Modified characteristics projection finite element method for time-dependent conduction-convection problems

Abstract

In this paper, we give a modified characteristics projection finite element method for the time-dependent conduction-convection problems, which is gotten by combining the modified characteristics finite element method and the projection method. The stability and the error analysis shows that our method is stable and has optimal convergence order. In order to show the effect of our method, some numerical results are presented. From the numerical results, we can see that the modified characteristics projection finite element method can simulate the fluid field, temperature field, and pressure field very well.

1 Introduction

The conduction-convection problem constitutes an important system of equations in atmospheric dynamics and dissipative nonlinear system of equations. There is a significant amount of literature as regards this problem. An optimizing reduced Petrov-Galerkin least squares mixed finite element (PLSMFE) [1] method for the non-stationary conduction-convection problems was given. An efficient sequential method was developed to estimate the unknown boundary condition on the surface of a body from transient temperature measurements inside the solid [2]. An analysis of conduction natural convection conjugate heat transfer in the gap between concentric cylinders under solar irradiation [3] was carried out by Kim et al., Boland and Layton [4] gave an error analysis for finite element methods for steady natural convection problems. Newton type iterative methods [5–7] and defect-correction methods [8–10] for the conduction-convection equations were presented.

The projection methods, which are efficient methods for solving the incompressible time-dependent fluid flow, were first introduced by Chorin [11] and Temam [12] in the late 1960s. This method is based on a special time-discretization of the Navier-Stokes equations. In this method, the convection-diffusion and the incompressibility are dealt with in two different sub-steps. The velocity obtained in the convection-diffusion sub-step is projected in order to satisfy the weak incompressibility condition. The projection methods can be classified into three families: the pressure-correction method [13, 14], the velocity-correction method [15], and the consistent splitting scheme [16, 17], which is called a gauge method also [18]. The convergence analysis of the semi-discrete projection methods can be found in Shen [19] and Guermond and Quartapelle [20]. In Guermond and Quartapelle [21], the projection method was implemented by the finite element method. It was used to solved the variable density Navier-Stokes equations in [22]. In [23], a gauge-Uzawa projection method was presented. Then it was applied to the conduction-convection equations [24] and incompressible flows with variable density [25].

As we know, the characteristics method is a highly effective method for the advection dominated problems. Douglas and Russell [26] presented the modified method of characteristics first. It was extended to nonlinear coupled systems by Russell [27] in two and three spatial dimensions. A detailed analysis for the Navier-Stokes equations has been done by Dawson et al. [28] and numerical tests have been presented by Buscagkia and Dari [29]. A second order MMOC mixed defect-correction finite element method [30] for time-dependent Navier-Stokes problems was given. Notsu et al. gave a single-step characteristics finite difference analysis for the convection-diffusion problems [31] and a single-step finite element method for the incompressible Navier-Stokes equations [32]. El-Amrani and Seaid gave the error estimates of the modified method of characteristics finite element methods for the Navier-Stokes [33], natural, and mixed convection flows [34]. In [35], Achdou and Guermond gave the projection/Lagrange-Galerkin method for the incompressible Navier-Stokes equations.

In this paper, we give the modified characteristics projection finite element method (MCPFEM) for the time-dependent conduction-convection problems, which is gotten by combining the modified characteristics finite element method and the projection method. We give stability and error analysis, which show that our method is stable and has optimal convergence order. In order to show the efficiency of our method, some numerical results are presented. At first, the numerical results of Bénard convection problems are given. Then we give some numerical results of the thermal driven cavity flow. From the numerical results, we can see that MCPFEM can simulate the fluid field, temperature field, and pressure field very well.

2 The modified characteristics projection finite element method for the time-dependent conduction-convection problems

In this paper, we consider the time-dependent conduction-convection problem in two dimensions whose coupled equations governing viscous incompressible flow and heat transfer for the incompressible fluid are Boussinesq approximations to the Navier-Stokes equations. For all $t\in(0,t_{1}]$, find $(u,p,T)\in X\times M\times W$ such that

$$ \left \{ \textstyle\begin{array}{l@{\quad}l} u_{t}-\nu\Delta u+(u\cdot\nabla)u+\nabla p=\kappa\nu^{2}gT+f, & x\in \Omega, \\ \operatorname{div} u=0, & x\in\Omega, \\ T_{t}-\lambda\nu\Delta T+u\cdot\nabla T=b, & x\in\Omega, \\ u(x,0)=u_{0},\qquad T(x,0)=T_{0}, & x\in\Omega, \\ u=0, \qquad T=0, & x\in\partial\Omega, \end{array}\displaystyle \right . $$

(1)

where Ω is a bounded domain in $\mathbb{R}^{2}$ assumed to have a Lipschitz continuous boundary ∂Ω. $u=(u_{1}(x,t),u_{2}(x,t))^{T}$ represents the velocity vector, $p(x,t)$ represents the pressure, $T(x,t)$ represents the temperature, κ represents the Grashoff number, $\lambda=\mathit{Pr}^{-1}$, Pr is the Prandtl number, g represents the vector of gravitational acceleration, $\nu =\mathit{Re}^{-1}$, Re is the Reynolds number, and f and b are the forcing functions.

In this section, we aim to describe some notations and materials which will be frequently used in this paper. For the mathematical setting of the conduction-convection problems (1), we introduce the Hilbert spaces

$$\begin{aligned}& X=H_{0}^{1}(\Omega)^{2},\qquad W=H^{1}(\Omega), \\& M=L_{0}^{2}(\Omega)=\biggl\{ \varphi\in L^{2}( \Omega);\int_{\Omega}\varphi \, dx=0\biggr\} . \end{aligned}$$

$\Im_{h}$ is a quasi-uniform partition of $\bar{\Omega}_{h}$ into non-overlapping triangles, indexed by a parameter $h=\max_{K\in\Im _{h}}\{h_{K};h_{K}=\operatorname{diam}(K)\}$. We introduce the finite element subspace $X_{h}\subset X$, $M_{h}\subset M$, $W_{h}\subset W$ as follows:

$$\begin{aligned}& X_{h} =\bigl\{ v_{h}\in X\cap C^{0}(\bar{ \Omega})^{2};v_{h}|_{K}\in P_{\ell }(K)^{2}, \forall K\in\Im_{h}\bigr\} , \\& M_{h} =\bigl\{ q_{h}\in M\cap C^{0}(\bar{ \Omega});q_{h}|_{K}\in P_{k}(K),\forall K\in \Im_{h}\bigr\} , \\& W_{h} =\bigl\{ \phi_{h}\in W\cap C^{0}(\bar{ \Omega});\phi_{h}|_{K}\in P_{j}(K),\forall K\in \Im_{h}\bigr\} , \end{aligned}$$

where $P_{\ell}(K)$ is the space of piecewise polynomials of degree ℓ on K, $\ell\geq1$, $k\geq1$, $j\geq1$ are three integers. $W_{0h}=W_{h}\cap H_{0}^{1}(\Omega)$, and assume that $(X_{h},M_{h})$ satisfies the discrete LBB condition, there exists $\beta>0$ such that

$$ \sup_{v_{h}\in X_{h}}\frac{d(v_{h},\varphi_{h})}{\Vert\nabla v_{h}\Vert _{0}}\geq\beta\Vert\varphi_{h} \Vert_{0},\quad \forall\varphi_{h}\in M_{h}, $$

where $d(v_{h},\varphi_{h})=-(\varphi_{h},\nabla\cdot v_{h})$. Let $V_{h}$ be the kernel of the discrete divergence operator,

$$ V_{h}= \bigl\{ v_{h}\in X_{h};(q_{h}, \nabla\cdot v_{h})=0,\forall q_{h}\in M_{h} \bigr\} . $$

For each positive integer N, let $\{\mathcal{J}_{n}: 1\leq n\leq N\}$ be a partition of $[0,t_{1}]$ into subintervals $\mathcal{J}_{n}=(t_{n-1},t_{n}]$, with $t_{n}=n\tau$, $\tau=T_{1}/N$. Set $u^{n}=u(\cdot,t_{n})$. The characteristic trace-back along the field $u^{n-1}$ of a point $x\in\Omega$ at time $t_{n}$ to $t_{n-1}$ is approximately

$$ \bar{x}(x,t_{n-1})=x-u^{n-1}\tau. $$

Consequently, the hyperbolic part in the first equation of (1) at time $t_{n}$ is approximated by

$$ u_{t}+u^{n-1}\cdot\nabla u^{n} \approx \frac{u^{n}-\bar {u}^{n-1}}{\triangle t}, $$

where

$$ \bar{u}^{n-1} =\left \{ \textstyle\begin{array}{l@{\quad}l} u^{n-1}(\bar{x}), & \bar{x}=x-u^{n-1}\tau\in\Omega, \\ 0, & \mbox{otherwise},\end{array}\displaystyle \right . $$

for any function w.

With the previous notations, we get the projection FEM for the time-dependent conduction-convection problem (1), which is a slight transmutation of the projection FEM [13, 19] for the time-dependent Navier-Stokes equations.

Algorithm 2.1

(Projection FEM)

Start $u_{h}^{0}$ as a solution of $(u_{h}^{0},v_{h})=(u_{0},v_{h})$ and $(T_{h}^{0},\psi_{h})=(T_{0},\psi_{h})$, $p_{h}^{0}=0$ for all $v_{h}\in V_{h}$, $\psi_{h}\in M_{h}$.

