In this section, we shall first formulate the variational multiscale method, and then develop the numerical scheme for the considered problem (2.1).
The variational multiscale method
Our variational multiscale method is based on two elliptic projection operators,
-
\(\Pi_{\mu}^{1}:X\rightarrow R_{1}=\{v\in H_{0}^{1}(\Omega)^{d}:v|_{K}\in (P_{1})^{d},\forall K\in E^{\mu}(\Omega)\}\),
-
\(\Pi_{\mu}^{2}:W\rightarrow R_{2}=\{S\in H_{0}^{1}(\Omega):S|_{K}\in P_{1},\forall K\in E^{\mu}(\Omega)\}\),
which can be defined as follows (see [22]):
$$\begin{aligned}& \bigl(\nabla\Pi_{\mu}^{1}u,\nabla v\bigr)=(\nabla u,\nabla v), \quad \forall u\in X,v\in R_{1}, \\& \bigl(\nabla\Pi_{\mu}^{2}T,\nabla S\bigr)=(\nabla T,\nabla S), \quad \forall T\in W,S\in R_{2}, \end{aligned}$$
and have the following properties:
$$\begin{aligned} \bigl\| \nabla\Pi_{\mu}^{1}u\bigr\| _{0}\leq\| \nabla u\|_{0},\quad \forall u\in X, \qquad \bigl\| \nabla \Pi_{\mu}^{2}T\bigr\| _{0}\leq\|\nabla T\|_{0}, \quad \forall T\in W, \end{aligned}$$
(3.1)
where \(R_{1}\) and \(R_{2}\) are two spaces of polynomials of degree less than or equal to one.
Based on projections \(\Pi_{\mu}^{1}\) and \(\Pi_{\mu}^{2}\), we define the following stabilization terms:
$$\begin{aligned}& G_{1}(u,v)=\alpha_{1}\bigl(\nabla\bigl(I- \Pi_{\mu}^{1}\bigr)u,\nabla\bigl(I-\Pi_{\mu}^{1} \bigr)v\bigr), \quad \forall u,v\in X, \end{aligned}$$
(3.2)
$$\begin{aligned}& G_{2}(T,S)=\alpha_{2}\bigl(\nabla\bigl(I- \Pi_{\mu}^{2}\bigr)T,\nabla\bigl(I-\Pi_{\mu}^{2} \bigr)S\bigr), \quad \forall T,S\in W, \end{aligned}$$
(3.3)
where \(\alpha_{1},\alpha_{2}>0\) are two user-defined stabilization parameters, typically chosen as \(\alpha_{1}=\alpha_{2}=\mathcal{O}(\mu)^{s}\) (here \(s>0\) is a constant related to the finite elements used for the discretization of the considered problem).
Thanks to (3.2) and (3.3), the finite element variational multiscale method for problem (2.1) reads as follows: Find \((u_{\mu},p_{\mu},T_{\mu})\in X_{\mu}\times M_{\mu}\times W_{\mu }\) such that
$$\begin{aligned} \left \{ \textstyle\begin{array}{@{}l} \operatorname{Pr} a(u_{\mu},v_{\mu})+c(u_{\mu},u_{\mu},v_{\mu})+b(v_{\mu},p_{\mu })-b(u_{\mu},q_{\mu})+G_{1}(u_{\mu},v_{\mu})\\ \quad=\operatorname{Pr} \operatorname{Ra}(T_{\mu}\zeta,v_{\mu})+(\gamma_{1},v_{\mu}),\\ k \bar{a}(T_{\mu},S_{\mu})+\bar{c}(u_{\mu},T_{\mu},S_{\mu})+G_{2}(T_{\mu },S_{\mu})=(\gamma_{2},S_{\mu}), \end{array}\displaystyle \right . \end{aligned}$$
(3.4)
for all \((v_{\mu},q_{\mu},S_{\mu})\in X_{\mu}\times M_{\mu}\times W_{\mu}\).
Stability and convergence of scheme (3.4)
The system (3.4) is nonlinear; it needs to be linearized in computations. A popular linearization process is the one based on the Newton iterative method. However, it is well known that the Newton iterative method is sensitive to the initial guess for the nonlinear iterations, i.e., to guarantee the convergence of the Newton iterations, the initial guess should be close enough to the solution \((u_{\mu},T_{\mu})\). Here we use the Oseen iterative method to solve (3.4).
