Appendices
1.1 Appendix A: Proof of Lemma 4.10
Lemma 4.10
Assume the analytical solutions \(\varvec{u}=(u^x,u^y)\) and p are smooth enough. \({\widetilde{u}}^{x}\), \({\widetilde{u}}^{y}\) are given in (4.3). \(U_H^x\), \(U_H^y\), \(P_H\) are obtained by Step 1 of the tg-MAC algorithm. Then there holds a positive constant C independent of H such that
$$\begin{aligned}&\parallel d_{x,H}(U_{H}^{x}-{\widetilde{u}}^{x})\parallel _{M} +\parallel D_{y,H}(U_{H}^{x}-{\widetilde{u}}^{x})\parallel _{T_{y}} +\parallel d_{y,H}(U_{H}^{y}-{\widetilde{u}}^{y})\parallel _{M}\nonumber \\&\quad +\parallel D_{x,H}(U_{H}^{y}-{\widetilde{u}}^{y})\parallel _{T_{x}}\le CH^{2}. \end{aligned}$$
(A.1)
Proof
From Step 1, the definition of \(\delta ^{\widetilde{\varvec{u}}}_{p}\), Lemma 4.2, Lemma 4.5, and Lemma 4.6, we have that for \(1\le \iota \le N_{x}-1\), \(0\le \sigma \le N_{y}-1\),
$$\begin{aligned}&-v[D_{x,H}d_{x,H}{\widetilde{u}}^{x}]_{\iota ,\sigma +1/2}-v[d_{y,H}D_{y,H}{\widetilde{u}}^{x}]_{\iota ,\sigma +1/2} +K_1^{-1}{\widetilde{u}}^{x}_{\iota ,\sigma +1/2}\nonumber \\&\quad \quad +F[S_{x}\widetilde{\varvec{u}}]{\widetilde{u}}^{x}_{\iota ,\sigma +1/2} +\Big [D_{x,H}(p-\delta ^{\widetilde{\varvec{u}}}_{p})\Big ]_{\iota ,\sigma +1/2}\nonumber \\&\quad =f^x_{\iota ,\sigma +1/2}-v\frac{\epsilon ^{x,x}_{\iota +1/2,\sigma +1/2}-\epsilon ^{x,x}_{\iota -1/2,\sigma +1/2}}{H_{\iota }^{x}}-v\frac{\epsilon ^{x,y}_{\iota ,\sigma +1}-\epsilon ^{x,y}_{\iota ,\sigma }}{H_{\sigma +1/2}^{y}}\nonumber \\&\quad \quad -r^{x}_{\iota ,\sigma +1/2}+R^{x}_{\iota ,\sigma +1/2}, \end{aligned}$$
(A.2)
where
$$\begin{aligned}&R^{x}_{\iota ,\sigma +1/2}=\Big [D_{x,H}(p-\delta ^{\widetilde{\varvec{u}}}_{p})\Big ]_{\iota ,\sigma +1/2}-\frac{\partial p}{\partial x}\Big |_{\iota ,\sigma +1/2} +F\Big ([S_{x}\widetilde{\varvec{u}}]_{\iota ,\sigma +1/2}-|\widetilde{\varvec{u}}|_{\iota ,\sigma +1/2}\Big ) {\widetilde{u}}^{x}_{\iota ,\sigma +1/2}\nonumber \\&\quad +\dfrac{1}{K_1}({\widetilde{u}}^{x}-u^{x})_{\iota ,\sigma +1/2}+F\Big (|\widetilde{\varvec{u}}| {\widetilde{u}}^{x}_{\iota ,\sigma +1/2}-|\varvec{u}|u^{x}_{\iota ,\sigma +1/2}\Big ). \end{aligned}$$
(A.3)
Similarly to the derivation of Lemma 4.2 in Rui [35], we can obtain that