Step 1::

Find $\hat{u}_{h}^{n+1}\in X_{h}$ as the solution of

$$\begin{aligned}& \biggl( \frac{\hat{u}_{h}^{n+1}-u_{h}^{n}}{\tau},v_{h} \biggr) +B\bigl({u}_{h}^{n},\hat{u}_{h}^{n+1},v_{h}\bigr)+\nu\bigl(\nabla \hat{u}_{h}^{n+1},\nabla v_{h}\bigr) \\& \quad =\kappa \nu^{2}\bigl(gT_{h}^{n},v_{h}\bigr)+ \bigl(f(t_{n+1}),v_{h}\bigr), \quad \forall v_{h} \in X_{h}, \end{aligned}$$

where $B(u_{h},v_{h},w_{h})=\frac{1}{2}((u_{h}\cdot\nabla )v_{h},w_{h})-\frac{1}{2}((u_{h}\cdot\nabla)w_{h},v_{h})$.

Step 2::

Find ${u}_{h}^{n+1}\in V_{h}$, $p_{h}^{n+1}\in M_{h}$ as the solution of

$$ \begin{aligned} & \biggl( \frac{{u}_{h}^{n+1}-\hat{u}_{h}^{n+1}}{\tau},v_{h} \biggr) +d\bigl(p_{h}^{n+1},v_{h}\bigr) =0,\quad \forall v_{h}\in V_{h}, \\ &d\bigl(q_{h},u_{h}^{n+1}\bigr) =0,\quad \forall q_{h}\in M_{h}. \end{aligned} $$

(2)

Step 3::

Compute $T_{h}^{n+1} \in W_{h}$ as the solution of the linear elliptic equation

$$\begin{aligned}& \biggl(\frac{T_{h}^{n+1}-T_{h}^{n}}{\tau}, \psi_{h} \biggr)+\bar{B}\bigl(u_{h}^{n+1},T_{h}^{n+1}, \psi_{h}\bigr)+\lambda\nu\bigl(\nabla T_{h}^{n+1}, \nabla \psi_{h}\bigr) \\& \quad =\bigl(b(t_{n+1}),\psi_{h} \bigr),\quad \forall\psi_{h}\in W_{0h}, \end{aligned}$$

(3)

$\bar{B}(u_{h},T_{h},\psi_{h})=\frac{1}{2}((u_{h}\cdot\nabla)T_{h},\psi_{h})-\frac{1}{2}((u_{h}\cdot\nabla)\psi_{h},T_{h})$.

Remark 2.1

Denote by $P_{h}$ the orthogonal projector in $(L^{2}(\Omega))^{2}$ onto V. We can readily check that (2) is equivalent to [19]

$$ u_{h}^{n+1}=P_{h}\hat{u}_{h}^{n+1}. $$

(4)

The MC time discretization, combined with the projection finite element method, leads to the following MC projection finite element method.

Algorithm 2.2

(MC projection FEM)

Start with $u_{h}^{0}$ as a solution of $(u_{h}^{0},v_{h})=(u_{0},v_{h})$ for all $v_{h}\in V_{h}$.

Step 1::

Find $\hat{u}_{h}^{n+1}\in X_{h}$ as the solution of

$$\begin{aligned}& \biggl(\frac{\hat{u}_{h}^{n+1}-\dot{u}_{h}^{n}}{\tau},v_{h} \biggr)+\nu\bigl(\nabla \hat{u}_{h}^{n+1},\nabla v_{h}\bigr) \\& \quad = \kappa\nu^{2}\bigl(gT_{h}^{n},v_{h} \bigr)+\bigl(f(t_{n+1}),v_{h}\bigr), \quad \forall v_{h}\in V_{h}, \end{aligned}$$

(5)

where

$$ \dot{u}_{h}^{n} =\left \{ \textstyle\begin{array}{l@{\quad}l} u_{h}^{n}(\dot{x}), & \dot{x}=x-u^{n}_{h}\tau\in\Omega, \\ 0, & \mbox{otherwise}.\end{array}\displaystyle \right . $$

Step 2::

Find ${u}_{h}^{n+1}\in V_{h}$, $p_{h}^{n+1}\in M_{h}$ as the solution of

$$ \begin{aligned} & \biggl(\frac{{u}_{h}^{n+1}-\hat{u}_{h}^{n+1}}{\tau},v_{h} \biggr)+b\bigl(p_{h}^{n+1},v_{h}\bigr)=0,\quad \forall v_{h}\in V_{h}, \\ &b\bigl(q_{h},u_{h}^{n+1}\bigr)=0,\quad \forall q_{h}\in M_{h}. \end{aligned} $$

(6)

Step 3::

Compute $T_{h}^{n+1} \in W_{h}$, the solution of the linear elliptic equation

$$ \biggl(\frac{T_{h}^{n+1}-\dot{T}_{h}^{n}}{\tau}, \psi_{h} \biggr)+\lambda\nu \bigl(\nabla T_{h}^{n+1},\nabla\psi_{h}\bigr)= \bigl(b(t_{n+1}),\psi_{h}\bigr),\quad \forall\psi_{h}\in W_{0h}, $$

(7)

where

$$ \dot{T}_{h}^{n} =\left \{ \textstyle\begin{array}{l@{\quad}l} T_{h}^{n}(\dot{x}), & \dot{x}=x-u^{n}_{h}\tau\in\Omega, \\ 0, & \mbox{otherwise}.\end{array}\displaystyle \right . $$

Remark 2.2

Define $\dot{\mathcal{X}}_{x}^{n+1}(t)=x-(t_{n+1}-t)u_{h}^{n}$, $\forall t\in[t_{n-1},t_{n+1}]$, $2\leq l\leq N$. Since $X_{h}$ is a subset of $W^{1,\infty}(\Omega)$, under the condition $\tau\leq\frac{1}{2L_{n}}$, $L_{n}=\max_{1\leq i\leq n}\|u_{h}^{n}\|_{W^{1,\infty}}$ on the time step it is an easy matter to verify that this mapping has a positive Jacobian, since $u_{h}^{n}$ vanishes on ∂Ω; this mapping is one-to-one and this is a change of variables from Ω onto Ω. This yields for any positive function ϕ on Ω the estimate (please see [36] for details)

$$ \int_{\Omega}\phi\bigl(\dot{\mathcal{X}}_{h}^{n+1}(t) \bigr)\, dx\leq C \int_{\Omega}\phi(x)\, dx. $$

3 Stability analysis

Theorem 3.1

(Stability)

If $\tau\leq\frac{1}{2L_{n}}$, $L_{n}=\max_{1\leq i\leq n}{\|u_{h}^{i}\|_{W^{1,\infty}}}$, the MC projection FEM is stable in the sense that

$$\begin{aligned}& \bigl\Vert u_{h}^{N+1}\bigr\Vert _{0}^{2} +\bigl\Vert T_{h}^{N+1}\bigr\Vert _{0}^{2}+2 \nu\tau\sum_{n=1}^{N}\bigl\Vert \nabla{u} _{h}^{n+1}\bigr\Vert _{0}^{2}+\lambda \nu\tau\sum_{n=1}^{N}\bigl\Vert \nabla T_{h}^{n+1}\bigr\Vert _{0}^{2} \\& \quad \leq C\bigl\Vert u_{h}^{0}\bigr\Vert _{0}^{2}+C\bigl\Vert T_{h}^{0}\bigr\Vert _{0}^{2}+C\frac{\tau}{2\nu}\sum _{n=1}^{N}\bigl\Vert f(t_{n+1}) \bigr\Vert _{-1}^{2} +2C\tau\sum _{n=1}^{N}\bigl\Vert b(t_{n+1})\bigr\Vert _{-1}. \end{aligned}$$

Remark 3.1

We will prove the boundary of $\|u_{h}^{n}\|_{W^{1,\infty}}$ in the next section. Here, we use mathematical induction method.

Proof

Let $v_{h}={u}_{h}^{n+1}$ in (5), we obtain

$$ \biggl( \frac{\hat{u}_{h}^{n+1}-\dot{u}_{h}^{n}}{\tau},u_{h}^{n+1} \biggr) +\nu\bigl( \nabla\hat{u}_{h}^{n+1},\nabla u_{h}^{n+1} \bigr)=\kappa\nu ^{2}\bigl(gT_{h}^{n},u_{h}^{n+1} \bigr)+\bigl(f(t_{n+1}),u_{h}^{n+1}\bigr). $$

Using (6), we deduce

$$ \bigl(\hat{u}_{h}^{n+1},v_{h}\bigr)= \bigl(u_{h}^{n+1},v_{h}\bigr)+\tau d(p_{h},v_{h}),\quad \forall v_{h}\in X_{h}. $$

Noting $\nabla\cdot u_{h}^{n+1}=0$, we get

$$\begin{aligned}& \bigl(\hat{u}_{h}^{n+1},u_{h}^{n+1} \bigr)=\bigl(u_{h}^{n+1},u_{h}^{n+1} \bigr), \end{aligned}$$

(8)

$$\begin{aligned}& \bigl(\nabla\hat{u}_{h}^{n+1},\nabla u_{h}^{n+1} \bigr)=\bigl(\nabla u_{h}^{n+1},\nabla u_{h}^{n+1} \bigr). \end{aligned}$$

(9)