From the definitions (3.2) and (3.3) of the projection operators \(\Pi_{\mu}^{1}\), \(\Pi_{\mu}^{2}\), we have
$$\begin{aligned}& \begin{aligned}[b] G_{1}(u,v)&=\alpha_{1}\bigl(\nabla\bigl(I- \Pi_{\mu}^{1}\bigr)u,\nabla\bigl(I-\Pi_{\mu}^{1} \bigr)v\bigr)\\ &=\alpha _{1}(\nabla u,\nabla v)-\alpha_{1}\bigl( \nabla\Pi_{\mu}^{1}u,\nabla v\bigr) , \quad \forall u,v\in X, \end{aligned} \end{aligned}$$
(3.5)
$$\begin{aligned}& \begin{aligned}[b] G_{2}(T,S)&=\alpha_{2}\bigl(\nabla\bigl(I- \Pi_{\mu}^{2}\bigr)T,\nabla\bigl(I-\Pi_{\mu}^{2} \bigr)S\bigr)\\ &=\alpha _{2}(\nabla T,\nabla S)-\alpha_{2}\bigl( \nabla\Pi_{\mu}^{2}T,\nabla S\bigr), \quad \forall T,S\in W. \end{aligned} \end{aligned}$$
(3.6)
By applying the Oseen iterative method to the nonlinear problem (3.4) and thanks to (3.5) and (3.6), we develop the variational multiscale Oseen iterative scheme for the natural convection problem: Find an iterative solution \((u_{\mu}^{n},p_{\mu}^{n},T_{\mu}^{n})\in X_{\mu }\times M_{\mu}\times W_{\mu}\) such that
$$\begin{aligned} \left \{ \textstyle\begin{array}{@{}l} (\operatorname{Pr}+\alpha_{1}) a(u_{\mu}^{n},v_{\mu})+c(u_{\mu}^{n-1},u_{\mu}^{n},v_{\mu })+b(v_{\mu},p_{\mu}^{n})-b(u_{\mu}^{n},q_{\mu}) \\ \quad=\operatorname{Pr} \operatorname{Ra}(T^{n}_{\mu}\zeta,v_{\mu})+(\gamma_{1},v_{\mu})+\alpha_{1}(\nabla\Pi _{\mu}^{1} u_{\mu}^{n-1},\nabla v_{\mu}),\\ (k+\alpha_{2}) \bar{a}(T_{\mu}^{n},S_{\mu})+\bar{c}(u_{\mu}^{n-1},T_{\mu }^{n},S_{\mu})=(\gamma_{2},S_{\mu})+\alpha_{2}(\nabla\Pi_{\mu}^{2} T_{\mu }^{n-1},\nabla S_{\mu}), \end{array}\displaystyle \right . \end{aligned}$$
(3.7)
for all \((v_{\mu},q_{\mu},S_{\mu})\in X_{\mu}\times M_{\mu}\times W_{\mu }\), \(n=1,2,\ldots\) , with \(u_{\mu}^{0}=0\), \(T_{\mu}^{0}=0\).
Throughout this paper, we assume that μ is small enough such that the iterative scheme (3.7) is convergent. In order to keep notation brief, we define
$$\begin{aligned} A=\operatorname{Pr}^{-1}\|\gamma_{1}\|_{-1}+ \operatorname{Ra} k^{-1}\|\gamma_{2}\|_{-1}, \qquad B=k^{-1}\|\gamma_{2}\|_{-1}. \end{aligned}$$
Theorem 3.1
The iterative solution
\((u_{\mu}^{n}, T_{\mu}^{n})\)
defined by (3.7) satisfies
$$\begin{aligned} \bigl\| \nabla u_{\mu}^{n}\bigr\| _{0}\leq A, \qquad \bigl\| \nabla T_{\mu}^{n}\bigr\| _{0}\leq B, \quad \forall n\geq1. \end{aligned}$$
(3.8)
Proof
Let \(S_{\mu}=T_{\mu}^{n}\) in the second equation of (3.7), we have
$$\begin{aligned} (k+\alpha_{2}) \bar{a}\bigl(T_{\mu}^{n},T_{\mu}^{n} \bigr)=\bigl(\gamma_{2},T_{\mu}^{n}\bigr)+\alpha _{2}\bigl(\nabla\Pi_{\mu}^{2} T_{\mu}^{n-1}, \nabla T_{\mu}^{n}\bigr). \end{aligned}$$
We use the Cauchy-Schwarz inequality and (3.1) to obtain
$$\begin{aligned} (k+\alpha_{2}) \bigl\| \nabla T_{\mu}^{n}\bigr\| _{0} \leq\|\gamma_{2}\|_{-1}+\alpha_{2}\bigl\| \nabla \Pi_{\mu}^{2}T_{\mu}^{n-1}\bigr\| _{0} \leq\|\gamma_{2}\|_{-1}+\alpha_{2}\bigl\| \nabla T_{\mu}^{n-1}\bigr\| _{0}. \end{aligned}$$
When \(n=1\), we get
$$\begin{aligned} (k+\alpha_{2}) \bigl\| \nabla T_{\mu}^{1}\bigr\| _{0} \leq\|\gamma_{2}\|_{-1}\quad\Rightarrow\quad \bigl\| \nabla T_{\mu}^{1}\bigr\| _{0}\leq\frac{1}{k+\alpha_{2}}\| \gamma_{2}\|_{-1}\leq k^{-1}\|\gamma_{2} \|_{-1}, \end{aligned}$$