$$\begin{aligned}&F\Big ([S_{x}\widetilde{\varvec{u}}]_{\iota ,\sigma +1/2}-|\widetilde{\varvec{u}}|_{\iota ,\sigma +1/2}\Big ) {\widetilde{u}}^x_{\iota ,\sigma +1/2}\nonumber \\&\quad =\dfrac{1}{4H_{\iota }}\left\{ \Big (F\dfrac{{\widetilde{u}}^{x}{\widetilde{u}}^{y}}{|\widetilde{\varvec{u}}|}\dfrac{\partial {\widetilde{u}}^{y}}{\partial x}\Big )_{\iota +1/2,\sigma +1/2}(H^{x}_{\iota +1/2})^{2}-\Big (F\dfrac{{\widetilde{u}}^{x}{\widetilde{u}}^{y}}{|\widetilde{\varvec{u}}|}\dfrac{\partial {\widetilde{u}}^{y}}{\partial x}\Big )_{\iota -1/2,\sigma +1/2}(H^{y}_{\iota -1/2})^{2}\right\} \nonumber \\&\quad \quad +O(H^{2}). \end{aligned}$$
(A.4)
Combining Lemma 4.2 and (A.4), we have that
$$\begin{aligned} \Big |\Big [D_{x,H}(p-\delta ^{\widetilde{\varvec{u}}}_{p})\Big ]_{\iota ,\sigma +1/2}-\frac{\partial p}{\partial x}\Big |_{\iota ,\sigma +1/2}+F\Big ([S_{x}\widetilde{\varvec{u}}]_{\iota ,\sigma +1/2}-|\widetilde{\varvec{u}}|_{\iota ,\sigma +1/2}\Big ) {\widetilde{u}}^{x}_{\iota ,\sigma +1/2}\Big |\le O(H^{2}). \end{aligned}$$
(A.5)
Using the similar estimation in Pan and Rui [35], we get
$$\begin{aligned} F\Big (|\widetilde{\varvec{u}}|{\widetilde{u}}^{x}_{\iota ,\sigma +1/2}-|\varvec{u}|u^{x}_{\iota ,\sigma +1/2}\Big )\le C(H^2). \end{aligned}$$
(A.6)
Then, from (A.3)-(A.6) and the definition of \(\tilde{\varvec{u}}\), we can easily obtain
$$\begin{aligned} \parallel \varvec{R} \parallel \le C(H^2), \end{aligned}$$
(A.7)
where \(\varvec{R}=(R^x,R^y)\).
Subtracting (A.2) from the second equation of (3.2), we have that
$$\begin{aligned}&-v[D_{x,H}d_{x,H}E^{x}_{\widetilde{\varvec{u}}}]_{\iota ,\sigma +1/2}-v[d_{y,H}D_{y,H}E^{y}_{\widetilde{\varvec{u}}}]_{\iota , \sigma +1/2}+K_1^{-1}E^{x}_{\widetilde{\varvec{u}},\iota ,\sigma +1/2}\nonumber \\&\quad +F\Big ([S_{x}\varvec{U}_{H}]U^{x}_{H,\iota ,\sigma +1/2}-[S_{x}\widetilde{\varvec{u}}]{\widetilde{u}}_{\iota ,\sigma +1/2}^{x}\Big ) +\Big [D_{x,H}(E_{p}+\delta ^{\widetilde{\varvec{u}}}_{p})\Big ]_{\iota ,\sigma +1/2}\nonumber \\&=v\dfrac{\epsilon ^{x,x}_{\iota +1/2,\sigma +1/2}-\epsilon ^{x,x}_{\iota -1/2,\sigma +1/2}}{H_{\iota }^{x}} +v\frac{\epsilon ^{x,y}_{\iota ,\sigma +1}-\epsilon ^{x,y}_{\iota ,\sigma }}{H_{\sigma +1/2}^{y}}+r^{x}_{\iota ,\sigma +1/2}-R^{x}_{\iota ,\sigma +1/2}. \end{aligned}$$
(A.8)