Then we deduce

$$ \biggl( \frac{{u}_{h}^{n+1}-\dot{u}_{h}^{n}}{\tau},u_{h}^{n+1} \biggr) +\nu \bigl\Vert \nabla u_{h}^{n+1}\bigr\Vert _{0}^{2}= \kappa\nu ^{2}\bigl(gT_{h}^{n},u_{h}^{n+1} \bigr)+\bigl(f(t_{n+1}),u_{h}^{n+1}\bigr). $$

We arrive at

$$\begin{aligned}& \bigl\Vert u_{h}^{n+1}\bigr\Vert _{0}^{2}-\bigl\Vert u_{h}^{n}\bigr\Vert _{0}^{2}+2\nu\tau \bigl\Vert \nabla \hat{u}_{h}^{n+1}\bigr\Vert _{0}^{2} \\& \quad \leq\bigl\Vert \dot {u}_{h}^{n}\bigr\Vert _{0}^{2}-\bigl\Vert u_{h}^{n}\bigr\Vert _{0}^{2}+2\kappa\nu^{2}\tau \bigl(gT_{h}^{n},\hat{u}_{h}^{n+1} \bigr)+2\tau\bigl(f(t_{n+1}),\hat{u}_{h}^{n+1}\bigr). \end{aligned}$$

(10)

Now, we estimate the bound of $\Vert\dot{u}_{h}^{n}\Vert_{0}^{2}-\Vert u_{h}^{n}\Vert_{0}^{2}$. By the definition of $\dot{\mathcal{X}}_{x}^{n}(t_{n-1})$, we have

$$ J\bigl(\dot{\mathcal{X}}_{x}^{n}(t_{n-1})\bigr)= \left ( \textstyle\begin{array}{@{}c@{\quad}c@{}} 1-\partial_{x}u_{h1}^{n-1}\tau& -\partial_{y}u_{h1}^{n-1}\tau \\ -\partial_{x}u_{h2}^{n-1}\tau& 1-\partial_{y}u_{h2}^{n-1}\tau \end{array}\displaystyle \right ) . $$

Hence,

$$ \det J\bigl(\dot{\mathcal{X}}_{x}^{n}(t_{n-1}) \bigr)=1+\mathcal{O}(\tau). $$

Then we get

$$\begin{aligned} \Vert\dot{u}_{h}^{n}\Vert_{0}^{2}- \Vert{u}_{h}^{n}\Vert_{0}^{2}& =\int _{\Omega}\bigl(\dot{u}_{h}^{n} \bigr)^{2}\, dx-\int_{\Omega}\bigl(u_{h}^{n} \bigr)^{2}\, dx \\ & =\int_{\Omega}\bigl(u_{h}^{n} \bigr)^{2}\bigl(1+\mathcal{O}(\tau)\bigr)\, dx-\int_{\Omega } \bigl(u_{h}^{n}\bigr)^{2}\, dx. \end{aligned}$$

We have

$$ \bigl\Vert \dot{u}_{h}^{n}\bigr\Vert _{0}^{2}-\bigl\Vert {u}_{h}^{n}\bigr\Vert _{0}^{2}\leq C\tau\bigl\Vert u_{h}^{n} \bigr\Vert _{0}^{2}. $$

(11)

On the other hand, by Cauchy-Schwarz inequality, we deduce

$$\begin{aligned} 2\kappa\nu^{2}\tau\bigl(gT_{h}^{n},u_{h}^{n+1} \bigr)& \leq2C\kappa\nu ^{2}\tau \bigl\Vert T_{h}^{n} \bigr\Vert _{0}\bigl\Vert {u}_{h}^{n+1}\bigr\Vert _{0} \\ & \leq C\kappa^{2}\nu^{4}\tau\bigl\Vert T_{h}^{n}\bigr\Vert _{0}^{2}+C\tau \bigl\Vert u_{h}^{n+1}\bigr\Vert _{0}^{2}. \end{aligned}$$

(12)

Combining (10), (11), and (12), we get

$$\begin{aligned}& \bigl\Vert u_{h}^{n+1}\bigr\Vert _{0}^{2} -\bigl\Vert u_{h}^{n}\bigr\Vert _{0}^{2}+2\nu\tau\bigl\Vert \nabla{u}_{h}^{n+1} \bigr\Vert _{0}^{2} \\& \quad \leq C\tau\bigl\Vert u_{h}^{n}\bigr\Vert _{0}^{2}+\frac{\tau}{2\nu}\bigl\Vert f(t_{n+1}) \bigr\Vert _{-1}^{2} +\frac{ 3\nu\tau}{2}\bigl\Vert \nabla{u}_{h}^{n+1}\bigr\Vert _{0}^{2} \\& \qquad {} + C\kappa^{2}\nu^{4}\tau\bigl\Vert T_{h}^{n}\bigr\Vert _{0}^{2}+C\tau \bigl\Vert u_{h}^{n+1}\bigr\Vert _{0}^{2}. \end{aligned}$$

(13)

Let $\psi_{h}=2\tau T_{h}^{n+1}$ in (7), we obtain

$$ 2 \bigl(T_{h}^{n+1}-\dot{T}_{h}^{n}, T_{h}^{n+1} \bigr)+2\tau\lambda\nu\bigl(\nabla T_{h}^{n+1},\nabla T_{h}^{n+1}\bigr)=2 \tau\bigl(b(t_{n+1}),T_{h}^{n+1}\bigr). $$

(14)

We deduce

$$ \bigl\Vert T_{h}^{n+1}\bigr\Vert _{0}^{2}- \bigl\Vert {T}_{h}^{n}\bigr\Vert _{0}^{2}+ \tau\lambda\nu\bigl\Vert \nabla T_{h}^{n+1}\bigr\Vert _{0}^{2}\leq2\tau\bigl\Vert b(t_{n+1})\bigr\Vert _{-1}+\bigl\Vert \dot{T}_{h}^{n}\bigr\Vert _{0}^{2}-\bigl\Vert T_{h}^{n}\bigr\Vert _{0}^{2}. $$

Similar to (11), we have

$$ \bigl\Vert \dot{T}_{h}^{n}\bigr\Vert _{0}^{2}-\bigl\Vert T_{h}^{n}\bigr\Vert _{0}^{2}\leq C\tau\bigl\Vert u_{h}^{n} \bigr\Vert _{0}^{2}. $$

Then we can get

$$ \bigl\Vert T_{h}^{n+1}\bigr\Vert _{0}^{2}-\bigl\Vert {T}_{h}^{n}\bigr\Vert _{0}^{2}+\tau\lambda\nu\bigl\Vert \nabla T_{h}^{n+1}\bigr\Vert _{0}^{2}\leq2\tau \bigl\Vert b(t_{n+1})\bigr\Vert _{-1}+C\tau\bigl\Vert u_{h}^{n}\bigr\Vert _{0}^{2}. $$

(15)

Adding (13) and (15), summing over all n from 0 to N, we can get

$$\begin{aligned}& \bigl\Vert u_{h}^{N+1}\bigr\Vert _{0}^{2}+ \bigl\Vert T_{h}^{N+1}\bigr\Vert _{0}^{2} +2\nu\tau\sum_{n=1}^{N}\bigl\Vert \nabla {u}_{h}^{n+1}\bigr\Vert _{0}^{2}+ \lambda\nu\tau\sum_{n=1}^{N}\bigl\Vert \nabla T_{h}^{n+1}\bigr\Vert _{0}^{2} \\ & \quad \leq \bigl\Vert u_{h}^{0}\bigr\Vert _{0}^{2}+\bigl\Vert T_{h}^{0}\bigr\Vert _{0}^{2}+ C\tau\sum_{n=1}^{N} \bigl\Vert u_{h}^{n+1}\bigr\Vert _{0}^{2} \\ & \qquad {} +\frac{\tau}{2\nu}\sum_{n=1}^{N} \bigl\Vert f(t_{n+1})\bigr\Vert _{-1}^{2} + C \kappa^{2}\nu^{4}\tau\sum_{n=1}^{N} \bigl\Vert T_{h}^{n}\bigr\Vert _{0}^{2}+2 \tau\sum_{n=1}^{N}\bigl\Vert b(t_{n+1})\bigr\Vert _{-1}. \end{aligned}$$

Using Gronwall lemma, we deduce

$$\begin{aligned}& \bigl\Vert u_{h}^{N+1}\bigr\Vert _{0}^{2} +\bigl\Vert T_{h}^{N+1}\bigr\Vert _{0}^{2}+2 \nu\tau\sum_{n=1}^{N}\bigl\Vert \nabla{u} _{h}^{n+1}\bigr\Vert _{0}^{2}+\lambda \nu\tau\sum_{n=1}^{N}\bigl\Vert \nabla T_{h}^{n+1}\bigr\Vert _{0}^{2} \\ & \quad \leq C\bigl\Vert u_{h}^{0}\bigr\Vert _{0}^{2}+C\bigl\Vert T_{h}^{0}\bigr\Vert _{0}^{2}+C\frac{\tau}{2\nu}\sum _{n=1}^{N}\bigl\Vert f(t_{n+1}) \bigr\Vert _{-1}^{2} +2C\tau\sum _{n=1}^{N}\bigl\Vert b(t_{n+1})\bigr\Vert _{-1}. \end{aligned}$$

 □

4 Error analysis

In order to get the error analysis, we give some lemmas first.