which shows that the second inequality of (3.8) holds for \(n=1\). We now assume it holds for \(n=J\) and prove it holds for \(n=J+1\),
$$ (k+\alpha_{2}) \bigl\| \nabla T_{\mu}^{J+1} \bigr\| _{0} \leq\|\gamma_{2}\|_{-1}+\alpha_{2} \bigl\| \nabla T_{\mu}^{J}\bigr\| _{0} \leq k^{-1}(k+\alpha_{2})\|\gamma_{2}\|_{-1}. $$
As a consequence one finds
$$\begin{aligned} \bigl\| \nabla T_{\mu}^{J+1}\bigr\| _{0}\leq k^{-1}\|\gamma_{2}\|_{-1}. \end{aligned}$$
(3.9)
Taking \(v_{\mu}=u_{\mu}^{n}\) and \(q_{\mu}=p_{\mu}^{n}\) in the first equation of (3.7), and using (3.1), we obtain
$$\begin{aligned}[b] (\operatorname{Pr}+\alpha_{1})\bigl\| \nabla u_{\mu}^{n} \bigr\| _{0}&\leq\|\gamma_{1}\|_{-1}+\alpha_{1} \bigl\| \nabla\Pi_{\mu}^{1} u_{\mu}^{n-1} \bigr\| _{0}+\operatorname{Pr} \operatorname{Ra}\bigl\| T_{\mu}^{n} \bigr\| _{-1} \\ &\leq\|\gamma_{1}\|_{-1} +\alpha_{1}\bigl\| \nabla u_{\mu}^{n-1}\bigr\| _{0}+ \operatorname{Pr} \operatorname{Ra} k^{-1}\|\gamma_{2}\|_{-1}. \end{aligned} $$
Due to the facts that \(\alpha_{1}>0\) and \(u^{0}_{\mu}=0\), we know that (3.8) holds for \(n=1\). Assume that the first inequality of (3.8) holds for \(n=J\). Taking \((v_{\mu},q_{\mu})=(u_{\mu}^{J+1},p_{\mu}^{J+1})\in X_{\mu }\times M_{\mu}\) in the first equation of (3.7) with \(n=J+1\), we get
$$\begin{aligned} (\operatorname{Pr}+\alpha_{1})\bigl\| \nabla u_{\mu}^{J+1} \bigr\| _{0}&\leq\|\gamma_{1}\|_{-1}+\alpha _{1} \bigl\| \nabla u_{\mu}^{J}\bigr\| _{0}+ \operatorname{Pr} \operatorname{Ra} k^{-1}\|\gamma_{2}\|_{-1} \\ &\leq\|\gamma_{1}\|_{-1}+ \alpha_{1}\bigl( \operatorname{Pr}^{-1}\|\gamma_{1}\|_{-1}+ \operatorname{Ra} k^{-1}\| \gamma_{2}\|_{-1}\bigr) + \operatorname{Pr} \operatorname{Ra} k^{-1}\|\gamma_{2} \|_{-1} \\ &\leq\frac{\operatorname{Pr}+\alpha_{1}}{\operatorname{Pr}}\|\gamma_{1}\|_{-1}+( \operatorname{Pr}+\alpha_{1})\operatorname{Ra} k^{-1}\| \gamma_{2}\|_{-1}. \end{aligned}$$
Consequently, we obtain
$$\begin{aligned} \bigl\| \nabla u_{\mu}^{J+1}\bigr\| _{0}\leq \operatorname{Pr}^{-1} \|\gamma_{1}\|_{-1}+ \operatorname{Ra} k^{-1}\| \gamma_{2}\|_{-1}, \end{aligned}$$
which with (3.9) completes the proof. □
Theorem 3.2
Assume that
\((u,T)\)
is the nonsingular solution to the natural convection problem (1.1), \(\alpha_{1}\)
and
\(\alpha_{2}\)
tend to zero as
μ
tends to zero. Under the condition of
$$\begin{aligned} \operatorname{Pr}+2\alpha_{1}-2N_{1}A-2(k+ \alpha_{2})^{-1}\cdot \operatorname{Pr} \operatorname{Ra} N_{2}B > 0, \end{aligned}$$
(3.10)
there exists
\(\mu_{0}>0\)
such that for all
\(\mu\leq\mu_{0}\), the solution
\((u_{\mu}^{n},p_{\mu}^{n},T_{\mu}^{n})\)
of (3.7) satisfies
$$\begin{aligned}& \begin{aligned}[b] &\frac{1}{2}\operatorname{Pr}\bigl\| \nabla \bigl(u-u_{\mu}^{n}\bigr)\bigr\| _{0} \\ &\quad\leq C(\beta) \biggl(\frac {3}{2}\operatorname{Pr}+ \alpha_{1}+2N_{1} A+\frac{\operatorname{Pr} \operatorname{Ra} N_{2} B}{k+\alpha_{2}}\biggr)\inf _{\bar {u}\in X_{\mu}} \bigl\| \nabla(u-\bar{u})\bigr\| _{0} \\ &\qquad{}+\sqrt{d}\inf_{\lambda_{\mu}\in M_{\mu}}\|p-\lambda_{\mu}\|_{0}+2\alpha_{1} A+\biggl(N_{1} A+ \frac{\operatorname{Pr} \operatorname{Ra} N_{2}B}{k+\alpha_{2}}\biggr) \bigl\| \nabla\bigl(u_{\mu}^{n}-u_{\mu }^{n-1} \bigr)\bigr\| _{0} \\ &\qquad{}+\frac{\operatorname{Pr} \operatorname{Ra}(2k+2\alpha_{2}+N_{2} A)}{k+\alpha_{2}}\inf_{\bar{T} \in W_{\mu}}{\bigl\| \nabla(T-\bar{T}) \bigr\| _{0}}+\frac{2\operatorname{Pr} \operatorname{Ra} \alpha_{2}B}{k+\alpha_{2}}, \end{aligned} \end{aligned}$$