For the discrete function \(\{w^{x}_{\iota ,\sigma +1/2}\}\) with \(w_{\iota ,\sigma +1/2}^{x}\Big |_{\partial \Omega _H}=0\), (A.8) times \(w^{x}_{\iota ,\sigma +1/2}H^{x}_{\iota }H^{y}_{\sigma +1/2}\) and making the summation for \(\iota ,\sigma \) with \(\iota =1,\ldots ,N_x-1\), \(\sigma =0,\ldots ,N_y-1\) we obtain that
$$\begin{aligned}&K_1^{-1}(E^{x}_{\widetilde{\varvec{u}}},w^{x})_{T,M}-(E_{p}+\delta ^{\widetilde{\varvec{u}}}_{p},d_{x,H}w^{x})_{M} +\Big (F[S_{x}\varvec{U}_{H}]U_{H}^{x}-F[S_{x}\widetilde{\varvec{u}}]{\widetilde{u}}^{x},w^{x}\Big )_{T,M}\nonumber \\&\quad +v(d_{x,H}E^{x}_{\widetilde{\varvec{u}}},d_{x,H}w^{x})_{M}+v(D_{y,H}E^{x}_{\widetilde{\varvec{u}}},D_{y,H}w^{x})_{T_{y}}\nonumber \\&=-v(\epsilon ^{x,x},d_{x,H}w^{x})_{M}-v(\epsilon ^{x,y},D_{y,H}w^{x})_{T_{y}}+(r^{x}-R^{x},w^{x})_{T,M}. \end{aligned}$$
(A.9)
Similarly for the discrete function \(\{w^y_{\iota +1/2,\sigma }\}\) with \(w^y_{\iota +1/2,\sigma }\Big |_{\partial \Omega _H}=0\), we obtain
$$\begin{aligned}&K_2^{-1}(E^{y}_{\widetilde{\varvec{u}}},w^{y})_{M,T}-(E_{p}+\delta ^{\widetilde{\varvec{u}}}_{p},d_{y,H}w^{y})_{M} +\Big (F[S_{x}\varvec{U}_{H}]U_{H}^{y}-F[S_{x}\widetilde{\varvec{u}}]{\widetilde{u}}^{y},w^{y}\Big )_{M,T}\nonumber \\&\quad +v(d_{y,H}E^{y}_{\widetilde{\varvec{u}}},d_{y,H}w^{y})_{M}+v(D_{x,H} E^{y}_{\widetilde{\varvec{u}}},D_{x,H}w^{y})_{T_{x}}\nonumber \\&=-v(\epsilon ^{y,y},d_{y,H}w^{y})_{M}-v(\epsilon ^{y,x},D_{x,H}w^{y})_{T_{x}}+(r^{y}-R^{y},w^{y})_{M,T}. \end{aligned}$$
(A.10)
Summing (A.9) and (A.10) we get
$$\begin{aligned}&v(d_{x,H}E^{x}_{\widetilde{\varvec{u}}},d_{x,H}w^{x})_{M}+v(D_{y,H}E^{x}_{\widetilde{\varvec{u}}}, D_{y,H}w^{x})_{T_{y}}+v(d_{y,H}E^{y}_{\widetilde{\varvec{u}}},d_{y,H}w^{y})_{M}\nonumber \\&\quad \quad +v(D_{x,H}E^{y}_{\widetilde{\varvec{u}}},D_{x,H}w^{y})_{T_{x}}-(E_{p}+\delta ^{\widetilde{\varvec{u}}}_{p},d_{x,H}w^{x} +d_{y,H}w^{y})_{M}+K_1^{-1}(E^{x}_{\widetilde{\varvec{u}}},w^{x})_{T,M}\nonumber \\&\quad \quad +K_2^{-1}(E^{y}_{\widetilde{\varvec{u}}},w^{y})_{M,T}+\Big (F[S_{x}\varvec{U}_{H}]U_{H}^{x}-F[S_{x} \widetilde{\varvec{u}}]{\widetilde{u}}^{x},w^{x}\Big )_{T,M}\nonumber \\&\quad \quad +\Big (F[S_{x}\varvec{U}_{H}]U_{H}^{y}-F[S_{x}\widetilde{\varvec{u}}]{\widetilde{u}}^{y},w^{y}\Big )_{M,T}\nonumber \\&\quad =-v(\epsilon ^{x,x},d_{x,H}w^{x})_{M}-v(\epsilon ^{x,y},D_{y,H}w^{x})_{T_{y}}-v(\epsilon ^{y,y},d_{y,H}w^{y})_{M}\nonumber \\&\quad \quad -v(\epsilon ^{y,x},D_{x,H}w^{y})_{T_{x}}+(r^{x}-R^{x},w^{x})_{T,M}+(r^{y}-R^{y},w^{y})_{M,T}. \end{aligned}$$