Lemma 4.1

[37, 38]

Let $e(x,n)= [\frac{u^{n}(x)-\bar{u}^{n-1}(x)}{\tau}- (\frac{\partial u}{\partial t}(x,t_{n})+u^{n}(x) \nabla u^{n}(x) ) ]$ and let $\tau>0$ be such that $u\in\mathscr{C}^{4}([\tau,T]; H^{3}(\Omega)^{2})$. For $t_{n}>\tau$, we have

$$ e(x,n)=-\tau \biggl(\frac{1}{2}\frac{d^{2}g_{x}^{n}}{dt^{2}}+ \frac{\partial u}{\partial t }\cdot\nabla u(x,t_{n}) \biggr)+O\bigl( \tau^{2}\bigr), $$

(16)

where $g_{x}^{n}(t)=u(x-(t_{n}-t)u^{n-1},t)$, $u^{n}(x)=u(x,t_{n})$.

Lemma 4.2

Let

$$ \zeta(x,n)= \biggl( \frac{T^{n}(x)-\dot{T}^{n-1}(x)}{\tau }-T_{t}(x,t)-u\cdot \nabla T \biggr) , $$

and let $\tau>0$ be such that $T\in\mathscr{C}^{4}([\tau ,T];H^{3}(\Omega))$. For $t_{n}>\tau$, we have

$$ \zeta(x,n)=-\tau \biggl( \frac{1}{2}\frac{d^{2}\gamma_{x}^{n}}{dt^{2}}+ \frac{\partial u}{\partial t} \cdot\nabla T(x,t_{n}) \biggr) +O\bigl(\tau^{2}\bigr), $$

where $\gamma_{x}^{n}(t)=T(x-(t_{n}-t)u_{h}^{n-1},t)$, $u^{n}(x)=u(x,t_{n})$.

Lemma 4.3

There exists $r_{h}:W\rightarrow W_{h}$; for all $\psi\in W$ we have

$$\begin{aligned}& \bigl(\nabla(\psi-r_{h}\psi),\phi_{h}\bigr)=0, \quad \forall\phi_{h}\in W_{h}, \end{aligned}$$

(17)

$$\begin{aligned}& \int_{\Omega}(\psi-r_{h}\psi)\, dx=0,\quad \|\nabla r_{h}\psi\|_{0}\leq\|\nabla\psi\|_{0}. \end{aligned}$$

(18)

When $\psi\in W^{r,q}(\Omega)$ ($1\leq q\leq\infty$), we have

$$ \|\psi-r_{h}\psi\|_{-s,q}\leq Ch^{r+s}|\psi|_{r,q},\quad -1\leq s\leq m,0\leq r\leq m+1. $$

(19)

There exists $\bar{r}_{h}:W_{0}\rightarrow W_{0h}$; for all $\psi\in W_{0}$ we have

$$ \bigl(\nabla(\psi-\bar{r}_{h}\psi),\phi_{h} \bigr)=0,\quad \forall\phi_{h}\in W_{0h}, \| \nabla \bar{r}_{h}\psi\|_{0}\leq\|\nabla\psi\|_{0}. $$

(20)

When $\psi\in W^{r,q}(\Omega)$ ($1\leq q\leq\infty$), we have

$$ \|\psi-\bar{r}_{h}\psi\|_{-s,q}\leq Ch^{r+s}|\psi|_{r,q}, \quad -1\leq s\leq m,0\leq r\leq m+1. $$

(21)

Then we define the Galerkin projection $(R_{h},Q_{h})=(R_{h}(u,p),Q_{h}(u,p)):(X,M)\rightarrow(X_{h}, M_{h})$, such that

$$\begin{aligned}& \nu a(R_{h}-u,v_{h})-d(Q_{h}-p,v_{h})+d( \varphi_{h},R_{h}-u)=0, \\& \quad \forall(u,p)\in(X,M),(v_{h},\varphi_{h}) \in(X_{h},M_{h}). \end{aligned}$$

(22)

Lemma 4.4

[39, 40]

The Galerkin projection $(R_{h},Q_{h})$ satisfies

$$\begin{aligned}& \|R_{h}-u\|_{0}+h\bigl(\bigl\Vert \nabla(R_{h}-u) \bigr\Vert _{0}+\|Q_{h}-p\|_{0}\bigr) \leq Ch^{k+1}\bigl(\nu \Vert u\Vert _{k+1}+\|p\|_{k} \bigr), \\& \quad k=1,2. \end{aligned}$$

(23)

4.1 Error estimate for velocity and temperature

Lemma 4.5

If $\tau\leq\frac{1}{2L_{n}}$, $L_{n}=\max_{1\leq i\leq n}{\Vert u_{h}^{i}\Vert_{W^{1,\infty}}}$, u, p, $u_{t}$, and $p_{t}$ are sufficiently smooth, we have

$$\begin{aligned}& \bigl\Vert u_{h}^{n+1}\bigr\Vert _{W^{1,\infty}}< + \infty, \\& \bigl\Vert {\xi}_{h}^{N+1}\bigr\Vert _{0}^{2}+\sum_{n=0}^{N} \bigl\Vert \hat{\xi}_{h}^{n+1}- \xi_{h}^{n}\bigr\Vert _{0}^{2}+ \frac{1}{2}\nu\tau \sum_{n=0}^{N}\bigl\Vert \nabla\hat{\xi}_{h}^{n+1}\bigr\Vert _{0}^{2}+\bigl\Vert \varepsilon_{h}^{N+1} \bigr\Vert _{0}^{2} \\& \quad {}+\sum_{n=0}^{N}\bigl\Vert \varepsilon_{h}^{n+1}-\xi_{h}^{n}\bigr\Vert _{0}^{2}+\lambda\nu\tau\sum _{n=0}^{N}\bigl\Vert \nabla\varepsilon _{h}^{n+1}\bigr\Vert _{0}^{2}\leq C \bigl(\tau^{2}+h^{2(k+1)}\bigr), \\& \bigl\Vert {\xi}_{h}^{N+1}\bigr\Vert _{0}^{2}+\sum_{n=0}^{N} \bigl\Vert \hat{\xi}_{h}^{n+1}- \xi_{h}^{n}\bigr\Vert _{0}^{2}+ \frac{1}{2}\nu\tau \sum_{n=0}^{N}\bigl\Vert \nabla{\xi}_{h}^{n+1}\bigr\Vert _{0}^{2}+ \bigl\Vert \varepsilon _{h}^{N+1}\bigr\Vert _{0}^{2} \\& \quad {}+\sum_{n=0}^{N}\bigl\Vert \varepsilon_{h}^{n+1}-\xi_{h}^{n}\bigr\Vert _{0}^{2}+\lambda\nu\tau\sum _{n=0}^{N}\bigl\Vert \nabla\varepsilon _{h}^{n+1}\bigr\Vert _{0}^{2}\leq C \bigl(\tau^{2}+h^{2(k+1)}\bigr), \end{aligned}$$

where $\hat{\xi}_{h}^{n}=\hat{u}_{h}^{n}-R_{h}^{n}$, ${\xi}_{h}^{n}={u}_{h}^{n}-R_{h}^{n}$, $\varepsilon_{h}^{n+1}=T_{h}^{n+1}-r_{h}T^{n+1}$, $R_{h}^{n}=R_{h}(u^{n},p^{n})$, C is a positive constant independent of τ and h.

Proof

Subtracting $(\frac{R_{h}^{n+1}-\dot{R}_{h}^{n}}{\tau},v_{h} )+\nu(\nabla R_{h}^{n+1},\nabla v_{h})$ from both sides of (5), we can get

$$\begin{aligned}& \biggl(\frac{(\hat{u}_{h}^{n+1}-R_{h}^{n+1})-(\dot{u}_{h}^{n}-\dot {R}_{h}^{n})}{\tau},v_{h} \biggr)+\nu \bigl(\nabla\bigl(\hat{u}_{h}^{n+1}-R_{h}^{n+1} \bigr),\nabla v_{h}\bigr) \\ & \quad =k\nu^{2}g\bigl(T_{h}^{n},v_{h} \bigr)+\bigl(f^{n},v_{h}\bigr)- \biggl(\frac{R_{h}^{n+1}-\dot {R}_{h}^{n}}{\tau},v_{h} \biggr)-\nu\bigl(\nabla R_{h}^{n+1},\nabla v_{h} \bigr). \end{aligned}$$

(24)

Defining $\eta^{n}=u^{n}-R_{h}^{n}$, we can get

$$\begin{aligned}& \biggl(\frac{\hat{\xi}_{h}^{n+1}-{\xi} _{h}^{n}}{\tau},v_{h} \biggr)+\nu \bigl(\nabla \hat{ \xi}_{h}^{n+1},\nabla v_{h}\bigr) \\ & \quad = - \biggl(\frac{u^{n+1}-\bar{u}^{n}}{\tau}-\nu\triangle u^{n+1}+\nabla p^{n+1}-k\nu^{2}gT^{n+1}-f^{n+1},v_{h} \biggr) \\ & \qquad {} + \biggl(\frac{\eta^{n+1}-\dot{\eta}^{n}}{\tau},v_{h} \biggr) + \biggl( \frac{\dot{u}^{n}-\bar{u}^{n}}{\tau },v_{h} \biggr) +\bigl(\nabla p^{n+1},v_{h} \bigr)+ \biggl(\frac{\dot{\xi}_{h}^{n}- \xi_{h}^{n}}{\tau},v_{h} \biggr) \\ & \qquad {} +\nu\bigl(\nabla\bigl(u^{n+1}-R_{h}^{n+1} \bigr),\nabla v_{h}\bigr)+k\nu^{2}g\bigl(T^{n+1}-T_{h}^{n},v_{h} \bigr) \end{aligned}$$

