q}}_1{\left(K\right)}^2\oplus \mathrm{span}\{{\varpi }_i·{\mathit{n}}_i:i=1,2,3,4\}$, ${\varpi }_1=$$(1-{\eta }^2)(1-\xi )$, ${\varpi }_2$=$(1-\eta )(1-{\xi }^2)$, ${\varpi }_3=(1-{\eta }^2)(1+\xi )$, ${\varpi }_4=(1+\eta )(1-{\xi }^2)$, and $(\xi ,\eta )={\mathit{F}}_K^{-1}(x_1,x_2)$ is the inverse of bilinear isoparametric transformation ${\mathit{F}}_K$ from ${[1,1]}^2$ to K (see [17]). The degrees of freedom of ${\mathbb{X}}_h$ are taken as the values of $\mathit{u}$ at the vertexes of quadrilateral element K and the flow rate through the four sides $F_i$$(i=1,2,3,4)$ of quadrilateral element K, the degrees of freedom of ${\mathbb{M}}_h$ are taken as the average ${\displaystyle {\int }_{K}p\mathrm{d}\mathit{x}/mes\left(K\right)}$ of p over the quadrilateral element K, and the degrees of freedom of ${\mathbb{W}}_h$ are taken as the values of T at the vertexes of element K. Noting that the ratio for the number $N_p$ of vertexes, the number $N_K$ of elements, and the number $N_s$ of sides in the quadrangle partition ${\Im }_h$ is $1:1:2$, we can calculate out that the number of total degrees of freedom for the first-order MFE format in the quadrangle partition ${\Im }_h$ is $3N_p+N_K+N_s=6N_p$, so that the format has fewer total degrees of freedom.

Example 5.

The MFE subspaces for the second-order MFE format in the quadrilateral partition ${\Im }_h$ are defined as follows:

$$\begin{array}{ccc}& & \left\{\begin{array}{c}{\mathbb{W}}_h=\{{\psi }_h\in \mathbb{W}\cap C^0\left(\overline{\mathsf{\Omega }}\right):{\psi }_h{|}_K\in Q_2\left(K\right),\forall K\in {\Im }_h\},\hfill \\ {\mathbb{M}}_h=\{{\phi }_h\in \mathbb{M}\cap C^0\left(\overline{\mathsf{\Omega }}\right):{\phi }_h{|}_K\in {\mathbb{Q}}_1\left(K\right),\forall K\in {\Im }_h\},\hfill \\ {\mathbb{X}}_h=\{{\mathit{v}}_h\in \mathbb{X}\cap C^0{\left(\overline{\mathsf{\Omega }}\right)}^2:{\mathit{v}}_h{|}_K\in {\mathbb{Q}}_K,\forall K\in {\Im }_h\},\hfill \end{array}\right.\hfill \end{array}$$

where ${\mathbb{Q}}_K={\mathbb{Q}}_1{\left(K\right)}^2\oplus \mathrm{span}\{\mathit{\alpha }D{F}_K{N}_0:\mathit{\alpha }\in {\mathbb{R}}^2\}\oplus \mathrm{span}\left\{(\eta ,\xi )D{F}_K{N}_0\right\}$, $N_0=(1-{\xi }^2)(1-{\eta }^2)$, $(\xi ,\eta )={\mathit{F}}_K^{-1}(x_1,x_2)$, and ${\mathbb{Q}}_2\left(K\right)$ is the incomplete biquadratic polynomial space on K. The degrees of freedom of ${\mathbb{X}}_h$, ${\mathbb{W}}_h$, and ${\mathbb{M}}_h$ may be taken the values of $\mathit{u}$, T, and p at the vertexes on quadrilateral K and the values of $\mathit{u}$ and T at midpoints on four sides of quadrilateral K, as well as the integral values ${\displaystyle {\int }_K\mathit{u}D{\mathit{F}}_K^{-1}\mathrm{d}\mathit{x}}$ and ${\displaystyle {\int }_K\mathit{u}D{\mathit{F}}_K^{-1}{(\eta ,\xi )}^T\mathrm{d}\mathit{x}}$. Thus, the number of total degrees of freedom of the format are $4N_p+3N_K+3N_s=13N_p$. It follows that this format is one of the least total degree of freedom among the existing second-order formats.