(A.11)
According to the estimate of Lemma 4.5 in Rui [35], we have
$$\begin{aligned} \Big (F[S_{x}\varvec{U}_{H}]U_{H}^{x}-F[S_{x}\widetilde{\varvec{u}}]{\widetilde{u}}^{x},w^{x}\Big )_{T,M}\ge 0,\quad \Big (F[S_{y}\varvec{U}_{H}]U_{H}^{y}-F[S_{y}\widetilde{\varvec{u}}]{\widetilde{u}}^{y},w^{y}\Big )_{M,T}\ge 0. \end{aligned}$$
(A.12)
Then using \(\varvec{w}=(w^x,w^y)\), \(\parallel \varvec{w} \parallel \le \parallel d_{x,H}w^{x}\parallel _{M}+\parallel d_{y,H}w^{y}\parallel _{M}\) and Cauchy-Schwartz inequality, (A.11) can be transformed into the following,
$$\begin{aligned}&-(E_{p}+\delta ^{\widetilde{\varvec{u}}}_{p},d_{x,H}w^{x}+d_{y,H}w^{y})_{M}\nonumber \\&\quad \le -v(d_{x,H}E^{x}_{\widetilde{\varvec{u}}},d_{x,H}w^{x})_{M}-v(D_{y,H}E^{x}_{\widetilde{\varvec{u}}},D_{y,H}w^{x})_{T_{y}} -v(d_{y,H}E^{y}_{\widetilde{\varvec{u}}},d_{y,H}w^{y})_{M}\nonumber \\&\quad \quad -v(D_{x,H}E^{y}_{\widetilde{\varvec{u}}},D_{x,H}w^{y})_{T_{x}}-v(\epsilon ^{x,x},d_{x,H}w^{x})_{M} -v(\epsilon ^{x,y},D_{y,H}w^{x})_{T_{y}}\nonumber \\&\quad \quad -v(\epsilon ^{y,y},d_{y,H}w^{y})_{M}-v(\epsilon ^{y,x},D_{x,H}w^{y})_{T_{x}}+(r^{x} -R^{x},w^{x})_{T,M}+(r^{y}-R^{y},w^{y})_{M,T}\nonumber \\&\quad \quad -K^{-1}(E^{x}_{\widetilde{\varvec{u}}},w_{x})_{T,M}-K^{-1}(E^{y}_{\widetilde{\varvec{u}}},w_{y})_{M,T}\nonumber \\&\quad \le C\Big (\parallel \epsilon ^{x,x}\parallel _{M}+\parallel \epsilon ^{x,y}\parallel _{T_{y}} +\parallel \epsilon ^{y,y}\parallel _{M}+\parallel \epsilon ^{y,x}\parallel _{T_{x}}+\parallel D_HE_{\widetilde{\varvec{u}}} \parallel \Big )\parallel D\varvec{w} \parallel \nonumber \\&\quad \quad +\Big (\parallel r^{x}-R^{x}\parallel _{T,M}+\parallel r^{y}-R^{y}\parallel _{M,T}+K_1^{-1} \parallel E^{x}_{\widetilde{\varvec{u}}}\parallel _{T,M}+K_2^{-1}\parallel E^{y}_{\widetilde{\varvec{u}}} \parallel _{M,T}\Big )\parallel \varvec{w} \parallel \nonumber \\&\quad \le C\Big (\parallel \epsilon ^{x,x}\parallel _{M}+\parallel \epsilon ^{x,y}\parallel _{T_{y}} +\parallel \epsilon ^{y,y}\parallel _{M}+\parallel \epsilon ^{y,x}\parallel _{T_{x}} +\parallel D_HE_{\widetilde{\varvec{u}}}\parallel +\parallel r^{x}-R^{x}\parallel _{T,M}\nonumber \\&\quad \quad +\parallel r^{y}-R^{y}\parallel _{M,T}+K_1^{-1}\parallel E^{x}_{\widetilde{\varvec{u}}} \parallel _{T,M}+K_2^{-1}\parallel E^{y}_{\widetilde{\varvec{u}}}\parallel _{M,T}\Big )\parallel D_H\varvec{w} \parallel . \end{aligned}$$