(25)

$$\begin{aligned}& \quad = - \biggl(\frac{u^{n+1}-\bar{u}^{n}}{\tau}-\nu\triangle u^{n+1}+\nabla p^{n+1}-k\nu^{2}gT^{n+1}-f^{n+1},v_{h} \biggr) \\ & \qquad {}+ \biggl(\frac{\dot{u}^{n}-\bar{u}^{n}}{\tau},v_{h} \biggr)+d \bigl(Q_{h}^{n+1}-p^{n+1}, v_{h}\bigr) + \biggl(\frac{\eta^{n+1}-\dot{\eta}^{n}}{\tau},v_{h} \biggr) \\ & \qquad {}+ \biggl(\frac{\dot{\xi}_{h}^{n}- \xi_{h}^{n}}{\tau},v_{h} \biggr)+\nu\bigl(\nabla \bigl(u^{n+1}-R_{h}^{n+1}\bigr),\nabla v_{h}\bigr)+k\nu^{2}g\bigl(T^{n+1}-T_{h}^{n},v_{h} \bigr) \\ & \quad = - \biggl(\frac{u^{n+1}-\bar{u}^{n}}{\tau}-\nu\triangle u^{n+1}+\nabla p^{n+1}-k\nu^{2}gT^{n+1}-f^{n+1},v_{h} \biggr) \\ & \qquad {}+ \biggl(\frac{\eta^{n+1}-\dot{\eta}^{n}}{\tau},v_{h} \biggr)+ \biggl( \frac{\dot{u}^{n}-\bar{u}^{n}}{\tau },v_{h} \biggr)+ \biggl(\frac{\dot{\xi }_{h}^{n}- \xi_{h}^{n}}{\tau},v_{h} \biggr) \\ & \qquad {}+k\nu^{2}g\bigl(T^{n+1}-T_{h}^{n},v_{h} \bigr). \end{aligned}$$

(26)

Let $v_{h}=2\tau\hat{\xi}_{h}^{n+1}$ in (26), we can get

$$\begin{aligned}& \bigl\Vert \hat{\xi}_{h}^{n+1}\bigr\Vert _{0}^{2}-\bigl\Vert \xi_{h}^{n}\bigr\Vert _{0}^{2}+\bigl\Vert \hat{\xi}_{h}^{n+1}- \xi _{h}^{n}\bigr\Vert _{0}^{2}+2 \nu\tau\bigl\Vert \nabla\hat{\xi}_{h}^{n+1}\bigr\Vert _{0}^{2} \\ & \quad = -2\tau \biggl(\frac{u^{n+1}-\bar{u}^{n}}{\tau}-\nu\triangle u^{n+1}+\nabla p^{n+1}-k\nu^{2}gT^{n+1}-f^{n+1},\hat{ \xi}_{h}^{n+1} \biggr) \\ & \qquad {} +2 \bigl({\eta^{n+1}-\dot{\eta}^{n}}, \hat{\xi}_{h}^{n+1} \bigr) +2 \bigl(\dot{u}^{n}- \bar {u}^{n},\hat{\xi}_{h}^{n+1} \bigr)+2 \bigl( \dot{\xi}_{h}^{n}- \xi_{h}^{n}, \hat{\xi}_{h}^{n+1} \bigr) \\ & \qquad {} +k\nu^{2}\tau g\bigl(T^{n+1}-T_{h}^{n}, \hat{\xi}_{h}^{n+1}\bigr) \\ & \quad \equiv \sum_{i=1}^{5} \mathcal{A}_{i}, \end{aligned}$$

(27)

where

$$\begin{aligned}& \mathcal{A}_{1} =-2\tau \biggl(\frac{u^{n+1}-\bar{u}^{n}}{\tau}-\nu \triangle u^{n+1}+\nabla p^{n+1}-k\nu^{2}gT^{n+1}-f^{n+1}-f^{n+1}, \hat{\xi}_{h}^{n+1} \biggr), \\& \mathcal{A}_{2} =2 \bigl(\dot{u}^{n}-\bar{u}^{n}, \hat{\xi}_{h}^{n+1} \bigr), \\& \mathcal{A}_{3} =2 \bigl({\eta^{n+1}-\dot{ \eta}^{n}},\hat{\xi }_{h}^{n+1} \bigr), \\& \mathcal{A}_{4} =2 \bigl(\dot{u}^{n}-\bar{u}^{n}, \hat{\xi}_{h}^{n+1} \bigr)+2 \bigl(\dot{\xi}_{h}^{n}- \xi_{h}^{n},\hat{\xi}_{h}^{n+1} \bigr), \\& \mathcal{A}_{5} =k\nu^{2}g\tau\bigl(T^{n+1}-T_{h}^{n}, \hat{\xi}_{h}^{n+1}\bigr). \end{aligned}$$

Now, we estimate each term $\mathcal{A}_{i}$, respectively. By Hölder inequality, we get

$$ \mathcal{A}_{1}\leq C\tau\bigl\Vert \bar{e}^{n+1}\bigr\Vert _{0}^{2}+\frac{\nu\tau}{8}\bigl\Vert \nabla \hat{ \xi}_{h}^{n+1}\bigr\Vert _{0}^{2}. $$

(28)

By the definition of ẋ and x̄, we can get

$$ \dot{x}(x,t_{n})-\bar{x}(x,t_{n})=\bigl(u_{h}^{n}-u^{n} \bigr)\tau. $$

Using Taylor’s formula, we obtain

$$\begin{aligned} \bigl\vert \dot{u}^{n}-\bar{u}^{n}\bigr\vert & =\bigl\vert u^{n}(\dot{x})-u^{n}(\bar{x})\bigr\vert \\ & \leq\tau\bigl\Vert \nabla u^{n}\bigr\Vert _{\infty}\bigl\vert u_{h}^{n}-u^{n}\bigr\vert \\ & \leq\tau\bigl\Vert \nabla u^{n}\bigr\Vert _{\infty }\bigl( \bigl\vert u_{h}^{n}-R_{h}^{n}\bigr\vert +\bigl\vert R_{h}^{n}-u^{n}\bigr\vert \bigr). \end{aligned}$$

Therefore, we have

$$\begin{aligned} \bigl\Vert \dot{u}^{n}-\bar{u}^{n}\bigr\Vert _{0}& \leq\tau\bigl\Vert \nabla u^{n}\bigr\Vert _{\infty}\bigl(\bigl\Vert u^{n}-R_{h}^{n} \bigr\Vert _{0}+\bigl\Vert R_{h}^{n}-u_{h}^{n} \bigr\Vert _{0}\bigr) \\ & \leq C\tau\bigl(h^{k+1}+\bigl\Vert \xi_{h}^{n} \bigr\Vert _{0}\bigr). \end{aligned}$$

Then we deduce

$$\begin{aligned} \mathcal{A}_{2}& \leq C\bigl\Vert \dot{u}^{n}- \bar{u}^{n}\bigr\Vert _{0}\bigl\Vert \nabla \hat{ \xi}_{h}^{n+1}\bigr\Vert _{0} \\ & \leq C\tau\bigl(h^{2(k+1)}+\bigl\Vert \xi_{h}^{n} \bigr\Vert _{0}^{2}\bigr)+\frac{\nu \tau}{8}\bigl\Vert \nabla\hat{\xi}_{h}^{n+1}\bigr\Vert _{0}^{2}. \end{aligned}$$

(29)

Now, we estimate the boundedness of $\mathcal{A}_{3}$. We have

$$\begin{aligned} \bigl\Vert \eta^{n+1}-\eta^{n}\bigr\Vert _{0}& = \biggl( \int_{\Omega}\bigl(\eta ^{n+1}-\eta^{n}\bigr)^{2}\, dx \biggr) ^{\frac{1}{2}}= \biggl( \int_{\Omega }\biggl\vert \int _{t_{n}}^{t_{n+1}}\frac{\partial\eta}{\partial t}(x,\theta )\, d\theta \biggr\vert ^{2}\, dx \biggr) ^{\frac{1}{2}} \\ & \leq\sqrt{\tau} \biggl( \int_{\Omega}\int_{t_{n}}^{t_{n+1}} \biggl\vert \frac{\partial\eta}{\partial t}\biggr\vert ^{2}(x,\theta)\, d\theta \, dx \biggr) ^{\frac{1}{2}} \\ & \leq\sqrt{\tau}\biggl\Vert \frac{\partial\eta}{\partial t} \biggr\Vert _{L^{2}([t_{n},t_{n+1}]:L^{2}(\Omega))}. \end{aligned}$$

(30)

By the definition of $\hat{\mathcal{X}}_{x}^{n+1}(t_{n})$, we can get

$$ J\bigl(\hat{\mathcal{X}}_{x}^{n+1}(t_{n})\bigr)= \left ( \textstyle\begin{array}{@{}c@{\quad}c@{}} 1-\partial_{x}u_{h1}^{n-1}\tau& -\partial_{y}u_{h1}^{n-1}\tau \\ -\partial_{x}u_{h2}^{n-1}\tau& 1-\partial_{y}u_{h2}^{n-1}\tau \end{array}\displaystyle \right ) . $$