(3.11)
$$\begin{aligned}& \begin{aligned}[b] \bigl\| p-p_{\mu}^{n} \bigr\| _{0} \leq{}& \biggl(\frac{\sqrt{d}}{\beta}+1\biggr) \inf_{\lambda_{\mu }\in M_{\mu}} \|p- \lambda_{\mu}\|_{0}+\frac{1}{\beta}(\operatorname{Pr}+ \alpha_{1}+2 N_{1} A)\bigl\| \nabla\bigl(u-u_{\mu}^{n} \bigr)\bigr\| _{0} \\ &{} +\frac{N_{1}A}{\beta}\bigl\| \nabla\bigl(u_{\mu}^{n}-u_{\mu}^{n-1} \bigr)\bigr\| _{0}+\frac {2\alpha_{1} A}{\beta}+\frac{\operatorname{Pr} \operatorname{Ra}}{\beta}\bigl\| \nabla \bigl(T-T_{\mu}^{n}\bigr)\bigr\| _{0}, \end{aligned} \end{aligned}$$
(3.12)
$$\begin{aligned}& \begin{aligned}[b] \bigl\| \nabla\bigl(T-T_{\mu}^{n}\bigr) \bigr\| _{0} \leq{}&\frac{2k+2\alpha_{2}+N_{2}A}{k+\alpha_{2}} \inf_{\bar{T} \in W_{\mu}} {\bigl\| \nabla(T-\bar{T})\bigr\| _{0}}+\frac{N_{2} B}{k+\alpha_{2}}\bigl\| \nabla\bigl(u-u_{\mu}^{n} \bigr)\bigr\| _{0} \\ &{}+\frac{N_{2}B}{k+\alpha_{2}}\bigl\| \nabla\bigl(u_{\mu}^{n}-u_{\mu}^{n-1} \bigr)\bigr\| _{0}+\frac {2\alpha_{2}B}{k+\alpha_{2}}. \end{aligned} \end{aligned}$$
(3.13)
Here
ū, \(\lambda_{\mu}\), and
T̄
are the approximations of
u, p, and
T
in
\(X_{\mu}\), \(M_{\mu}\), and
\(W_{\mu}\), respectively.
Remark 3.1
As the velocity u and temperature T are coupled in system (2.1), the error estimation for \(T-T_{\mu }^{n}\) is coupled with the error \(u-u_{\mu}^{n}\).
Proof
Setting \((e_{\mu}^{n},\eta_{\mu}^{n},\phi_{\mu}^{n})=(u-u_{\mu }^{n},p-p_{\mu}^{n},T-T_{\mu}^{n})\) and subtracting (3.7) from (2.1), we obtain the following error equations for the variational multiscale Oseen iterative scheme:
$$\begin{aligned} \left \{ \textstyle\begin{array}{@{}l} (\operatorname{Pr}+\alpha_{1})(\nabla e_{\mu}^{n},\nabla v_{\mu})+c(u,u,v_{\mu})-c(u_{\mu }^{n-1},u_{\mu}^{n},v_{\mu})+b(v_{\mu},\eta_{\mu}^{n})-b(e_{\mu}^{n},q_{\mu})\\ \quad=\operatorname{Pr} \operatorname{Ra}(\phi^{n}_{\mu}\zeta,v_{\mu}) +\alpha_{1}(\nabla u,\nabla v_{\mu })-\alpha_{1}(\nabla\Pi_{\mu}^{1} u_{\mu}^{n-1},\nabla v_{\mu}),\\ (k+\alpha_{2}) \bar{a}(\phi_{\mu}^{n},S_{\mu})+\bar{c}(u,T,S_{\mu})-\bar {c}(u_{\mu}^{n-1},T_{\mu}^{n},S_{\mu})\\ \quad=\alpha_{2}(\nabla T,\nabla S_{\mu})-\alpha_{2}(\nabla\Pi_{\mu}^{2} T_{\mu }^{n-1},\nabla S_{\mu}), \end{array}\displaystyle \right . \end{aligned}$$
(3.14)
for all \((v_{\mu},q_{\mu},S_{\mu})\in X_{\mu}\times M_{\mu}\times W_{\mu}\). Taking \(v_{\mu}\in V_{\mu}\) and \(q_{\mu}=0\), we get for any \(\lambda_{\mu }\in M_{\mu}\)
$$\begin{aligned} &(\operatorname{Pr}+\alpha_{1}) \bigl(\nabla e_{\mu}^{n},\nabla v_{\mu}\bigr)+ c(u,u,v_{\mu })-c\bigl(u_{\mu}^{n-1},u_{\mu}^{n},v_{\mu} \bigr)-(\nabla\cdot v_{\mu},p-\lambda _{\mu}) \\ &\quad=\operatorname{Pr} \operatorname{Ra}\bigl(\phi^{n}_{\mu} \zeta,v_{\mu}\bigr)+\alpha_{1}(\nabla u,\nabla v_{\mu })-\alpha_{1}\bigl(\nabla\Pi_{\mu}^{1} u_{\mu}^{n-1},\nabla v_{\mu}\bigr). \end{aligned}$$