Example 6.

A set of new MFE subspaces in the triangulation ${\Im }_h$ are defined as follows:

$$\begin{array}{ccc}& & \left\{\begin{array}{c}{\mathbb{W}}_h=\{{\psi }_h\in \mathbb{W}\cap C^0\left(\overline{\mathsf{\Omega }}\right):{\psi }_h{|}_K\in {\mathbb{P}}_m\left(K\right),\forall K\in {\Im }_h\},\hfill \\ {\mathbb{M}}_h=\{{\phi }_h\in \mathbb{M}:{\phi }_h{|}_K\in {\mathbb{P}}_{m-1}\left(K\right),\forall K\in {\Im }_h\},\hfill \\ {\displaystyle {\mathbb{X}}_h={\{{(v_{1h},v_{2h})}^T\in L^2{\left(\mathsf{\Omega }\right)}^2:v_{jh}|}_K\in {\mathbb{P}}_m\left(K\right)\cap H^1\left(K\right),\forall K\in {\Im }_h,}\hfill \\ {\displaystyle v_{jh}{n}_{i1}{{|}_S=v_{jh}{n}_{i2}|}_S,S=\partial K_1\cap \partial K_2\subset \mathsf{\Omega },}\hfill \\ {\displaystyle v_{jh}{n}_j{|}_S=0,S\subset \partial K\cap \partial \mathsf{\Omega },i,j=1,2\}.}\hfill \end{array}\right.\hfill \end{array}$$

where ${\Im }_h$ is the quasi-uniform triangle subdivision on $\overline{\mathsf{\Omega }}$ and ${\mathbb{P}}_l\left(K\right)$ represents the polynomial space of degree $⩽l$ on K.

It can be proven that ${\mathbb{X}}_h\subset \mathbb{X}$, and there exists ${\mathsf{\Pi }}_h:\mathbb{X}\to {\mathbb{X}}_h$, such that, $\forall \mathit{v}\in \mathbb{X}$, there hold the following equalities:

$$\begin{array}{ccc}& & (\mathrm{div}(\mathit{v}-{\mathsf{\Pi }}_h\mathit{v}),{\phi }_h)=0,\forall {\phi }_h\in {\mathbb{M}}_h,\mathrm{div}{\mathsf{\Pi }}_h\mathit{v}=P_h\mathrm{div}\mathit{v},\forall \mathit{v}\in \mathbb{X},\hfill \end{array}$$

where $P_h$ is the $L^2$ projection. Additionally, when $\mathit{v}\in W^{r,q}{\left(\mathsf{\Omega }\right)}^2$, there hold the following error estimates:

$$\begin{array}{ccc}& & \parallel \mathit{v}-{\mathsf{\Pi }}_h{\mathit{v}\parallel }_{0,q}⩽C{h}^r{\parallel \mathit{v}\parallel }_{r,q},1⩽r⩽m+1,1⩽q⩽\infty .\hfill \end{array}$$

More MFE subspaces ${\mathbb{X}}_h$, ${\mathbb{W}}_h$, and ${\mathbb{M}}_h$ can refer to [17,18]. Thus, we can establish the following MFE format for the steady Boussinesq equation:

Problem 4.