Hence,

$$ \det J\bigl(\hat{\mathcal{X}}_{x}^{n+1}(t_{n}) \bigr)=1+\mathcal{O}(\tau). $$

Then we get

$$\begin{aligned} \bigl\Vert \eta^{n}-\dot{\eta}^{n}\bigr\Vert _{-1} =& \sup_{v\in V} \bigl( \Vert \nabla v \Vert_{0}^{-1}\bigl(\eta^{n}-\hat{ \eta}^{n},v\bigr) \bigr) \\ =& \sup_{v\in V} \biggl[ \Vert\nabla v\Vert_{0}^{-1} \biggl( \int_{\Omega }\eta^{n}(x)v(x)\, dx-\int _{\Omega}{\eta}^{n}(z)v\bigl(\hat{\mathcal{X}}_{x}^{n}(t_{n})^{-1} \bigr) \bigl(1+\mathcal{O}\bigl(\triangle t^{2}\bigr)\bigr)\, dz \biggr) \biggr] \\ \leq& \sup_{v\in V} \biggl( \Vert\nabla v\Vert_{0}^{-1} \int_{\Omega }\eta ^{n-1}(x) \bigl(v(x)-v\bigl(\hat{ \mathcal{X}}_{x}^{n}(t_{n})^{-1}\bigr) \bigr)\, dx \biggr) \\ &{} +\sup_{v\in V} \biggl( C\tau^{2}\Vert\nabla v \Vert_{0}^{-1}\int_{\Omega}{\eta}^{n-1}(z)v\bigl(\hat{\mathcal{X}}_{x}^{n}(t_{n})^{-1} \bigr)\, dz \biggr) . \end{aligned}$$

Let $G(x)=x-\hat{\mathcal{X}}_{x}^{n+1}(t_{n})^{-1}$, then $|G(x)|\leq C\tau$, and

$$\begin{aligned} \bigl\Vert v(x)-v\bigl(\hat{\mathcal{X}}_{x}^{n+1}(t_{n})^{-1} \bigr)\bigr\Vert _{0}^{2}& \leq \int_{\Omega} \biggl( \int_{t_{n}}^{t_{n+1}}\frac{d}{dt}v\bigl( \hat{\mathcal {X}}_{x}^{n+1}(t)^{-1} \bigr)\, dt \biggr) ^{2}\, dx \\ & \leq C\tau^{2}\Vert\nabla v\Vert_{0}^{2}. \end{aligned}$$

Similarly, we have

$$ \bigl\Vert v\bigl(\hat{\mathcal{X}}_{x}^{n+1}(t_{n})^{-1} \bigr)\bigr\Vert \leq\Vert v\Vert _{0}^{2}(1+C\tau). $$

Then we deduce

$$ \bigl\Vert \eta^{n}-\dot{\eta}^{n}\bigr\Vert _{-1}\leq C\tau\bigl\Vert \eta^{n}\bigr\Vert _{0}. $$

(31)

By (30) and (31), we have

$$ \bigl\Vert \eta^{n+1}-\dot{\eta}^{n}\bigr\Vert _{-1}\leq C\tau h^{k+1}\bigl\Vert u_{h}^{n}\bigr\Vert _{\infty}+C{h^{k+1}} \sqrt{\tau}. $$

(32)

Therefore, we get

$$\begin{aligned} \mathcal{A}_{3}& \leq\bigl\Vert \eta^{n+1}-\dot{ \eta}^{n}\bigr\Vert _{-1}\bigl\Vert \nabla\hat{ \xi}_{h}^{n+1}\bigr\Vert _{0} \\ & \leq C\tau^{2}h^{2(k+1)}+C\tau{h^{2(k+1)}}+ \frac{\nu\tau}{16}\bigl\Vert \nabla\hat{\xi}_{h}^{n+1}\bigr\Vert _{0}^{2}. \end{aligned}$$

(33)

Similarly, we obtain

$$ \mathcal{A}_{4}\leq C\tau\bigl\Vert \xi_{h}^{n} \bigr\Vert _{0}^{2}+\frac{\nu \tau}{16}\bigl\Vert \nabla \hat{\xi}_{h}^{n+1}\bigr\Vert _{0}^{2}. $$

For the term $\mathcal{A}_{5}$, by Taylor’s formula, we can get $\Vert T^{n+1}-T^{n}\Vert_{0}\leq C\tau$, then

$$\begin{aligned} \mathcal{A}_{5}& =k\nu^{2}g\tau\bigl(T^{n+1}-T_{h}^{n}, \hat{\xi}_{h}^{n+1}\bigr) \\ & =k\nu^{2}g\tau\bigl(T^{n+1}-T^{n},\hat{ \xi}_{h}^{n+1}\bigr)+k\nu^{2}g\tau \bigl(T^{n}-T_{h}^{n},\hat{\xi}_{h}^{n+1} \bigr) \\ & \leq Ck^{2}\nu^{4}\tau^{3}+Ck^{2} \nu^{4}\tau\bigl\Vert T^{n}-T_{h}^{n} \bigr\Vert _{0}^{2}+\frac{\nu\tau}{8}\bigl\Vert \nabla \xi_{h}^{n+1}\bigr\Vert _{0}^{2}. \end{aligned}$$

(34)

Combining (27), (28), (20), and (34), we arrive at

$$\begin{aligned}& \bigl\Vert \hat{\xi}_{h}^{n+1}\bigr\Vert _{0}^{2}-\bigl\Vert \xi_{h}^{n}\bigr\Vert _{0}^{2}+\bigl\Vert \hat{\xi}_{h}^{n+1}- \xi_{h}^{n}\bigr\Vert _{0}^{2}+ \frac {1}{2}\nu \tau\bigl\Vert \nabla\hat{\xi}_{h}^{n+1} \bigr\Vert _{0}^{2}+\nu\tau \bigl( \bigl\Vert s_{h}^{n+1}\bigr\Vert _{0}^{2}-\bigl\Vert s_{h}^{n}\bigr\Vert _{0}^{2} \bigr) \\& \quad \leq C\tau\bigl(\tau^{2}+h^{2(k+1)}\bigr)+C\tau\bigl\Vert \xi_{h}^{n}\bigr\Vert _{0}^{2}+Ck^{2} \nu^{4}\tau\bigl\Vert T^{n}-T_{h}^{n} \bigr\Vert _{0}^{2}. \end{aligned}$$

(35)

Subtracting $\tau^{-1}(r_{h}T^{n+1}-r_{h}\dot{T}^{n},\psi _{h})+\lambda\nu (\nabla r_{h}T^{n+1},\nabla\psi_{h})$ from both sides of (7), we can get

$$\begin{aligned}& \biggl( \frac{(T_{h}^{n+1}-r_{h}T^{n+1})-(\dot{T}_{h}^{n}-r_{h}\dot {T}^{n})}{\tau},\psi_{h} \biggr) +\lambda\nu\bigl(\nabla \bigl(T_{h}^{n+1}-r_{h}T^{n+1}\bigr), \nabla\psi_{h}\bigr) \\& \quad = - \biggl( \frac{r_{h}T^{n+1}-r_{h}\dot{T}^{n}}{\tau},\psi_{h} \biggr) -\lambda\nu \bigl(\nabla r_{h}T^{n+1},\nabla\psi_{h}\bigr)+ \bigl(b(t_{n+1}),\nabla \psi _{h}\bigr) \\& \quad = \biggl( \frac{(T^{n+1}-r_{h}T^{n+1})-(\dot{T}^{n}-r_{h}\dot {T}^{n})}{\tau},\psi_{h} \biggr) + \lambda\nu\bigl(\nabla\bigl(T^{n+1}-r_{h}T^{n+1} \bigr),\nabla \psi _{h}\bigr) \\& \qquad {} +\bigl(\zeta(t_{n+1}),\psi_{h}\bigr). \end{aligned}$$

(36)

Letting $\dot{\varepsilon}_{h}^{n+1}=\dot{T}_{h}^{n+1}-r_{h}\dot {T}^{n+1}$, $\theta^{n+1}=T^{n+1}-r_{h}T^{n+1}$, $\dot{\theta}^{n}=\dot {T}^{n}-r_{h}\dot{T}^{n}$, and $\psi_{h}=2\tau\varepsilon_{h}^{n+1}$ in (36), we can get

$$\begin{aligned}& 2 \bigl( \varepsilon_{h}^{n+1}-\varepsilon_{h}^{n}, \varepsilon _{h}^{n+1} \bigr) +2\lambda\nu\tau\bigl(\nabla \varepsilon _{h}^{n+1},\nabla \varepsilon_{h}^{n+1} \bigr) \\& \quad =2 \bigl( \theta^{n+1}-\dot{\theta}^{n},\varepsilon _{h}^{n+1} \bigr) +2\bigl(\zeta(t_{n+1}), \varepsilon_{h}^{n+1}\bigr)+2 \bigl( \varepsilon_{h}^{n}- \dot{\varepsilon}_{h}^{n},\varepsilon _{h}^{n+1} \bigr). \end{aligned}$$

Similarly to (32), we get

$$\begin{aligned}& \bigl\Vert \varepsilon_{h}^{n}-\dot{ \varepsilon}_{h}^{n}\bigr\Vert _{0} \leq C\tau \bigl\Vert \varepsilon_{h}^{n}\bigr\Vert _{0}, \\& \bigl\Vert \theta^{n+1}-\dot{\theta}^{n}\bigr\Vert _{-1} \leq C\tau h^{k+1}\bigl\Vert u_{h}^{n} \bigr\Vert _{\infty}+C{h^{k+1}}\sqrt{\tau}. \end{aligned}$$