(A.13)
Using Lemma 4.9 and (A.13), we can obtain
$$\begin{aligned}&\beta \parallel E_{p}+\delta ^{\widetilde{\varvec{u}}}_{p}\parallel _{M}\nonumber \\&\quad \le \mathop {sup}\limits _{\varvec{w}\le \varvec{V}_{H}} \dfrac{-(E_{p}+\delta ^{\widetilde{\varvec{u}}}_{p},d_{x,H}w^{x}+d_{y,H}w^{y})_{M}}{\parallel D_H\varvec{w}\parallel }\nonumber \\&\quad \le C\Big (\parallel D_HE_{\widetilde{\varvec{u}}}\parallel +\parallel \epsilon ^{x,x}\parallel _{M} +\parallel \epsilon ^{x,y}\parallel _{T_{y}}+\parallel \epsilon ^{y,y}\parallel _{M}+\parallel \epsilon ^{y,x}\parallel _{T_{x}}\nonumber \\&\quad \quad +\parallel r^{x}-R^{x}\parallel _{T,M}+\parallel r^{y}-R^{y}\parallel _{M,T}+K_1^{-1} \parallel E^{x}_{\widetilde{\varvec{u}}}\parallel _{T,M}+K_2^{-1}\parallel E^{y}_{\widetilde{\varvec{u}}}\parallel _{M,T}\Big )\nonumber \\&\quad \le C\Big (\parallel D_HE_{\widetilde{\varvec{u}}}\parallel +H^{2}\Big ). \end{aligned}$$
(A.14)
Setting \(w_{\iota ,\sigma +1/2}^{x}=E_{\widetilde{\varvec{u}},\iota ,\sigma +1/2}^{x}\) and \(w_{\iota +1/2,\sigma }^{y}=E_{\widetilde{\varvec{u}},\iota +1/2,\sigma }^{y}\) in (A.11), we have
$$\begin{aligned}&v\parallel D_HE_{\widetilde{\varvec{u}}}\parallel ^{2}\nonumber \\&\quad =v\parallel d_{x,H}E^{x}_{\widetilde{\varvec{u}}}\parallel _{M} +v\parallel D_{y,H}E^{x}_{\widetilde{\varvec{u}}}\parallel _{T_{y}} +v\parallel d_{y,H}E^{y}_{\widetilde{\varvec{u}}}\parallel _{M} +v\parallel D_{x,H}E^{y}_{\widetilde{\varvec{u}}}\parallel _{T_{x}}\nonumber \\&\quad =-v(\epsilon ^{x,x},d_{x,H}E^{x}_{\widetilde{\varvec{u}}})_{M}-v(\epsilon ^{x,y},D_{y,H}E^{x}_{\widetilde{\varvec{u}}})_{T_{y}} -v(\epsilon ^{y,y},d_{y,H}E^{y}_{\widetilde{\varvec{u}}})_{M}-v(\epsilon ^{y,x},D_{x,H}E^{y}_{\widetilde{\varvec{u}}})_{T_{x}}\nonumber \\&\quad \quad +(r^{x}-R^{x},E^{x}_{\widetilde{\varvec{u}}})_{T,M}+(r^{y}-R^{y},E^{y}_{\widetilde{\varvec{u}}})_{M,T} -K_1^{-1}(E^{x}_{\widetilde{\varvec{u}}},E^{x}_{\widetilde{\varvec{u}}})_{T,M} -K_2^{-1}(E^{y}_{\widetilde{\varvec{u}}},E^{y}_{\widetilde{\varvec{u}}})_{M,T}\nonumber \\&\quad \quad -\Big (F[S_{x}\varvec{U}_{H}]U_{H}^{x}-F[S_{x}\widetilde{\varvec{u}}]{\widetilde{u}}^{x},E^{x}_{\widetilde{\varvec{u}}}\Big )_{T,M} -\Big (F[S_{y}\varvec{U}_{H}]U_{H}^{y}-F[S_{y}\widetilde{\varvec{u}}]{\widetilde{u}}^{y},E^{y}_{\widetilde{\varvec{u}}}\Big )_{M,T}\nonumber \\&\quad \quad +(E_{p}+\delta ^{\widetilde{\varvec{u}}}_{p},d_{x}E^{x}_{\widetilde{\varvec{u}}}+d_{y}E^{y}_{\widetilde{\varvec{u}}})_{M}. \end{aligned}$$
(A.15)
Noting (4.10), (4.13), (4.17) and (4.18), we can get
$$\begin{aligned}&-v(\epsilon ^{x,x},d_{x,H}E^{x}_{\widetilde{\varvec{u}}})_{M}-v(\epsilon ^{x,y},D_{y,H}E^{x}_{\widetilde{\varvec{u}}})_{T_{y}} -v(\epsilon ^{y,y},d_{y,H}E^{y}_{\widetilde{\varvec{u}}})_{M}-v(\epsilon ^{y,x},D_{x,H}E^{y}_{\widetilde{\varvec{u}}})_{T_{x}}\nonumber \\&\quad \quad +(r^{x}-R^{x},E^{x}_{\widetilde{\varvec{u}}})_{T,M}+(r^{y}-R^{y},E^{y}_{\widetilde{\varvec{u}}})_{M,T}\nonumber \\&\quad \le \dfrac{v}{4}\Big (\parallel d_{x,H}E^{x}_{\widetilde{\varvec{u}}}\parallel _{M}^{2} +\parallel D_{y,H}E^{x}_{\widetilde{\varvec{u}}}\parallel _{T_{y}}^{2} +\parallel d_{y,H}E^{y}_{\widetilde{\varvec{u}}}\parallel _{M}^{2} +\parallel D_{x,H}E^{y}_{\widetilde{\varvec{u}}}\parallel _{T_{x}}^{2}\Big )+CH^{4}. \end{aligned}$$
(A.16)
Using Lemma 4.4 and (A.14), we have that
$$\begin{aligned}&(E_{p}+\delta ^{\widetilde{\varvec{u}}}_{p},d_{x,H}E^{x}_{\widetilde{\varvec{u}}}+d_{y,H}E^{y}_{\widetilde{\varvec{u}}})_{M}\le \frac{v}{4}\parallel D_HE_{\widetilde{\varvec{u}}}\parallel ^{2}+CH^{4}. \end{aligned}$$
(A.17)
Then, from (A.15), (A.16) and (A.17), we easily get
$$\begin{aligned}&\parallel D_HE_{\widetilde{\varvec{u}}}\parallel ^{2}\nonumber \\&\quad =\parallel d_{x,H}E^{x}_{\widetilde{\varvec{u}}}\parallel _{M}^2 +\parallel D_{y,H}E^{x}_{\widetilde{\varvec{u}}}\parallel _{T_{y}}^2 +\parallel d_{y,H}E^{y}_{\widetilde{\varvec{u}}}\parallel _{M}^2 +\parallel D_{x,H}E^{y}_{\widetilde{\varvec{u}}}\parallel _{T_{x}}^2\nonumber \\&\quad \le CH^{4}. \end{aligned}$$
(A.18)
Therefore, the Lemma holds. \(\square \)
1.2 Appendix B: Proof of Lemma 4.11
Lemma 4.11
Define the interpolation operator \(\hat{\varvec{U}}_H: \Omega _{H}\rightarrow \Omega _h\). And \({\tilde{u}}^x\), \({\tilde{u}}^y\) are given in (4.3), then there holds a positive constant C independent of H such that