(3.15)
Splitting \(e_{\mu}^{n}=\rho_{1}-\xi_{\mu}^{n}\) into \(\rho_{1}=u-\bar{u}\) and \(\xi_{\mu}^{n}=u_{\mu}^{n}-\bar{u}\), ū is an approximation of u in \(V_{\mu}\), splitting \(\phi_{\mu} ^{n}=\rho_{2}-\varepsilon_{\mu}^{n}\) with \(\rho_{2}=T-\bar {T}\) and \(\varepsilon_{\mu}^{n}=T_{\mu}^{n}-\bar{T}\), T̄ is an approximation of T in \(W_{\mu}\), noting that
$$\begin{aligned} &c(u,u,v_{\mu})-c\bigl(u_{\mu}^{n-1},u_{\mu}^{n},v_{\mu}\bigr) \\ &\quad=c\bigl(u,u-u_{\mu}^{n}+u_{\mu}^{n},v_{\mu}\bigr)-c\bigl(u_{\mu}^{n-1},u_{\mu}^{n},v_{\mu}\bigr) \\ &\quad=c\bigl(u,e_{\mu}^{n},v_{\mu}\bigr)+c \bigl(e_{\mu}^{n},u_{\mu}^{n},v_{\mu}\bigr)+c\bigl(u_{\mu}^{n}-u_{\mu}^{n-1},u_{\mu}^{n},v_{\mu}\bigr) \\ &\quad=c(u,\rho_{1},v_{\mu})-c\bigl(u,\xi_{\mu}^{n},v_{\mu}\bigr)+c\bigl(\rho_{1},u_{\mu}^{n},v_{\mu}\bigr) \\ &\qquad{}-c\bigl(\xi _{\mu}^{n},u_{\mu}^{n},v_{\mu}\bigr)+c\bigl(u_{\mu}^{n}-u_{\mu}^{n-1},u_{\mu}^{n},v_{\mu}\bigr). \end{aligned}$$
(3.16)
In the same way, we have
$$\begin{aligned} &\bar{c}(u,T,S_{\mu})-\bar{c}\bigl(u_{\mu}^{n-1},T_{\mu}^{n},S_{\mu}\bigr) \\ &\quad=\bar{c}(u,\rho_{2},S_{\mu})-\bar{c}\bigl(u, \varepsilon_{\mu}^{n},S_{\mu}\bigr)+\bar {c} \bigl(e_{\mu}^{n},T_{\mu}^{n},S_{\mu}\bigr)+\bar{c}\bigl(u_{\mu}^{n}-u_{\mu}^{n-1},T_{\mu}^{n},S_{\mu}\bigr). \end{aligned}$$
(3.17)
From the second equation of (3.14), one finds
$$\begin{aligned} (k+\alpha_{2}) \bigl(\nabla\varepsilon_{\mu}^{n}, \nabla S_{\mu}\bigr) =&(k+\alpha _{2}) (\nabla \rho_{2},\nabla S_{\mu})+\bar{c}(u,T,S_{\mu})-\bar{c} \bigl(u,T_{\mu}^{n},S_{\mu}\bigr)\\ &{}+\alpha_{2}\bigl(\nabla\Pi_{\mu}^{2} T_{\mu}^{n-1},\nabla S_{\mu}\bigr)- \alpha_{2}(\nabla T,\nabla S_{\mu}). \end{aligned}$$
We take \(S_{\mu}=\varepsilon_{\mu}^{n}\) using (3.17) to obtain
$$\begin{aligned} (k+\alpha_{2}) \bigl(\nabla\varepsilon_{\mu}^{n}, \nabla\varepsilon_{\mu}^{n}\bigr) =&(k+\alpha_{2}) \bigl(\nabla\rho_{2},\nabla\varepsilon_{\mu}^{n} \bigr)+\bar {c}\bigl(u,\rho_{2},\varepsilon_{\mu}^{n} \bigr)+\bar{c}\bigl(e_{\mu}^{n},T_{\mu}^{n}, \varepsilon _{\mu}^{n}\bigr) \\ &{}+\bar{c}\bigl(u_{\mu}^{n}-u_{\mu}^{n-1},T_{\mu}^{n}, \varepsilon_{\mu}^{n}\bigr) +\alpha_{2}\bigl(\nabla \Pi_{\mu}^{2} T_{\mu}^{n-1},\nabla \varepsilon_{\mu}^{n}\bigr)-\alpha _{2}\bigl(\nabla T,\nabla\varepsilon_{\mu}^{n}\bigr). \end{aligned}$$
Thanks to Theorem 2.2, (3.1) and Theorem 3.1 we derive that
$$\begin{aligned}[b] (k+\alpha_{2})\bigl\| \nabla\varepsilon_{\mu}^{n} \bigr\| _{0} \leq{}&(k+\alpha_{2}+N_{2} A)\|\nabla \rho_{2}\|_{0}+N_{2} B\bigl\| \nabla e_{\mu}^{n} \bigr\| _{0}\\ &{}+N_{2} B \bigl\| \nabla\bigl(u_{\mu}^{n}-u_{\mu}^{n-1} \bigr)\bigr\| _{0}+2\alpha_{2}B. \end{aligned} $$
With the help of the triangle inequality one finds
$$\begin{aligned}[b] (k+\alpha_{2})\bigl\| \nabla\phi_{\mu}^{n} \bigr\| _{0} \leq{}&(2k+2\alpha_{2}+N_{2} A)\|\nabla \rho_{2}\|_{0}+N_{2} B\bigl\| \nabla e_{\mu}^{n} \bigr\| _{0}\\ &{}+N_{2} B \bigl\| \nabla\bigl(u_{\mu}^{n}-u_{\mu}^{n-1}\bigr)\bigr\| _{0} +2\alpha_{2}B. \end{aligned} $$
We can get the error estimation (3.13) for \(T-T_{\mu}^{n}\) by taking the infimum over \(W_{\mu}\).