Then we deduce

$$\begin{aligned}& \bigl\Vert \varepsilon_{h}^{n+1}\bigr\Vert _{0}^{2}-\bigl\Vert \varepsilon _{h}^{n} \bigr\Vert _{0}^{2}+\bigl\Vert \varepsilon_{h}^{n+1}- \varepsilon_{h}^{n}\bigr\Vert _{0}+2\lambda\nu \tau\bigl\Vert \nabla\varepsilon_{h}^{n+1}\bigr\Vert _{0}^{2} \\& \quad \leq C\tau h^{r+1}+C\tau^{3}+\lambda\nu\tau\bigl\Vert \nabla\varepsilon _{h}^{n+1}\bigr\Vert _{0}^{2}+C\tau\bigl\Vert \varepsilon_{h}^{n} \bigr\Vert _{0}^{2}. \end{aligned}$$

Namely,

$$\begin{aligned}& \bigl\Vert \varepsilon_{h}^{n+1}\bigr\Vert _{0}^{2}-\bigl\Vert \varepsilon _{h}^{n} \bigr\Vert _{0}^{2}+\bigl\Vert \varepsilon_{h}^{n+1}- \varepsilon_{h}^{n}\bigr\Vert _{0}+\lambda \nu \tau\bigl\Vert \nabla\varepsilon_{h}^{n+1}\bigr\Vert _{0}^{2} \\& \quad \leq C\tau h^{2(r+1)}+C\tau^{3}+C \tau\bigl\Vert \varepsilon_{h}^{n}\bigr\Vert _{0}^{2}. \end{aligned}$$

(37)

Adding (35) to (37), we get

$$\begin{aligned}& \bigl\Vert \xi_{h}^{n+1}\bigr\Vert _{0}^{2}- \bigl\Vert \xi_{h}^{n}\bigr\Vert _{0}^{2}+ \bigl\Vert \xi_{h}^{n+1}-\xi_{h}^{n} \bigr\Vert _{0}^{2}+\frac{1}{2}\nu\tau\bigl\Vert \nabla \hat{\xi}_{h}^{n+1}\bigr\Vert _{0}^{2} \\& \qquad {} +\bigl\Vert \varepsilon_{h}^{n+1}\bigr\Vert _{0}^{2}-\bigl\Vert \varepsilon_{h}^{n} \bigr\Vert _{0}^{2}+\bigl\Vert \varepsilon _{h}^{n+1}-\varepsilon_{h}^{n}\bigr\Vert _{0}+\lambda\nu\tau\bigl\Vert \nabla \varepsilon_{h}^{n+1} \bigr\Vert _{0}^{2} \\& \quad \leq C\tau h^{2(r+1)}+C\tau^{3}+C\tau\bigl\Vert \xi_{h}^{n}\bigr\Vert _{0}^{2}+C\tau \bigl\Vert \varepsilon_{h}^{n}\bigr\Vert _{0}^{2}. \end{aligned}$$

Summing over n from 0 to N gives

$$\begin{aligned}& \bigl\Vert \hat{\xi}_{h}^{N+1}\bigr\Vert _{0}^{2}+\sum_{n=0}^{N} \bigl\Vert \hat{\xi}_{h}^{n+1}- \xi_{h}^{n}\bigr\Vert _{0}^{2}+2\sum _{n=0}^{N}\bigl\Vert \nabla\rho _{h}^{n+1}\bigr\Vert _{0}^{2} \\& \qquad {} +\frac{1}{2}\nu\tau\sum_{n=0}^{N} \bigl\Vert \nabla\hat{\xi}_{h}^{n+1}\bigr\Vert _{0}^{2}+\bigl\Vert \varepsilon_{h}^{N+1} \bigr\Vert _{0}^{2} \\& \qquad {}+\sum_{n=0}^{N} \bigl\Vert \varepsilon_{h}^{n+1}-\varepsilon _{h}^{n}\bigr\Vert _{0}^{2} +\lambda\nu\tau\sum_{n=0}^{N} \bigl\Vert \nabla\varepsilon _{h}^{n+1}\bigr\Vert _{0}^{2} \\& \quad \leq C\bigl(\tau^{2}+h^{2(k+1)} \bigr)+C\tau\sum_{n=0}^{N} \bigl( \bigl\Vert \varepsilon_{h}^{n}\bigr\Vert _{0}^{2}+ \bigl\Vert \xi_{h}^{n}\bigr\Vert _{0}^{2} \bigr) . \end{aligned}$$

By Gronwall lemma, we obtain

$$\begin{aligned}& \bigl\Vert \hat{\xi}_{h}^{N+1}\bigr\Vert _{0}^{2}+\sum_{n=0}^{N} \bigl\Vert \hat{\xi}_{h}^{n+1}- \xi_{h}^{n}\bigr\Vert _{0}^{2}+ \frac{1}{2}\nu\tau \sum_{n=0}^{N}\bigl\Vert \nabla\hat{\xi}_{h}^{n+1}\bigr\Vert _{0}^{2}+\bigl\Vert \varepsilon_{h}^{N+1} \bigr\Vert _{0}^{2} \\& \quad {} +\sum_{n=0}^{N}\bigl\Vert \varepsilon_{h}^{n+1}-\xi_{h}^{n}\bigr\Vert _{0}^{2}+\lambda\nu\tau\sum _{n=0}^{N}\bigl\Vert \nabla\varepsilon _{h}^{n+1}\bigr\Vert _{0}^{2} \leq C \bigl(\tau^{2}+h^{2(k+1)}\bigr). \end{aligned}$$

Using (6), we get

$$ \biggl( \frac{\xi_{h}^{n+1}-\hat{\xi}_{h}^{n+1}}{\tau},v_{h} \biggr) -b\bigl(p_{h}^{n+1},v_{h} \bigr)=0. $$

Let $v_{h}=2\tau{\xi}_{h}^{n+1}$, we can get

$$ \bigl\Vert {\xi}_{h}^{n+1}\bigr\Vert _{0}^{2}-\bigl\Vert \hat{\xi}_{h}^{n+1} \bigr\Vert _{0}^{2}+\bigl\Vert {\xi}_{h}^{n+1}- \hat{\xi}_{h}^{n+1}\bigr\Vert _{0}^{2}=0. $$

Then we have

$$\begin{aligned}& \bigl\Vert {\xi}_{h}^{N+1}\bigr\Vert _{0}^{2}+\sum_{n=0}^{N} \bigl\Vert \hat{\xi}_{h}^{n+1}- \xi_{h}^{n}\bigr\Vert _{0}^{2}+ \frac{1}{2}\nu\tau \sum_{n=0}^{N}\bigl\Vert \nabla\hat{\xi}_{h}^{n+1}\bigr\Vert _{0}^{2}+\bigl\Vert \varepsilon_{h}^{N+1} \bigr\Vert _{0}^{2} \\& \quad {} +\sum_{n=0}^{N}\bigl\Vert \varepsilon_{h}^{n+1}-\xi_{h}^{n}\bigr\Vert _{0}^{2}+\lambda\nu\tau\sum _{n=0}^{N}\bigl\Vert \nabla\varepsilon _{h}^{n+1}\bigr\Vert _{0}^{2} \leq C \bigl(\tau^{2}+h^{2(k+1)}\bigr). \end{aligned}$$

Using the inequality (see [41], Remark 1.6 and [19]),

$$ \Vert\nabla P_{h}u\Vert_{0}\leq C\Vert\nabla u \Vert_{0} $$

we have

$$\begin{aligned}& \bigl\Vert {\xi}_{h}^{N+1}\bigr\Vert _{0}^{2}+\sum_{n=0}^{N} \bigl\Vert \hat{\xi}_{h}^{n+1}- \xi_{h}^{n}\bigr\Vert _{0}^{2}+ \frac{1}{2}\nu\tau \sum_{n=0}^{N}\bigl\Vert \nabla{\xi}_{h}^{n+1}\bigr\Vert _{0}^{2}+ \bigl\Vert \varepsilon _{h}^{N+1}\bigr\Vert _{0}^{2} \\& \quad {} +\sum_{n=0}^{N}\bigl\Vert \varepsilon_{h}^{n+1}-\xi_{h}^{n}\bigr\Vert _{0}^{2}+\lambda\nu\tau\sum _{n=0}^{N}\bigl\Vert \nabla\varepsilon _{h}^{n+1}\bigr\Vert _{0}^{2} \leq C \bigl(\tau^{2}+h^{2(k+1)}\bigr). \end{aligned}$$

(38)

Using the triangle inequality, we deduce

$$ \bigl\Vert u_{h}^{n+1}\bigr\Vert _{W^{1,\infty}}\leq \bigl\Vert u_{h}^{n+1}-R_{h}^{n+1}\bigr\Vert _{W^{1,\infty}}+\bigl\Vert R_{h}^{n+1}\bigr\Vert _{W^{1,\infty}}. $$

Via the inverse inequality, $\Vert v_{h}\Vert_{W^{1,\infty}}\leq Ch^{-1}\Vert\nabla v_{h}\Vert_{0}$ (see [36]), we can get

$$ \bigl\Vert u_{h}^{n+1}\bigr\Vert _{W^{1,\infty}}\leq Ch^{-1}\bigl\Vert \nabla \bigl(u_{h}^{n+1}-R_{h}^{n+1} \bigr)\bigr\Vert _{0}+\bigl\Vert R_{h}^{n+1} \bigr\Vert _{\infty}. $$