Find $({\mathit{u}}_h,p_h,T_h)\in {\mathbb{X}}_h\times {\mathbb{M}}_h\times {\mathbb{W}}_h$ that satisfies $T_h{|}_{\partial \mathsf{\Omega }}=T_0$ and

$$\left\{\begin{array}{cc}a({\mathit{u}}_h,{\mathit{v}}_h)+a_1({\mathit{u}}_h,{\mathit{u}}_h,{\mathit{v}}_h)-b(p_h,{\mathit{v}}_h)={\gamma }_0(T_h\mathit{j},{\mathit{v}}_h),\hfill & \forall {\mathit{v}}_h\in {\mathbb{X}}_h,\hfill \\ b({\phi }_h,{\mathit{u}}_h)=0,\hfill & \forall {\phi }_h\in {\mathbb{M}}_h,\hfill \\ d(T_h,{\psi }_h)+{\gamma }_0{a}_2({\mathit{u}}_h,T_h,{\psi }_h)=0,\hfill & \forall {\psi }_h\in {\mathbb{W}}_{0h},\hfill \end{array}\right.$$

where $T_h{|}_{\partial \mathsf{\Omega }}=T_0$ represents the interpolation of $T_0$ on ${\overline{\mathsf{\Omega }}}_h$ or $\overline{\mathsf{\Omega }}$, which satisfies $\left(A_2\right)$, and the equality holds on $\partial \mathsf{\Omega }$, and ${\mathbb{W}}_{0h}={\mathbb{W}}_h\cap H_0^1\left(\mathsf{\Omega }\right)$.

In the following, we employ the fixed point theorem and the Lax-Milgram theorem (see [13] or [14] or [15]) to prove the existence and uniqueness of MFE solution for Problem 4. We have the following result:

Theorem 2.

If the conditions $\left(A_1\right)$–$\left(A_5\right)$ hold, then Problem 4 possesses a unique solution $({\mathit{u}}_h,p_h,T_h)$$\in {\mathbb{X}}_h\times {\mathbb{M}}_h\times {\mathbb{W}}_h$ satisfying $T_{h|\partial \mathsf{\Omega }}=T_0$ and the following boundedness:

$$\begin{array}{c}\hfill \parallel \nabla {\mathit{u}}_h{\parallel }_0⩽A,{\parallel \nabla T_h\parallel }_0⩽B.\end{array}$$

(44)

Proof. 

For ${\tilde{\mathit{u}}}_h\in {\mathbb{X}}_h$, by the EF method for elliptic equation, we can conclude that:

$$\begin{array}{c}\hfill d(T_h,{\psi }_h)+{\gamma }_0{a}_2({\tilde{\mathit{u}}}_h,T_h,{\psi }_h)=0,\forall {\psi }_h\in {\mathbb{W}}_{0h}\end{array}$$

(45)

possesses a unique solution $T_h\in {\mathbb{W}}_h$, which satisfies $T_{h|\partial \mathsf{\Omega }}=T_0$. For the obtained $T_h$, we consider the following problem:

$$\begin{array}{ccc}\hfill & & a({\mathit{u}}_h^{*},{\mathit{v}}_h)+a_1({\mathit{u}}_h^{*},{\mathit{u}}_h^{*},{\mathit{v}}_h)-b(p_h,{\mathit{v}}_h)={\gamma }_0(T_h\mathit{j},{\mathit{v}}_h),\forall {\mathit{v}}_h\in {\mathbb{X}}_h,\hfill \end{array}$$

(46)

$$\begin{array}{ccc}\hfill & & b({\phi }_h,{\mathit{u}}_h^{*})=0,\forall {\phi }_h\in {\mathbb{M}}_h.\hfill \end{array}$$

(47)

Using the same method as proving that there exists unique solution of Equation (13), we can prove that there exists unique solution $({\mathit{u}}_h^{*},$$p_h)$$\in {\mathbb{X}}_h\times {\mathbb{M}}_h$, which implies that, for ${\tilde{\mathit{u}}}_h$, by (45)–(47), there exists a unique ${\mathit{u}}_h^{*}$ that corresponds to it, and this correspondence is written as ${\ell }_h$:

$$\begin{array}{ccc}\hfill & & {\ell }_h{\tilde{\mathit{u}}}_h={\mathit{u}}_h^{*}.\hfill \end{array}$$

(48)

(1)

Estimate $\parallel \nabla T_h{\parallel }_0$.

