Numerical simulation on staggered grids of three-dimensional brinkman-forchheimer flow and heat transfer in porous media

Abstract

In this paper, three-dimensional numerical algorithm is constructed to simulate the behavior of the Brinkman-Forchheimer flow and thermal fields. Numerical results of velocity, pressure and temperature are obtained by applying the efficient modified two-grid marker and cell (MAC) algorithm on staggered grids with the second-order backward difference formula (BDF2) time approximation. The modified-upwind idea is introduced to convective heat transfer equations for improving accuracy without any numerical oscillation. The second-order convergence rate can be achieved for pressure, velocity and temperature of considered three-dimensional model. Some numerical experiments are presented to illustrate the efficiency of algorithm. The numerical example with analytical solution is used to validate the effectiveness and accuracy of the algorithm by comparing with the results of traditional MAC algorithm. A time-dependent test is proposed to show a detailed sensitivity analysis to indicate the influence of parameters including the $\varepsilon$, Forchheimer number, Brinkman number and thermal diffusivity on the physical properties of Brinkman-Forchheimer flow and heat transfer in porous media.

Appendix: error proof

1.1 Error proof for the Lemma 11

Here we show the error proof for the Lemma 11.

Proof

Set

$$\begin{aligned}&E_{p}=P_{H}-p,\quad E_{\tilde{\textbf{u}}}=(E_{\tilde{\textbf{u}}}^{x},E_{\tilde{\textbf{u}}}^{y},E_{\tilde{\textbf{u}}}^{z})=\textbf{U}_{H}-{\tilde{\textbf{u}}},\\&E_{T}=Z_{H}-T,\quad E_{\textbf{w}}=(E_{\textbf{w}}^{x},E_{\textbf{w}}^{y},E_{\textbf{w}}^{z})=\textbf{W}_{H}-\textbf{w},\\&H=\text {max}_{i,j,l}\{H_{i}^{x},H_{i+1/2}^{x},H_{j}^{y},H_{j+1/2}^{y},H_{l}^{z},H_{l+1/2}^{z}\}. \end{aligned}$$

We define

$$\begin{aligned}&\delta _{p,i+1/2,j+1/2,l+1/2}^{\tilde{\textbf{u}}}\\&=\frac{(h_{i+1/2}^{x})^2}{8}\frac{\partial ^{2}p_{i+1/2,j+1/2,l+1/2}^{K+1}}{\partial x^2}\nonumber \\&\quad +\frac{(h_{j+1/2}^{y})^2}{8}\frac{\partial ^{2}p_{i+1/2,j+1/2,l+1/2}^{K+1}}{\partial y^2}\nonumber \\&\quad +\frac{(h_{l+1/2}^{z})^2}{8}\frac{\partial ^{2}p_{i+1/2,j+1/2,l+1/2}^{K+1}}{\partial z^2}\\&\quad +\frac{F}{8}\frac{[\tilde{u}^{x}\tilde{u}^{y}\tilde{u}^{z}]}{|{\tilde{\textbf{u}}}_{i+1/2,j+1/2,l+1/2}|}\big (\frac{\partial \tilde{u}_{i+1/2,j+1/2,l+1/2}^y}{\partial x}(h_{i+1/2}^x)^{2}\nonumber \\&\quad +\frac{\partial \tilde{u}_{i+1/2,j+1/2,l+1/2}^z}{\partial x}(h_{i+1/2}^x)^{2}\\&+\frac{\partial \tilde{u}_{i+1/2,j+1/2,l+1/2}^x}{\partial y}(h_{j+1/2}^y)^{2}\nonumber \\&\quad +\frac{\partial \tilde{u}_{i+1/2,j+1/2,l+1/2}^z}{\partial y}(h_{j+1/2}^y)^{2}\\&+\frac{\partial \tilde{u}_{i+1/2,j+1/2,l+1/2}^x}{\partial z}(h_{l+1/2}^z)^{2}\nonumber \\&\quad +\frac{\partial \tilde{u}_{i+1/2,j+1/2,l+1/2}^y}{\partial z}(h_{l+1/2}^z)^{2}\big ). \end{aligned}$$