$$\begin{aligned}&\parallel {\tilde{u}}^{x}-{\widehat{U}}_{H}^{x}\parallel _{T,M}^2+ \parallel {\tilde{u}}^{y}-{\widehat{U}}_{H}^{y}\parallel _{M,T}^2 \le CH^{4}. \end{aligned}$$
(A.19)
Proof
From (3.7), we obtain
$$\begin{aligned}&({\tilde{u}}^x-{\hat{U}}_H^x)_{\varsigma ,\kappa +1/2}\nonumber \\&\quad =\dfrac{\mu \nu }{IJ}({\tilde{u}}^x-U_H^x)_{\iota +1,\sigma +3/2}+\dfrac{\mu (J-\nu )}{IJ}({\tilde{u}}^x-U_H^x)_{\iota +1,\sigma +1/2}\nonumber \\&\quad \quad +\dfrac{(I-\mu )\nu }{IJ}({\tilde{u}}^x-U_H^x)_{\iota ,\sigma +3/2}+\dfrac{(I-\mu )(J-\nu )}{IJ}({\tilde{u}}^x-U_H^x)_{\iota ,\sigma +1/2}\nonumber \\&\quad \quad +O(H^2). \end{aligned}$$
(A.20)
Next, we get
$$\begin{aligned}&\Vert {\tilde{u}}^x-{\hat{U}}_H^x\Vert _{M}^{2}\nonumber \\&\quad \leqslant {C}\Big (\dfrac{\mu ^{2}\nu ^2}{I^{2}J^2}\Vert {\tilde{u}}^x-U_H^x\Vert _{T,M}^{2}+\dfrac{\mu ^{2}(J-\nu )^2}{I^{2}J^{2}}\Vert {\tilde{u}}^x -U_H^x\Vert _{T,M}^{2}\nonumber \\&\quad \quad +\dfrac{(I-\mu )^{2}\nu ^2}{I^{2}J^{2}}\Vert {\tilde{u}}^x-U_H^x\Vert _{T,M}^{2} +\dfrac{(I-\mu )^{2}(J-\nu )^{2}}{I^{2}J^{2}}\Vert {\tilde{u}}^x-U_H^x\Vert _{T,M}^{2}\nonumber \\&\quad \quad +O(H^4)\Big ). \end{aligned}$$
(A.21)
It follows from the definition of \({\tilde{u}}^x\) that
$$\begin{aligned} \parallel {\tilde{u}}^{x}-{\widehat{U}}_{H}^{x}\parallel _{T,M}^2\le CH^{4}. \end{aligned}$$
(A.22)
The proof of \(\parallel {\tilde{u}}^{y}-{\widehat{U}}_{h}^{y}\parallel _{M,T}^2\) is similar to \(\parallel {\tilde{u}}^{x}-{\widehat{U}}_{h}^{x}\parallel _{T,M}^2\). Therefore, the Lemma holds. \(\square \)
1.3 Appendix C: Proof of Lemma 4.12
Lemma 4.12
Suppose \( U_h^x, U_h^y\) are the nonlinear solutions on fine grid, then we have
$$\begin{aligned}&\parallel U_h^x-{\widehat{U}}_{H}^{x}\parallel _{T,M}+ \parallel U_h^y-{\widehat{U}}_{H}^{y}\parallel _{M,T} \le C(H^{2}+h^2). \end{aligned}$$
(A.23)
Proof
From the definition of \({\tilde{u}}^x\), we get
$$\begin{aligned} \parallel U_{H}^{x}-{\tilde{u}}^{x}\parallel _{T,M}\le CH^{2}. \end{aligned}$$
(A.24)
Similarly we can get
$$\begin{aligned} \parallel U_{h}^{x}-{\tilde{u}}^{x}\parallel _{T,M}\le Ch^{2}. \end{aligned}$$
(A.25)
Combining the Lemma 4.11, (A.24) and (A.25), we obtain
$$\begin{aligned} \parallel U_h^x-{\widehat{U}}_{H}^{x}\parallel _{T,M}\le C(H^{2}+h^2). \end{aligned}$$
(A.26)