For $w_h\in {\mathbb{W}}_{0h}$, taking $T_h=w_h+T_0$ and ${\psi }_h=w_h$ in (45), we obtain:

$$\begin{array}{c}\hfill d(w_h,w_h)=-{\gamma }_0{a}_2({\tilde{\mathit{u}}}_h,w_h,T_0)-d(T_0,w_h).\end{array}$$

Thus, by the condition (A${}_3$), we obtain:

$$\begin{array}{c}\hfill \parallel \nabla w_h{\parallel }_0^2⩽\parallel \nabla T_0{\parallel }_0\parallel \nabla w_h{\parallel }_0+{\gamma }_0\overline{N}\parallel \nabla {\tilde{\mathit{u}}}_h{\parallel }_0\parallel \nabla w_h{\parallel }_0{\parallel \nabla T_0\parallel }_0.\end{array}$$

Therefore, by the condition (A${}_2)$, we obtain:

$$\begin{array}{c}\hfill \parallel \nabla w_h{\parallel }_0⩽\parallel \nabla T_0{\parallel }_0+{\gamma }_0\overline{N}\epsilon {\parallel \nabla {\tilde{\mathit{u}}}_h\parallel }_0.\end{array}$$

Thereupon, we obtain:

$$\begin{array}{ccc}\hfill {\displaystyle \parallel \nabla T_h{\parallel }_0}& ⩽& {\displaystyle \parallel \nabla w_h{\parallel }_0+{\parallel \nabla T_0\parallel }_0}\hfill \\ \hfill & ⩽& {\displaystyle 2\parallel T_0{\parallel }_1+{\gamma }_0\overline{N}\epsilon \parallel \nabla {\tilde{\mathit{u}}}_h{\parallel }_0\equiv \widehat{B}(\parallel \nabla {\tilde{\mathit{u}}}_h{\parallel }_0).}\hfill \end{array}$$

(49)

(2)

Estimate $\parallel \nabla {\mathit{u}}_h^{*}{\parallel }_0$.

Taking ${\mathit{v}}_h={\mathit{u}}_h^{*}$ in (46), by (47), we obtain:

$$\begin{array}{ccc}& & \nu \parallel \nabla {\mathit{u}}_h^{*}{\parallel }_0^2=|{\gamma }_0(T_h\mathit{j},{\mathit{u}}_h^{*})|⩽{\gamma }_0\parallel T_h{\parallel }_0\parallel {\mathit{u}}_h^{*}{\parallel }_0⩽{\gamma }_0{c}_0\parallel T_0{\parallel }_0{\parallel \nabla {\mathit{u}}_h^{*}\parallel }_0.\hfill \end{array}$$

Thus, by (A${}_1)$ and (49), we obtain:

$$\begin{array}{ccc}\hfill {\displaystyle \parallel \nabla {\mathit{u}}_h^{*}{\parallel }_0}& ⩽& {\displaystyle {\nu }^{-1}{\gamma }_0{C}_0\parallel T_h{\parallel }_0⩽{\nu }^{-1}{\gamma }_0{C}_0[\parallel w_h{\parallel }_0+\parallel T_0{\parallel }_0]}\hfill \\ \hfill & ⩽& {\displaystyle {\nu }^{-1}{\gamma }_0{C}_0[C_0\parallel \nabla w_h{\parallel }_0+\parallel T_0{\parallel }_0]}\hfill \\ \hfill & ⩽& {\displaystyle {\nu }^{-1}{\gamma }_0{C}_0(3C_0+1)\parallel T_0{\parallel }_1+{\nu }^{-1}{\gamma }_0^2{C}_0^2\overline{N}\epsilon {\parallel \nabla {\tilde{\mathit{u}}}_h\parallel }_0}\hfill \\ \hfill & \equiv & {\displaystyle \widehat{A}(\parallel \nabla {\tilde{\mathit{u}}}_h{\parallel }_0).}\hfill \end{array}$$

(50)

(3)

Discuss the continuity of ${\ell }_h$.