From Lemma 4 and 5 and Eq. 2.26, we obtain that for $i=1,...,N_{x}-1$, $j=0,...,N_{y}-1$, $l=0,...,N_{z}-1$,

$$\begin{aligned}&[D_{t}E_{\tilde{\textbf{u}}}^{x}]_{i,j+1/2,l+1/2}^{K+1} -vD_{x}[d_{x}E_{\tilde{\textbf{u}}}^{x}]_{i,j+1/2,l+1/2}^{K+1}\nonumber \\&\quad -vd_{y}[D_{y}E_{\tilde{\textbf{u}}}^{x}]_{i,j+1/2,l+1/2}^{K+1}-vd_{z}[D_{z}E_{\tilde{\textbf{u}}}^{x}]_{i,j+1/2,l+1/2}^{K+1}\nonumber \\&\quad +F([S_{x}\textbf{U}_{H}]_{i,j+1/2,l+1/2}^{K+1}U_{H,i,j+1/2,l+1/2}^{x,K+1}\nonumber \\&\quad -[S_{x}{\tilde{\textbf{u}}}]_{i,j+1/2,l+1/2}^{K+1}\tilde{u}_{i,j+1/2,l+1/2}^{x,K+1})\nonumber \\&\quad +[D_{x}(E_{p}+\delta _{p}^{\tilde{\textbf{u}}})]_{i,j+1/2,l+1/2}^{K+1}\nonumber \\&=vD_{x}\epsilon _{i,j+1/2,l+1/2}^{x,x,K+1}+vd_{y}\epsilon _{i,j+1/2,l+1/2}^{x,y,K+1}\nonumber \\&\quad +vd_{z}\epsilon _{i,j+1/2,l+1/2}^{x,z,K+1}+g^{x}\Pi _{h}(E_{T}+\delta _{T})_{i,j+1/2,l+1/2}^{K+1}\nonumber \\&\quad +R_{i,j+1/2,l+1/2}^{x,K+1}. \end{aligned}$$

(1.1)

where,

$$\begin{aligned}&\Vert \textbf{R}^{1}\Vert = O(\Delta T+(H^{x})^{2}+(H^{y})^{2}+(H^{z})^{2}),\end{aligned}$$

(1.2)

$$\begin{aligned}&\Vert \textbf{R}^{K+1}\Vert = O((\Delta T)^{2}+(H^{x})^{2}+(H^{y})^{2}+(H^{z})^{2}). \end{aligned}$$

(1.3)

Noting Lemma 2 and multiplying Eq. 6.1 by $v_{i.j+1/2,l+1/2}^{x}H_{i}^{x}H_{j+1/2}^{y}H_{l+1/2}^{z}$ with discrete function $\textbf{v}=(v^{x},v^{y},v^{z})\in \textbf{V}_{H}$ making summation on $i=1,\cdots ,N_{x}-1$, $j=0,...,N_{y}-1$, $l=0,...,N_{z}-1$, and similarly in y and z direction, we have

$$\begin{aligned}&(D_{t}E_{\tilde{\textbf{u}}}^{K+1},\textbf{v})+v(\textbf{D}E_{\tilde{\textbf{u}}}^{K+1},\textbf{Dv})\nonumber \\&\quad +(F[\textbf{SU}^{K+1}]\textbf{U}^{K+1}-F[\textbf{S}\tilde{\textbf{u}}^{K+1}]{\tilde{\textbf{u}}}^{K+1},\textbf{v})\nonumber \\&\quad -((E_{p}+\delta _{p}^{\tilde{\textbf{u}}})^{K+1},d_{x}v^{x}+d_{y}v^{y}+d_{z}v^{z})_{M}\nonumber \\&=-v(\epsilon ^{x,x,K+1},d_{x}v^{x})_{M}-v(\epsilon ^{x,y,K+1},D_{y}v^{x})_{T_{y}}\nonumber \\&\quad -v(\epsilon ^{x,z,K+1},D_{z}v^{x})_{T_{z}}\nonumber \\&\quad -v(\epsilon ^{y,x,K+1},D_{x}v^{y})_{T_{x}}-v(\epsilon ^{y,y,K+1},d_{y}v^{y})_{M}\nonumber \\&\quad -v(\epsilon ^{y,z,K+1},D_{z}v^{y})_{T_{z}}\nonumber \\&\quad -v(\epsilon ^{z,x,K+1},D_{x}v^{z})_{T_{x}}-v(\epsilon ^{z,y,K+1},D_{y}v^{z})_{T_{y}}\nonumber \\&\quad -v(\epsilon ^{z,z,K+1},d_{z}v^{z})_{M}\nonumber \\&\quad +(\textbf{R}^{K+1},\textbf{v})_{M}+(g\Pi _{h}(E_{T}+\delta _{T})^{K+1},\textbf{v})_{M}. \end{aligned}$$