Now let us to prove (3.11). The applications of (3.15) and (3.16) lead to
$$\begin{aligned} (\operatorname{Pr}+\alpha_{1}) \bigl(\nabla\xi_{\mu}^{n}, \nabla v_{\mu}\bigr) =&(\operatorname{Pr}+\alpha_{1}) (\nabla \rho_{1},\nabla v_{\mu})+c(u,\rho_{1},v_{\mu})-c \bigl(u,\xi _{\mu}^{n},v_{\mu}\bigr)+c\bigl( \rho_{1},u_{\mu}^{n},v_{\mu}\bigr) \\ &{}-c\bigl(\xi_{\mu}^{n},u_{\mu}^{n},v_{\mu}\bigr)+c\bigl(u_{\mu}^{n}-u_{\mu}^{n-1},u_{\mu}^{n},v_{\mu}\bigr)-\operatorname{Pr}\ \operatorname{Ra}\bigl(\phi^{n}_{\mu} \zeta,v_{\mu}\bigr) \\ &{}-\alpha_{1}(\nabla u,\nabla v_{\mu})+\alpha_{1} \bigl(\nabla\Pi_{\mu}^{1} u_{\mu }^{n-1}, \nabla v_{\mu}\bigr)-(\nabla\cdot v_{\mu},p- \lambda_{\mu}). \end{aligned}$$
Taking \(v_{\mu}=\xi_{\mu}^{n}\), we get
$$\begin{aligned} &(\operatorname{Pr}+\alpha_{1}) \bigl(\nabla\xi_{\mu}^{n}, \nabla\xi_{\mu}^{n}\bigr) \\ &\quad=(\operatorname{Pr}+ \alpha_{1}) \bigl(\nabla\rho_{1},\nabla\xi_{\mu}^{n} \bigr)+c\bigl(u,\rho_{1},\xi_{\mu}^{n}\bigr)+c\bigl( \rho_{1},u_{\mu}^{n},\xi_{\mu}^{n} \bigr)-c\bigl(\xi_{\mu}^{n},u_{\mu}^{n}, \xi_{\mu}^{n}\bigr) \\ &\qquad{}+c\bigl(u_{\mu}^{n}-u_{\mu}^{n-1},u_{\mu}^{n}, \xi_{\mu}^{n}\bigr)-\operatorname{Pr} \operatorname{Ra}\bigl( \phi^{n}_{\mu}\zeta,\xi ^{n}_{\mu}\bigr)- \alpha_{1}\bigl(\nabla u,\nabla\xi_{\mu}^{n}\bigr) \\ &\qquad{}+\alpha_{1}\bigl(\nabla\Pi_{\mu}^{1} u_{\mu}^{n-1},\nabla\xi_{\mu}^{n}\bigr)- \bigl(\nabla \cdot\xi_{\mu}^{n},p-\lambda_{\mu}\bigr). \end{aligned}$$
Thus
$$\begin{aligned} &(\operatorname{Pr}+\alpha_{1})\bigl\| \nabla\xi_{\mu}^{n} \bigr\| _{0} \\ &\quad\leq \bigl(\operatorname{Pr}+\alpha_{1}+N_{1}\|\nabla u\|_{0}+N_{1}\bigl\| \nabla u_{\mu}^{n} \bigr\| _{0}\bigr)\|\nabla \rho_{1}\|_{0}+N_{1} \bigl\| \nabla u_{\mu}^{n}\bigr\| _{0}\bigl\| \nabla \bigl(u_{\mu}^{n}-u_{\mu}^{n-1}\bigr) \bigr\| _{0} \\ &\qquad{}+\operatorname{Pr} \operatorname{Ra}\bigl\| \nabla \phi_{\mu}^{n} \bigr\| _{0}+\alpha_{1}\|\nabla u\|_{0}+ \alpha_{1}\bigl\| \nabla \Pi_{\mu}^{1} u_{\mu}^{n-1} \bigr\| _{0} \\ &\qquad{}+N_{1}\bigl\| \nabla u_{\mu}^{n} \bigr\| _{0}\bigl\| \nabla\xi_{\mu}^{n}\bigr\| _{0}+\sqrt{d} \|p-\lambda_{\mu}\|_{0}. \end{aligned}$$
Making use of Theorem 2.2 and Theorem 3.1, we obtain
$$\begin{aligned} (\operatorname{Pr}+\alpha_{1})\bigl\| \nabla \xi_{\mu}^{n}\bigr\| _{0} \leq&(\operatorname{Pr}+ \alpha_{1}+2N_{1}A)\|\nabla\rho_{1} \|_{0}+N_{1}A\bigl\| \nabla\bigl(u_{\mu}^{n}-u_{\mu}^{n-1}\bigr)\bigr\| _{0} \\ &{}+\operatorname{Pr} \operatorname{Ra}\bigl\| \nabla\phi_{\mu}^{n} \bigr\| _{0}+2\alpha_{1}A+N_{1}A\bigl\| \nabla \xi_{\mu}^{n}\bigr\| _{0}+\sqrt{d}\|p- \lambda_{\mu}\|_{0}. \end{aligned}$$
(3.18)