Let ${\tilde{\mathit{u}}}_h^1,{\tilde{\mathit{u}}}_h^2\in {\mathbb{X}}_h$. Then, by (45)–(47), we can uniquely solve out $({\mathit{u}}_h^{*1},p_h^1,T_h^1)$, $({\mathit{u}}_h^{*2},p_h^2,$$T_h^2)\in {\mathbb{X}}_h\times {\mathbb{M}}_h\times {\mathbb{W}}_h$, which satisfies $T_h^1{{|}_{\partial \mathsf{\Omega }}=T_h^2|}_{\partial \mathsf{\Omega }}=T_0$. We easily obtain:

$$\begin{array}{c}\hfill \parallel \nabla (T_h^1-T_h^2){\parallel }_0⩽{\gamma }_0\overline{N}\parallel \nabla ({\tilde{\mathit{u}}}_h^1-{\tilde{\mathit{u}}}_h^2){\parallel }_0{\parallel \nabla T_h^1\parallel }_0,\end{array}$$

(51)

$$\begin{array}{ccc}\hfill {\displaystyle \parallel \nabla ({\mathit{u}}_h^{*1}-{\mathit{u}}_h^{*2}){\parallel }_0}& ⩽& {\displaystyle {\nu }^{-1}{N}_0\parallel \nabla {\mathit{u}}_h^{*1}{\parallel }_0{\parallel \nabla ({\mathit{u}}_h^{*1}-{\mathit{u}}_h^{*2})\parallel }_0}\hfill \\ \hfill & & {\displaystyle +{\nu }^{-1}{c}_0^2{\gamma }_0^2\overline{N}\parallel \nabla ({\tilde{\mathit{u}}}_h^1-{\tilde{\mathit{u}}}_h^2){\parallel }_0{\parallel \nabla T_h^1\parallel }_0.}\hfill \end{array}$$

(52)

By (49) and (50) we can conclude that, if $\parallel \nabla {\tilde{\mathit{u}}}_h^i{\parallel }_0$ is bounded, so are $\parallel \nabla {\mathit{u}}_h^{*i}{\parallel }_0$$⩽\widehat{A}(\parallel \nabla {\tilde{\mathit{u}}}_h^i{\parallel }_0)$ and $\parallel \nabla T_h^i{\parallel }_0⩽\widehat{B}(\parallel \nabla {\tilde{\mathit{u}}}_h^i{\parallel }_0)$ ($i=1,2$). Therefore, if there holds

$$\begin{array}{ccc}\hfill & & {\nu }^{-1}{N}_0\widehat{A}⩽1-{\delta }_1,0<{\delta }_1⩽1,\hfill \end{array}$$

(53)

then, by (52), we obtain:

$$\begin{array}{ccc}\hfill & & \parallel \nabla ({\mathit{u}}_h^{*1}-{\mathit{u}}_h^{*2}){\parallel }_0⩽{\delta }_1^{-1}{\nu }^{-1}{c}_0^2{\gamma }_0^2\overline{N}\widehat{B}{\parallel \nabla ({\tilde{\mathit{u}}}_h^1-{\tilde{\mathit{u}}}_h^2)\parallel }_0.\hfill \end{array}$$

(54)

Therefore, ${\mathit{u}}_h^{*}$ is continuously dependent on ${\tilde{\mathit{u}}}_h$.