(1.4)

Choosing $\textbf{v}=E_{\tilde{\textbf{u}}}^{K+1}$ in Eq. 6.4, and then since $F([\textbf{SU}^{K+1}]\textbf{U}^{K+1}-[\textbf{S}\tilde{\textbf{u}}^{K+1}]\tilde{\textbf{u}}^{K+1},E_{\tilde{\textbf{u}}}^{K+1})\ge 0$ and noting Lemma 10, we have

$$\begin{aligned}&\beta \Vert (E_{p}+\delta _{p}^{\tilde{{\textbf {u}}}})^{K+1}\Vert _{M}\nonumber \\&\quad \le \sup \dfrac{\!-((E_{p}\!+\!\delta _{p}^{\tilde{{\textbf {u}}}})^{K\!+\!1}\!,d_{x}E_{\tilde{{\textbf {u}}}}^{x,K\!+\!1}\!+\!d_{y}E_{\tilde{{\textbf {u}}}}^{y,K\!+\!1}\!+\!d_{z}E_{\tilde{{\textbf {u}}}}^{z,K\!+\!1})}{\Vert {{\textbf {D}}}E_{\tilde{{\textbf {u}}}}^{K+1}\Vert }\nonumber \\&\quad \le C(\Vert D_{t}E_{\tilde{{\textbf {u}}}}^{K+1}\Vert _{M}+\Vert {{\textbf {D}}}E_{\tilde{{\textbf {u}}}}^{K+1}\Vert _{M}+\Vert E_{\tilde{{\textbf {u}}}}^{K+1}\Vert _{M})\nonumber \\&\qquad +C\Vert (E_{T}+\delta _{T})^{K+1}\Vert _{M}+C((\Delta T)^{2}+H^{2}), \quad K\ge 1, \end{aligned}$$

(1.5)

and

$$\begin{aligned}&\beta \Vert (E_{p}+{\delta _{p}^{\tilde{{\textbf {u}}}}})^{1}\Vert _{M}\nonumber \\&\quad \le \sup \dfrac{-((E_{p}+{\delta _{p}^{\tilde{{\textbf {u}}}}})^{1},d_{x}E_{\tilde{{\textbf {u}}}}^{x,1}+d_{y}E_{\tilde{{\textbf {u}}}}^{y,1}+d_{z}E_{\tilde{{\textbf {u}}}}^{z,1})}{\Vert {{\textbf {D}}}E_{\tilde{{\textbf {u}}}}^{1}\Vert }\nonumber \\&\quad \le C(\Vert d_{t}E_{\tilde{{\textbf {u}}}}^{1}\Vert _{M}+\Vert {{\textbf {D}}}E_{\tilde{{\textbf {u}}}}^{1}\Vert _{M}+\Vert E_{\tilde{{\textbf {u}}}}^{1}\Vert _{M})\nonumber \\&\qquad +C\Vert (E_{T}+\delta _{T})^{1}\Vert _{M}+C(\Delta T+H^{2}). \end{aligned}$$

(1.6)

Therefore, we have

$$\begin{aligned}&\Vert E_{\tilde{{\textbf {u}}}}^{q+1}\Vert ^{2}+\sum _{K=0}^{q}\Delta T\Vert {{\textbf {D}}}E_{\tilde{{\textbf {u}}}}^{K+1}\Vert ^{2}\nonumber \\&\quad \le C\sum _{K=0}^{q}\Delta T\Vert (E_{T}+\delta _{T})^{K+1}\Vert _{M}^{2}+O((\Delta T)^{4}+H^{4}). \end{aligned}$$