We thus finish the proof. □

Theorem 4.6

(Error estimates for the velocity and temperature)

If $\tau\leq\frac{1}{2L_{n}}$, u, p, $u_{t}$, and $p_{t}$ are sufficiently smooth, we have

$$\begin{aligned}& \tau\sum_{n=0}^{N}\bigl\Vert u^{N+1}-u_{h}^{N+1}\bigr\Vert _{0}^{2} \leq C\bigl(\tau ^{2}+h^{2(k+1)}\bigr), \end{aligned}$$

(39)

$$\begin{aligned}& \frac{1}{2}\nu\tau\sum_{n=0}^{N} \bigl\Vert \nabla \bigl(u^{n+1}-u_{h}^{n+1}\bigr) \bigr\Vert _{0}^{2} \leq C\bigl(\tau^{2}+h^{2k} \bigr), \end{aligned}$$

(40)

$$\begin{aligned}& \tau\sum_{n=0}^{N}\bigl\Vert T^{N+1}-T_{h}^{N+1}\bigr\Vert _{0}^{2} \leq C\bigl(\tau ^{2}+h^{2(k+1)}\bigr), \end{aligned}$$

(41)

$$\begin{aligned}& \frac{1}{2}\nu\tau\sum_{n=0}^{N} \bigl\Vert \nabla \bigl(T^{n+1}-T_{h}^{n+1}\bigr) \bigr\Vert _{0}^{2} \leq C\bigl(\tau^{2}+h^{2k} \bigr). \end{aligned}$$

(42)

Proof

Using triangle inequality, (23), and Lemma 4.3, we can get this theorem. □

4.2 Error estimates for the pressure

The following theorem on the pressure is a consequence of the previous theorem on the velocity.

Theorem 4.7

(Error estimate for pressure)

If $\tau\leq\frac{1}{2L_{n}}$, u, p, $u_{t}$, and $p_{t}$ are sufficiently smooth, we have for all $1\leq n\leq N$,

$$ \bigl\Vert p^{n+1}-p_{h}^{n+1}\bigr\Vert _{0}\leq C\bigl(\tau+h^{k}\bigr). $$

Proof

By (6), we deduce

$$\begin{aligned}& \bigl(p^{n+1}-p_{h}^{n+1},\nabla v_{h} \bigr) \\& \quad = \biggl( \frac{u^{n+1}-\bar{u}^{n}}{\tau}-\nu\triangle u^{n+1}+\nabla p^{n+1}-k\nu ^{2}gT^{n+1}-f^{n+1},v_{h} \biggr)- \biggl( \frac{{\xi}_{h}^{n+1}-{\xi} _{h}^{n}}{\tau},v_{h} \biggr) \\& \qquad {} -\nu \bigl(\nabla\hat{\xi}_{h}^{n+1},\nabla v_{h}\bigr) + \biggl( \frac{\eta^{n+1}-\dot {\eta}^{n}}{\tau},v_{h} \biggr) \\& \qquad {} + \biggl( \frac{\dot{u}^{n}-\bar{u}^{n}}{\tau},v_{h} \biggr) + \biggl( \frac{\dot {\xi}_{h}^{n}-\xi_{h}^{n}}{\tau},v_{h} \biggr) \\& \qquad {} +\nu\bigl(\nabla \bigl(u^{n+1}-R_{h}^{n+1} \bigr),\nabla v_{h}\bigr)+k\nu^{2}g\bigl(T^{n+1}-T_{h}^{n},v_{h} \bigr). \end{aligned}$$

By the LBB condition and Cauchy-Schwarz inequality, we get

$$\begin{aligned}& \bigl\Vert p^{n+1}-p_{h}^{n+1}\bigr\Vert _{0} \\& \quad \leq \biggl\Vert \frac{u^{n+1}-\bar{u}^{n}}{\tau}-\nu\triangle u^{n+1}+\nabla p^{n+1}-k\nu^{2}gT^{n+1}-f^{n+1}\biggr\Vert _{0} \\& \qquad {} +C\tau^{-1}\bigl\Vert \xi_{h}^{n+1}- \xi_{h}^{n}\bigr\Vert _{0}+\nu\bigl\Vert \hat{ \xi}_{h}^{n+1}\bigr\Vert _{0}+C\biggl\Vert \frac{\dot{u}^{n}-\bar {u}^{n}}{\tau}\biggr\Vert _{0} \\& \qquad {} +C\biggl\Vert \frac{\eta^{n+1}-\dot{\eta}^{n}}{\tau}\biggr\Vert _{0}+C \biggl\Vert \frac{\dot{\xi}_{h}^{n}-\xi_{h}^{n}}{\tau}\biggr\Vert _{0}-\nu\bigl\Vert \nabla\bigl(u^{n+1}-R_{h}^{n+1}\bigr)\bigr\Vert _{0} \\& \qquad {} +k\nu^{2}g\bigl\Vert T^{n+1}-T_{h}^{n} \bigr\Vert _{0}. \end{aligned}$$

Using (27), (28), (20), and (34), we arrive at

$$ \bigl\Vert p^{n+1}-p_{h}^{n+1}\bigr\Vert _{0}\leq C\bigl(\tau+h^{k}\bigr). $$

Thus, we finish the proof. □

5 Numerical experiments

In order to show the effect of our method, we give some numerical results in this section.

5.1 Bénard convection problem

The first experiments is Bénard convection problem in the domain $\Omega =[0,5]\times[0,1]$ with the forcing $f=0$ and $b=0$. Figure 1 displays the initial and boundary conditions for velocity u and temperature T. It means that the boundary conditions for the velocity are the no-slip boundary condition $u=0$ on ∂Ω, thermal insulation $\partial_{\upsilon}T=0$ on the lateral boundaries, and a fixed temperature difference between top and bottom boundaries. Here, we choose $h=1/16$, $\tau=0.01$, and the finite element space is a Taylor-Hood finite element space. Here, we use the software package FreeFEM++ [42] for our program.

Figure 1

Physics model of Bénard convection problem.

First, we set $\kappa=10^{4}$, $\lambda=1.0$, $\nu=1.0$. Figure 2 gives the numerical temperature at $t=0.05, 0.1, 0.15,\mbox{and }1.0$. Figure 3 gives the numerical pressure at $t=0.05, 0.1, 0.15, \mbox{and }1.0$. Figure 4 gives the numerical streamline at $t=0.05, 0.1, 0.15, \mbox{and }1.0$.

Figure 2

Numerical temperatures of Bénard convection problem with $\pmb{\kappa=1\times10^{4}}$ at different times.

Figure 3

Numerical pressures of Bénard convection problem with $\pmb{\kappa=1\times10^{4}}$ at different times.

Figure 4

Numerical streamlines of Bénard convection problem with $\pmb{\kappa=1\times10^{4}}$ at different times.

Then we set $\kappa=10^{5}$, $\lambda=1.0$, $\nu=1.0$. Figure 5 gives the numerical temperature at $t=0.05, 0.1, 0.15, \mbox{and }1.0$. Figure 6 gives the numerical pressure at $t=0.05, 0.1, 0.15, \mbox{and }1$. Figure 7 gives the numerical streamline at $t=0.05, 0.1, 0.15, \mbox{and }1$. From the numerical results, we can see that MCPFEM can simulate the fluid field, temperature field and pressure field very well, and it works well for a high Grashoff number κ.

Figure 5

Numerical temperatures of Bénard convection problem with $\pmb{\kappa=1\times10^{5}}$ at different times.

Figure 6

Numerical pressures of Bénard convection problem with $\pmb{\kappa=1\times10^{5}}$ at different times.

Figure 7

Numerical streamline of Bénard convection problem with $\pmb{\kappa=1\times10^{5}}$ at different times.

5.2 Thermal driven cavity flow problem

Here, we consider the thermal driven flow in an enclosed square $\Omega=[0,1]^{2}$ with the forcing $f=0$ and $b=0$, and the initial and boundary conditions are given by Figure 8. It means that the boundary conditions for velocity is no-slip boundary condition $u=0$ on ∂Ω, and thermal insulation $\partial_{\upsilon}T=0$ on the top and bottom boundaries, and a fixed temperature difference between left and right boundaries. Here, we choose $h=1/32$, $\tau=10^{-4}$, and the finite element space is a Taylor-Hood finite element space.

Figure 8

Physics model of thermal driven cavity flow.

We choose $\lambda=1$, $\nu=1$, $\kappa=10^{5}\mbox{ and }10^{6}$ respectively. Figures 9 and 10 give the numerical results for $\kappa=10^{5}\mbox{ and }10^{6}$, respectively. From the numerical results, we can see that MCPFEM can simulate the fluid field, temperature field, and pressure field very well. The numerical experiments confirm our theoretical analysis and demonstrate the efficiency of our method.

Figure 9

Numerical results of thermal driven cavity flow with $\pmb{\kappa=1\times10^{5}}$ at different time, left panels the numerical streamlines, middle panels the numerical pressures, and right panels the numerical temperatures.

Figure 10

Numerical results of thermal driven cavity flow with $\pmb{\kappa=1\times10^{6}}$ at different times, left panels the numerical streamlines, middle panels the numerical pressures, and right panels the numerical temperatures.