To complete the proof, the bound (3.13) is inserted into the one of (3.18), this gives
$$\begin{aligned} &(\operatorname{Pr}+\alpha_{1})\bigl\| \nabla\xi_{\mu}^{n} \bigr\| _{0} \\ &\quad\leq(\operatorname{Pr}+\alpha_{1}+2N_{1}A) \|\nabla\rho_{1}\|_{0}+N_{1}A\bigl\| \nabla \bigl(u_{\mu}^{n}-u_{\mu}^{n-1}\bigr) \bigr\| _{0}+2\alpha_{1}A+N_{1}A\bigl\| \nabla \xi_{\mu}^{n}\bigr\| _{0} \\ &\qquad{}+\sqrt{d}\|p-\lambda_{\mu}\|_{0}+\operatorname{Pr} \operatorname{Ra} {\biggl[}\frac{2k+2\alpha_{2}+N_{2} A}{k+\alpha_{2}} \inf_{\bar{T} \in W_{\mu}} {\bigl\| \nabla(T-\bar{T})\bigr\| _{0}} \\ &\qquad{}+\frac{N_{2} B}{k+\alpha_{2}}\bigl\| \nabla\bigl(u-u_{\mu}^{n}\bigr) \bigr\| _{0}+\frac{N_{2}\ B}{k+\alpha_{2}}\bigl\| \nabla\bigl(u_{\mu}^{n}-u_{\mu}^{n-1} \bigr)\bigr\| _{0}+\frac{2\alpha _{2}B}{k+\alpha_{2}} {\biggr]}, \end{aligned}$$
thus
$$\begin{aligned} &\biggl(\operatorname{Pr}+\alpha_{1}-N_{1} A- \frac{\operatorname{Pr} \operatorname{Ra} N_{2} B}{k+\alpha_{2}}\biggr) \bigl\| \nabla\xi_{\mu}^{n}\bigr\| _{0} \\ &\quad\leq \biggl(\operatorname{Pr}+\alpha_{1}+2N_{1} A+ \frac{\operatorname{Pr} \operatorname{Ra} N_{2} B}{k+\alpha_{2}}\biggr)\|\nabla \rho_{1}\|_{0} + \biggl(N_{1} A+\frac{\operatorname{Pr} \operatorname{Ra} N_{2} B}{k+\alpha_{2}}\biggr) \bigl\| \nabla\bigl(u_{\mu}^{n}-u_{\mu }^{n-1} \bigr)\bigr\| _{0} \\ &\qquad{}+2\alpha_{1} A +\sqrt{d}\|p-\lambda_{\mu}\|_{0} \\ &\qquad{}+\frac{\operatorname{Pr} \operatorname{Ra}(2k+2\alpha_{2}+N_{2} A)}{k+\alpha_{2}}\inf_{\bar{T} \in W_{\mu}}{\bigl\| \nabla(T- \bar{T})\bigr\| _{0}}+\frac{2\operatorname{Pr} \operatorname{Ra} \alpha_{2} B}{k+\alpha _{2}}. \end{aligned}$$
(3.19)
Under the condition of (3.10) we find that
$$\begin{aligned} \frac{1}{2}\operatorname{Pr}\bigl\| \nabla e_{\mu}^{n} \bigr\| _{0} \leq& \frac{1}{2}\operatorname{Pr}\bigl(\|\nabla \rho_{1}\| _{0}+\bigl\| \nabla\xi_{\mu}^{n} \bigr\| _{0}\bigr) \\ \leq& \biggl(\frac{3}{2}\operatorname{Pr}+\alpha_{1}+2N_{1} A+\frac{\operatorname{Pr} \operatorname{Ra} N_{2} B}{k+\alpha _{2}}\biggr)\|\nabla\rho_{1}\|_{0}+\sqrt{d} \|p-\lambda_{\mu}\|_{0} \\ &{}+2\alpha_{1} A+\frac{2\operatorname{Pr} \operatorname{Ra} \alpha_{2} B}{k+\alpha_{2}}+\biggl(N_{1} A+ \frac{\operatorname{Pr}\ \operatorname{Ra} N_{2} B}{k+\alpha_{2}}\biggr)\bigl\| \nabla\bigl(u_{\mu}^{n}-u_{\mu}^{n-1} \bigr)\bigr\| _{0} \\ &{}+\frac{\operatorname{Pr} \operatorname{Ra}(2k+2\alpha_{2}+N_{2} A)}{k+\alpha_{2}}\inf_{\bar{T} \in W_{\mu}}{\bigl\| \nabla(T-\bar{T}) \bigr\| _{0}}. \end{aligned}$$
(3.20)
Taking the infimum over \(\bar{u}\in V_{\mu}\), \(\lambda_{\mu}\in M_{\mu}\), and using (2.7), we get the required result (3.11).