(2.1)

Choosing ${\textbf {v}}= D_{t}{E_{\tilde{{\textbf {u}}}}^{K+1}}$ in Eq. 6.4, and deducing in a similar way then we can get

$$\begin{aligned}&\Vert {{\textbf {D}}}E_{\tilde{{\textbf {u}}}}^{q+1}\Vert ^{2}+\sum _{K=1}^{q}\Delta T\Vert {D_{t}}E_{\tilde{{\textbf {u}}}}^{K+1}\Vert ^{2}+\Delta T\Vert { d_{t}}E_{\tilde{{\textbf {u}}}}^{1}\Vert ^{2}\nonumber \\&\le C\sum _{K=0}^{q}\Delta T\Vert (E_{T}+\delta _{T})^{K+1}\Vert _{M}^{2}+O((\Delta T)^{4}+H^{4}). \end{aligned}$$

(2.2)

and

$$\begin{aligned}&\sum _{K=0}^{q}\Delta T\Vert (E_{p}+\delta _{p}^{\tilde{{\textbf {u}}}})^{K+1}\Vert _{M}^{2}\nonumber \\&\quad \le C\sum _{K=0}^{q}\Delta T\Vert (E_{T}+\delta _{T})^{K+1}\Vert _{M}^{2}+O((\Delta T)^{4}+H^{4}). \end{aligned}$$

(2.3)

In what follows, we establish the error analysis for the temperature. Recalling Eq. 2.29, we have for $i=0,...,N_{x}-1$, $j=0,...,N_{y}-1$ and $l=0,...,N_{z}-1$,

$$\begin{aligned}&[D_{t}(E_{T}+\delta _{T})]_{i+1/2,j+1/2,l+1/2}^{K+1}+[d_{x}E_{w}^{x}]_{i+1/2,j+1/2,l+1/2}^{K+1}\nonumber \\&\quad +[d_{y}E_{w}^{y}]_{i+1/2,j+1/2,l+1/2}^{K+1}\nonumber \\&+[d_{z}E_{w}^{z}]_{i+1/2,j+1/2,l+1/2}^{K+1}=R_{T,i+1/2,j+1/2,l+1/2}^{K+1}, \end{aligned}$$

(2.4)

where

$$\begin{aligned} R_{T,i+1/2,j+1/2,l+1/2}^{1}=&d_{t}\delta _{T,i+1/2,j+1/2,l+1/2}^{1}\nonumber \\&+O(\Delta T+H^2),\end{aligned}$$

(2.5)

$$\begin{aligned} R_{T,i+1/2,j+1/2,l+1/2}^{K+1}=&d_{t}\delta _{T,i+1/2,j+1/2,l+1/2}^{K+1}\nonumber \\&+O((\Delta T)^{2}+H^2). \end{aligned}$$

(2.6)

Recalling Eq. 2.28 we can get

$$\begin{aligned} E_{{\textbf {w}}}^{K+1}=&{{\textbf {U}}}^{K+1}(E_{T}+\delta _{T})^{k+1}+E_{\tilde{{\textbf {u}}}}^{K+1}(T-\delta _{T})^{K+1}\nonumber \\&-\lambda [\nabla _{h}(E_{T}+\delta _{T})]^{K+1}+R_{w}^{K+1}, \end{aligned}$$

(2.7)

where $\nabla _{h}=(D_{x},D_{y},D_{z})$, and

$$\begin{aligned} R_{w}^{K+1}=&(\tilde{{\textbf {u}}}^{K+1}-{{\textbf {u}}}^{K+1})(T^{K+1}-\delta _{T}^{K+1})-{{\textbf {u}}}^{K+1}\delta _{T}^{K+1}\nonumber \\&+\lambda \epsilon ^{K+1}(T)+O(H^2). \end{aligned}$$

(2.8)