(4)

Consider the following equation.

$$\begin{array}{ccc}\hfill & & {\mathit{u}}_h=s{\ell }_h{\mathit{u}}_h,0⩽s⩽1.\hfill \end{array}$$

(55)

Obviously, when $s=0$, we obtain ${\mathit{u}}_h=0$. Therefore, we only consider the case when $0<s⩽1$. Inputting $s^{-1}{\mathit{u}}_h={\ell }_h{\mathit{u}}_h$ into (50), we obtain:

$$\begin{array}{c}\hfill s^{-1}\parallel \nabla {\mathit{u}}_h{\parallel }_0⩽{\nu }^{-1}{\gamma }_0{c}_0(3c_0+1)\parallel T_0{\parallel }_1+{\nu }^{-1}{\gamma }_0^2{c}_0^2\overline{N}\epsilon {\parallel \nabla {\mathit{u}}_h\parallel }_0.\end{array}$$

(56)

Noting that, when $0<s⩽1$, $s{(2-s)}^{-1}⩽1$ holds when taking

$$\begin{array}{ccc}\hfill & & \epsilon ={\left(2{\nu }^{-1}{\gamma }_0^2{c}_0^2\overline{N}\right)}^{-1},\hfill \end{array}$$

(57)

then, by (56), we obtain:

$$\begin{array}{ccc}\hfill & & \parallel \nabla {\mathit{u}}_h{\parallel }_0⩽2c_0{\gamma }_0{\nu }^{-1}(3c_0+1){\parallel T_0\parallel }_1\equiv A,\hfill \end{array}$$

(58)

which implies that $\parallel \nabla {\mathit{u}}_h{\parallel }_0$ is bounded. Inputting (58) into (49) and using (57), we obtain:

$$\begin{array}{ccc}\hfill & & \parallel \nabla T_h{\parallel }_0⩽2{\parallel T_0\parallel }_1+{\left(2{\gamma }_0{c}_0^2\right)}^{-1}\nu A\equiv B.\hfill \end{array}$$

(59)

Thus, when ${\delta }_1^{-1}{\nu }^{-1}{c}_0^2{\gamma }_0^{2}B\overline{N}⩽1-{\delta }_2$$(0<{\delta }_2⩽1)$, ${\ell }_h$ is the contraction operator. Thereupon, by the Banach fixed point theorem (see [13]), we can conclude that Problem 4 possesses a unique solution. Theorem 2 is thus proven. □

4. Discussions

We have proven theoretically that the existence and uniqueness of generalized solution and MFE solution for the two-dimensional steady Boussinesq equation by means of the completeness of Hilbert’s space, the fixed point theorem, and the Lax-Milgram theorem. From the discussion in Section 2 and Section 3, we can conclude that if the conditions (A${}_1$)–(A${}_5$) hold, the method herein can be used to discuss the three-dimensional problem.

The vector $\mathit{j}={(0,1)}^T$ in the first equation on the right hand in Problem 1 may be changed into the general continuous or integrable vector function $\mathit{g}$, and all results are true, but it has the specific physical meaning in the steady square cavity flow (see, e.g., [18]).

The process of proof in Section 2 and Section 3 has provided the cyclic iterative method to find the approximated solutions for the steady Boussinesq equation, which is also different from the existing method that the existence and uniqueness of generalized solutions of Navier-Stokes equation were proved by means of the spatial sequence approximation (see [16]), as mentioned in Remark 1. Herein, the existence and uniqueness of the mixed generalized solutions of the steady Boussinesq equation are proven by the Banach compression fixed point theorem and the completeness of Hilbert’s space, which can be easily applied to proving the existence and uniqueness of generalized solutions of complex hydrodynamics problems.

Herein, we have created several sets of MFE spaces. Some of them are new and have been proposed for the first time and can be applied other fluid mechanics equations, which are also some of highlights of this article.

Owing to this paper having been very long, we will discus the convergence of MFE solution and conduct numerical simulations in next paper.

5. Conclusions

Herein, we have employed the fixed point theorem and Lax-Milgram theorem to discuss the existence and uniqueness of generalized solution and MFE solution for the two-dimensional steady Boussinesq equation. This is a new attempt to discuss the existence and uniqueness of generalized solution and numerical solution for nonlinear partial differential equations. It can be generalized to the more complex with actual engineering problems.