Now, we present the convergence of \(p-p_{\mu}^{n}\). Letting \(\lambda_{\mu}\) be an approximation of p in \(M_{\mu}\) and setting \(p-p_{\mu}^{n}=(p-\lambda_{\mu})-(p_{\mu}^{n}-\lambda_{\mu})\), from the first equation of (3.14) and (3.16) we get
$$\begin{aligned} &\bigl(\nabla\cdot v_{\mu},p_{\mu}^{n} - \lambda_{\mu}\bigr) \\ &\quad=(\nabla\cdot v_{\mu},p-\lambda_{\mu})-( \operatorname{Pr}+\alpha_{1}) \bigl(\nabla e_{\mu }^{n}, \nabla v_{\mu}\bigr)-c\bigl(u,e_{\mu}^{n},v_{\mu} \bigr)-c\bigl(e_{\mu}^{n},u_{\mu}^{n},u_{\mu} \bigr) \\ &\qquad{}-c\bigl(u_{\mu}^{n}-u_{\mu}^{n-1},u_{\mu}^{n},v_{\mu} \bigr)+\alpha_{1}(\nabla u,\nabla v_{\mu})-\alpha_{1} \bigl(\nabla\Pi_{\mu}^{1} u_{\mu}^{n-1}, \nabla v_{\mu}\bigr) +\operatorname{Pr} \operatorname{Ra}\bigl( \phi^{n}_{\mu}\zeta,v_{\mu}\bigr) \\ &\quad\leq {\bigl[}\sqrt{d}\|p-\lambda_{\mu}\|_{0}+\bigl( \operatorname{Pr}+\alpha_{1}+N_{1}\|\nabla u\| _{0}+N_{1}\bigl\| \nabla u_{\mu}^{n} \bigr\| _{0}\bigr)\bigl\| \nabla e_{\mu}^{n}\bigr\| _{0} + \operatorname{Pr} \operatorname{Ra}\bigl\| \nabla\phi_{\mu}^{n} \bigr\| _{0} \\ &\qquad{}+N_{1}\bigl\| \nabla u_{\mu}^{n}\bigr\| _{0} \bigl\| \nabla\bigl(u_{\mu}^{n}-u_{\mu}^{n-1} \bigr)\bigr\| _{0}+ \alpha_{1}\bigl(\|\nabla u\|_{0}+\bigl\| \nabla u_{\mu}^{n-1}\bigr\| _{0}\bigr) {\bigr]}\|\nabla v_{\mu}\|_{0}, \end{aligned}$$
which together with the inf-sup condition and the bounds of \(\|\nabla u\|_{0}\), \(\|\nabla u_{\mu}^{n-1}\|_{0}\), \(\|\nabla u_{\mu}^{n}\|_{0}\) yields
$$\begin{aligned} &\beta\bigl\| p_{\mu}^{n} -\lambda_{\mu}\bigr\| _{0} \\ &\quad\leq \sqrt{d}\|p-\lambda_{\mu}\|_{0}+\bigl( \operatorname{Pr}+\alpha_{1}+N_{1}\|\nabla u\| _{0}+N_{1}\bigl\| \nabla u_{\mu}^{n} \bigr\| _{0}\bigr)\bigl\| \nabla e_{\mu}^{n}\bigr\| _{0} \\ &\qquad{}+N_{1}\bigl\| \nabla u_{\mu}^{n}\bigr\| _{0} \bigl\| \nabla\bigl(u_{\mu}^{n}-u_{\mu}^{n-1} \bigr)\bigr\| _{0}+\alpha_{1}\bigl(\|\nabla u\|_{0}+\bigl\| \nabla u_{\mu}^{n-1}\bigr\| _{0}\bigr) +\operatorname{Pr} \operatorname{Ra}\bigl\| \nabla\phi_{\mu}^{n}\bigr\| _{0} \\ &\quad\leq \sqrt{d}\|p-\lambda_{\mu}\|_{0}+( \operatorname{Pr}+\alpha_{1}+2 N_{1} A)\bigl\| \nabla \bigl(u-u_{\mu}^{n}\bigr)\bigr\| _{0} \\ &\qquad{}+N_{1} A\bigl\| \nabla\bigl(u_{\mu}^{n}-u_{\mu}^{n-1} \bigr)\bigr\| _{0} +2\alpha_{1} A+\operatorname{Pr} \operatorname{Ra}\bigl\| \nabla\bigl(T-T_{\mu}^{n}\bigr) \bigr\| _{0}. \end{aligned}$$
Using the triangle inequality and taking the infimum over \(\lambda_{\mu } \in M_{\mu}\), we finish the proof. □
Corollary 3.2
Under the condition of Theorem
3.2, the solution
\((u_{\mu}^{n},p_{\mu}^{n},T_{\mu}^{n})\in X_{\mu}\times M_{\mu}\times W_{\mu}\)
defined by (3.7) satisfies
$$\begin{aligned} &\bigl\| \nabla\bigl(u-u_{\mu}^{n}\bigr)\bigr\| _{0}+ \bigl\| p-p_{\mu}^{n} \bigr\| _{0}+\bigl\| \nabla\bigl(T-T_{\mu}^{n} \bigr)\bigr\| _{0}\\ &\quad\leq c\mu^{s}+C_{1} \alpha_{1}+C_{2}\alpha_{2}+C_{3} \bigl\| \nabla \bigl(u_{\mu}^{n}-u_{\mu }^{n-1}\bigr) \bigr\| _{0}. \end{aligned}$$
Remark 3.3
Corollary 3.2 shows that to ensure a good approximation, the variational multiscale Oseen iterations should converge sufficiently. Moreover, the estimator suggests the choice of the stabilization parameters \(\alpha_{1}=\mathcal{O}(\mu^{s})\) and \(\alpha_{2}=\mathcal{O}(\mu^{s})\) which ensure optimal convergence.