Combining Lemma 2, Eqs. 6.10 and 6.13, we get

$$\begin{aligned}&\Vert (E_{T}+\delta _{T})^{q+1}\Vert _{M}^{2}+\sum _{K=0}^{q}\Delta T\Vert \nabla _{h}(E_{T}+\delta _{T})^{K+1}\Vert _{M}^{2}\nonumber \\&\le C\sum _{K=0}^{q}\Delta T\Vert E_{\tilde{{\textbf {u}}}}^{K+1}\Vert ^{2}+O((\Delta T)^{4}+H^{4}). \end{aligned}$$

(2.9)

We are now in position to prove our main results. Combining Eqs. 6.7, 6.9 and 6.15 we obtain

$$\begin{aligned} \Vert E_{\tilde{{\textbf {u}}}}^{q+1}\Vert ^{2}+&\sum _{K=0}^{q}\Delta T\Vert {{\textbf {D}}}E_{\tilde{{\textbf {u}}}}^{K+1}\Vert ^{2}\nonumber \\&+\sum _{K=0}^{q}\Delta T\Vert (E_{p}+\delta _{p}^{\tilde{{\textbf {u}}}})^{K+1}\Vert _{M}^{2}\nonumber \\&+\Vert (E_{T}+\delta _{T})^{q+1}\Vert _{M}^{2}\le O((\Delta T)^{4}+H^{4}). \end{aligned}$$

(2.10)

According to the definition of ${\delta _{p}^{\tilde{{\textbf {u}}}},{K+1}}$ and $\delta _{T}^{K+1}$, we can complete the proof.

1.2 Error proof for the Lemma 16

Here we show the error proof for the lemma 16.

Proof

Set

$$\begin{aligned}&e_{p}=P_{h}-p,\quad e_{\tilde{\textbf{u}}}=(e_{\tilde{\textbf{u}}}^{x},e_{\tilde{\textbf{u}}}^{y},e_{\tilde{\textbf{u}}}^{z})={{\textbf {U}}}_{h}-{\tilde{\textbf{u}}},\\&e_{T}=Z_{h}-T,\quad e_{\textbf{w}}=(e_{\textbf{w}}^{x},e_{\textbf{w}}^{y},e_\textbf{w}^{z})=\textbf{W}_{h}-\textbf{w},\\&h=\text {max}_{m,n,b}\{h_{m}^{x},h_{m+1/2}^{x},h_{n}^{y},h_{n+1/2}^{y},h_{b}^{z},h_{b+1/2}^{z}\}. \end{aligned}$$

Similar to the proof of Lemma 11, the definition of $\delta _{p}^{\tilde{\textbf{u}}}$, Lemmas 4, 5 and Eq. 2.30, we can get that for $m=1,\cdots ,n_{x}-1$, $n=0,\cdots ,n_{y}-1$, $b=0,\cdots ,n_{z}-1$,

$$\begin{aligned}&[D_{t}e_{\tilde{{\textbf {u}}}}^{x}]_{m,n+1/2,b+1/2}^{k+1} -vD_{x}[d_{x}e_{\tilde{\textbf{u}}}^{x}]_{m,n+1/2,b+1/2}^{k+1}\nonumber \\&\quad -vd_{y}[D_{y}E_{\tilde{{\textbf {u}}}}^{x}]_{m,n+1/2,b+1/2}^{k+1}\nonumber \\&\quad -vd_{z}[D_{z}e_{\tilde{{\textbf {u}}}}^{x}]_{m,n+1/2,b+1/2}^{k+1}+Q_{\varepsilon }^{x}(\textbf{U}_{h})_{m,n+1/2,b+1/2}^{k+1}\nonumber \\&\quad -q^{x}({\tilde{\textbf{u}}})_{m,n+1/2,b+1/2}^{k+1}\nonumber \\&\quad +[D_{x}(e_{p}+\delta _{p}^{\tilde{\textbf{u}}})]_{m,n+1/2,b+1/2}^{k+1}\nonumber \\&=vD_{x}\epsilon _{m,n+1/2,b+1/2}^{x,x,k+1}+vd_{y}\epsilon _{m,n+1/2,b+1/2}^{x,y,k+1}\nonumber \\&\quad +vd_{z}\epsilon _{m,n+1/2,b+1/2}^{x,z,k+1}\nonumber \\&\quad +g^{x}\Pi _{h}(E_{T}+\delta _{T})_{m,n+1/2,b+1/2}^{k+1}+R_{m,n+1/2,b+1/2}^{x,k+1}. \end{aligned}$$

(2.11)

Combining the Taylor expansion, the definition of $Q_{\varepsilon }^{x}({{\textbf {U}}}_{h})_{m,n+1/2,b+1/2}^{k+1}$ and Eq. 6.17, we can obtain

$$\begin{aligned}{} & {} (D_{t}e_{\tilde{{\textbf {u}}}}^{x,k+1},{\textbf {v}})_{m} +v({{\textbf {D}}}e_{\tilde{{\textbf {u}}}}^{k+1},{\textbf {Dv}})_{m}\nonumber \\{} & {} +(q_{\varepsilon }^{x}({\tilde{{\textbf {u}}}})^{k+1}-q^{x}({\tilde{{\textbf {u}}}})^{k+1},v^{x})_{Tmm}\nonumber \\{} & {} +(q^{x}(\hat{{\textbf {U}}}_{H})^{k+1}-q_{\varepsilon }^{x}(\hat{{\textbf {U}}}_{H})^{k+1},v^{x})_{Tmm}\nonumber \\{} & {} +(q_{\varepsilon }^{y}({\tilde{{\textbf {u}}}})^{k+1}-q^{y}({\tilde{{\textbf {u}}}})^{k+1},v^{y})_{Tmm}\nonumber \\{} & {} +(q^{y}(\hat{{\textbf {U}}}_{H})^{k+1}-q_{\varepsilon }^{y}(\hat{{\textbf {U}}}_{H})^{k+1},v^{y})_{Tmm}\nonumber \\{} & {} +(q_{\varepsilon }^{z}({\tilde{{\textbf {u}}}})^{k+1}-q^{z}({\tilde{{\textbf {u}}}})^{k+1},v^{z})_{Tmm}\nonumber \\{} & {} +(q^{z}(\hat{{\textbf {U}}}_{H})^{k+1}-q_{\varepsilon }^{z}(\hat{{\textbf {U}}}_{H})^{k+1},v^{z})_{Tmm}\nonumber \\{} & {} +(\dfrac{\partial q_{\varepsilon }^{x}}{\partial u^{x}}(\hat{{\textbf {U}}}_{H})^{k+1}e_{\tilde{{\textbf {u}}}}^{x,k+1},v^{x})_{Tmm}\nonumber \\{} & {} +(\dfrac{\partial q_{\varepsilon }^{y}}{\partial u^{y}}(\hat{{\textbf {U}}}_{H})^{k+1}e_{\tilde{{\textbf {u}}}}^{y,k+1},v^{y})_{mTm}\nonumber \\{} & {} +(\dfrac{\partial q_{\varepsilon }^{z}}{\partial u^{z}}(\hat{{\textbf {U}}}_{H})^{k+1}e_{\tilde{{\textbf {u}}}}^{z,k+1},v^{z})_{mmT}\nonumber \\{} & {} -((e_{p}+\delta _{p}^{\tilde{{\textbf {u}}}})^{k+1},d_{x}v_{x}+d_{y}v_{y}+d_{z}v_{z})_{m}\nonumber \\{} & {} +((\dfrac{\partial q_{\varepsilon }^{x}}{\partial u_{1}^{y}}(\hat{{\textbf {U}}}_{H})^{k+1}+\dfrac{\partial q_{\varepsilon }^{x}}{\partial u_{2}^{y}}(\hat{{\textbf {U}}}_{H})^{k+1}\nonumber \\{} & {} +\dfrac{\partial q_{\varepsilon }^{x}}{\partial u_{3}^{y}}(\hat{{\textbf {U}}}_{H})^{k+1}+\dfrac{\partial q_{\varepsilon }^{x}}{\partial u_{4}^{y}}(\hat{{\textbf {U}}}_{H})^{k+1})e_{\tilde{{\textbf {u}}}}^{y,k+1},v^{x})_{T,m,m}\nonumber \\{} & {} +((\dfrac{\partial q_{\varepsilon }^{x}}{\partial u_{1}^{z}}(\hat{{\textbf {U}}}_{H})^{k+1}+\dfrac{\partial q_{\varepsilon }^{x}}{\partial u_{2}^{z}}(\hat{{\textbf {U}}}_{H})^{k+1}\nonumber \\{} & {} +\dfrac{\partial q_{\varepsilon }^{x}}{\partial u_{3}^{z}}(\hat{{\textbf {U}}}_{H})^{k+1}+\dfrac{\partial q_{\varepsilon }^{x}}{\partial u_{4}^{z}}(\hat{{\textbf {U}}}_{H})^{k+1})e_{\tilde{{\textbf {u}}}}^{z,k+1},v^{x})_{Tmm}\nonumber \\{} & {} +((\dfrac{\partial q_{\varepsilon }^{y}}{\partial u_{1}^{x}}(\hat{{\textbf {U}}}_{H})^{k+1}+\dfrac{\partial q_{\varepsilon }^{y}}{\partial u_{2}^{x}}(\hat{{\textbf {U}}}_{H})^{k+1}\nonumber \\{} & {} +\dfrac{\partial q_{\varepsilon }^{y}}{\partial u_{3}^{x}}(\hat{{\textbf {U}}}_{H})^{k+1}+\dfrac{\partial q_{\varepsilon }^{y}}{\partial u_{4}^{x}}(\hat{{\textbf {U}}}_{H})^{k+1})e_{\tilde{{\textbf {u}}}}^{x,k+1},v^{y})_{mTm}\nonumber \\{} & {} +((\dfrac{\partial q_{\varepsilon }^{y}}{\partial u_{1}^{z}}(\hat{{\textbf {U}}}_{H})^{k+1}+\dfrac{\partial q_{\varepsilon }^{y}}{\partial u_{2}^{z}}(\hat{{\textbf {U}}}_{H})^{k+1}\nonumber \\{} & {} +\dfrac{\partial q_{\varepsilon }^{y}}{\partial u_{3}^{z}}(\hat{{\textbf {U}}}_{H})^{k+1}+\dfrac{\partial q_{\varepsilon }^{y}}{\partial u_{4}^{z}}(\hat{{\textbf {U}}}_{H})^{k+1})e_{\tilde{{\textbf {u}}}}^{z,k+1},v^{y})_{mTm}\nonumber \\{} & {} +((\dfrac{\partial q_{\varepsilon }^{z}}{\partial u_{1}^{x}}(\hat{{\textbf {U}}}_{H})^{k+1}+\dfrac{\partial q_{\varepsilon }^{z}}{\partial u_{2}^{x}}(\hat{{\textbf {U}}}_{H})^{k+1}\nonumber \\{} & {} +\dfrac{\partial q_{\varepsilon }^{z}}{\partial u_{3}^{x}}(\hat{{\textbf {U}}}_{H})^{k+1}+\dfrac{\partial q_{\varepsilon }^{z}}{\partial u_{4}^{x}}(\hat{{\textbf {U}}}_{H})^{k+1})e_{\tilde{{\textbf {u}}}}^{x,k+1},v^{z})_{mmT}\nonumber \\{} & {} +((\dfrac{\partial q_{\varepsilon }^{z}}{\partial u_{1}^{y}}(\hat{{\textbf {U}}}_{H})^{k+1}+\dfrac{\partial q_{\varepsilon }^{z}}{\partial u_{2}^{y}}(\hat{{\textbf {U}}}_{H})^{k+1}\nonumber \\{} & {} +\dfrac{\partial q_{\varepsilon }^{z}}{\partial u_{3}^{y}}(\hat{{\textbf {U}}}_{H})^{k+1}+\dfrac{\partial q_{\varepsilon }^{z}}{\partial u_{4}^{y}}(\hat{{\textbf {U}}}_{H})^{k+1})e_{\tilde{{\textbf {u}}}}^{y,k+1},v^{z})_{mmT}\nonumber \\{} & {} =({{\textbf {g}}}\Pi _{h}(E_{T}+\delta _{T})^{k+1},v_{x}+v_{y}+v_{z})_{m}\nonumber \\{} & {} -(v\epsilon ^{x,x,k+1},d_{x}v_{x})_{m}-(v\epsilon ^{x,y,k+1},D_{y}v_{x})_{m}\nonumber \\{} & {} -(v\epsilon ^{x,z,k+1},D_{z}v_{x})_{m}\nonumber \\{} & {} -(v\epsilon ^{y,x,k+1},D_{x}v_{x})_{m}-(v\epsilon ^{y,y,k+1},d_{y}v_{x})_{m}\nonumber \\{} & {} -(v\epsilon ^{y,z,k+1},D_{z}v_{x})_{m}\nonumber \\{} & {} -(v\epsilon ^{z,x,k+1},D_{x}v_{x})_{m}-(v\epsilon ^{z,y,k+1},D_{y}v_{x})_{m}\nonumber \\{} & {} -(v\epsilon ^{z,z,k+1},d_{z}v_{x})_{m}\nonumber \\{} & {} +(R^{x,k+1},v_{x})_{Tmm}+(R^{y,k+1},v_{y})_{mTm}\nonumber \\{} & {} +(R^{z,k+1},v_{z})_{mmT}+(O((\Delta T)^{4}+H^{4}),{{\textbf {v}}}). \end{aligned}$$

(2.12)

By Eq. 6.18, Lemmas 10, 14 and Cauchy-Schwartz inequality, we obtain

$$\begin{aligned}&\beta \Vert (e_{p}+\delta _{p}^{\tilde{\textbf{u}}})^{k+1}\Vert _{m}\nonumber \\&\quad \le \sup \dfrac{-((e_{p}+\delta _{p}^{\tilde{\textbf{u}}})^{k+1},d_{x}v_{x}+d_{y}v_{y}+d_{z}v_{z})}{\Vert {{\textbf {Dv}}}\Vert }\nonumber \\&\quad \le C(\Vert D_{t}e_{\tilde{\textbf{u}}}^{k+1}\Vert +\Vert {{\textbf {D}}}e_{\tilde{\textbf{u}}}^{k+1}\Vert +\Vert e_{\tilde{\textbf{u}}}^{k+1}\Vert )\nonumber \\&\qquad +C\Vert (e_{T}+\delta _{T})^{k+1}\Vert _{m}\nonumber \\&\qquad +C(\varepsilon +(\Delta t)^{2}+(\Delta T)^{4}+h^{2}+H^{4}), \quad K\ge 1. \end{aligned}$$

(2.13)

Similarly we can get

$$\begin{aligned} \beta \Vert (e_{p}+\delta _{p}^{\tilde{\textbf{u}}})^{1}\Vert _{m}&\le \sup \dfrac{-((e_{p}\!+\!\delta _{p}^{\tilde{\textbf{u}}})^{1},d_{x}v_{x}+d_{y}v_{y}+d_{z}v_{z})}{\Vert {{\textbf {Dv}}}\Vert }\nonumber \\&\le C(\Vert d_{t}e_{\tilde{\textbf{u}}}^{1}\Vert +\Vert {{\textbf {D}}}e_{\tilde{\textbf{u}}}^{1}\Vert +\Vert e_{\tilde{\textbf{u}}}^{1}\Vert )\nonumber \\&\quad +C\Vert (e_{T}+\delta _{T})^{1}\Vert _{m}\nonumber \\&\quad +C(\varepsilon +\Delta t+(\Delta T)^{4}+h^{2}+H^{4}). \end{aligned}$$

(2.14)

And the rest proof process is similar to the Lemma 11, we shall omit it for simplicity. Finally, we can get,

$$\begin{aligned}&\Vert e_{\tilde{\textbf{u}}}^{q+1}\Vert ^{2}+\sum _{k=0}^{q}\Delta t\Vert {{\textbf {D}}}e_{\tilde{\textbf{u}}}^{k+1}\Vert ^{2}+\sum _{k=0}^{q}\Delta t\Vert (e_{p}+\delta _{p}^{\tilde{\textbf{u}}})^{k+1}\Vert _{m}^{2}\nonumber \\&+\Vert (e_{T}+\delta _{T})^{q+1}\Vert _{m}^{2}\nonumber \\&\le O(\varepsilon ^{2}+(\Delta t)^{4}+(\Delta T)^{8}+h^{4}+H^{8}). \end{aligned}$$

(2.15)