The proof of \(\parallel U_h^y-{\widehat{U}}_{H}^{y}\parallel _{M,T}\) is similar to \(\parallel U_h^x-{\widehat{U}}_{H}^{x}\parallel _{T,M}\). Therefore, the Lemma can be established. \(\square \)
1.4 Appendix D: Proof of Lemma 4.15
Lemma 4.15
Let \(U_h^x\), \(U_h^y\) \(P_h\) be obtained by Step 2 of the tg-MAC algorithm. Suppose the analytical solutions \(\varvec{u}=(u^x,u^y)\) and p are smooth enough. \({\tilde{u}}^x\), \({\tilde{u}}^y\) are denoted in a similar way with (4.3), then there holds a positive constant C independent of h such that
$$\begin{aligned}&\parallel d_{x,h}(U_{h}^{x}-{\tilde{u}}^{x})\parallel _{M}+ \parallel D_{y,h}(U_{h}^{x}-{\tilde{u}}^{x})\parallel _{T_{y}}+ \parallel d_{y,h}(U_{h}^{y}-{\tilde{u}}^{y})\parallel _{M}\nonumber \\&\quad +\parallel D_{x,h}(U_{h}^{y}-{\tilde{u}}^{y})\parallel _{T_{x}} \le C(\varepsilon +H^{4}+h^2). \end{aligned}$$
(A.27)
Proof
From (1.1), the definition of \(\delta ^{\widetilde{\varvec{u}}}_{p}\), Lemmas 4.2, 4.5 and 4.6, for \(1\le \varsigma \le n_{x}-1\), \(0\le \kappa \le n_{y}-1\), we have
$$\begin{aligned}&-v[D_{x,h}d_{x,h}{\tilde{u}}^{x}]_{\varsigma ,\kappa +1/2}-v[d_{y,h}D_{y,h}{\tilde{u}}^{x}]_{\varsigma ,\kappa +1/2} +l_x(\varvec{{\tilde{u}}})_{\varsigma ,\kappa +1/2}+\Big [D_{x,h}(p-\delta ^{\varvec{{\tilde{u}}}}_{p})\Big ]_{\varsigma ,\kappa +1/2}\nonumber \\&=f^x_{\varsigma ,\kappa +1/2}+O(h^2). \end{aligned}$$
(A.28)
Subtracting (A.28) from the second equation of (3.4), we have
$$\begin{aligned}&-v[D_{x,h}d_{x,h}e_{\varvec{{\tilde{u}}}}^{x}]_{\varsigma ,\kappa +1/2}-v[d_{y,h}D_{y,h}e_{\varvec{{\tilde{u}}}}^{x}]_{\varsigma ,\kappa +1/2} -l_x(\varvec{{\tilde{u}}})_{\varsigma ,\kappa +1/2}+L_x^\varepsilon (\varvec{U}_h)_{\varsigma ,\kappa +1/2} \nonumber \\&+\Big [D_{x,h}(e_p+\delta ^{\varvec{{\tilde{u}}}}_{p})\Big ]_{\varsigma ,\kappa +1/2}=O(h^2). \end{aligned}$$
(A.29)
Combining this with Taylor expansion and (3.9), we can get
$$\begin{aligned}&-v[D_{x,h}d_{x,h}e_{\varvec{{\tilde{u}}}}^{x}]_{\varsigma ,\kappa +1/2}-v[d_{y,h}D_{y,h}e_{\varvec{{\tilde{u}}}}^{x}]_{\varsigma ,\kappa +1/2} -l_x(\tilde{\varvec{u}})_{\varsigma ,\kappa +1/2}+l_x^\varepsilon (\tilde{\varvec{u}})_{\varsigma ,\kappa +1/2}\nonumber \\&\quad -l_x^\varepsilon (\widehat{\varvec{U}}_{H})_{\varsigma ,\kappa +1/2}+l_x(\widehat{\varvec{U}}_{H})_{\varsigma ,\kappa +1/2} +\dfrac{\partial l^{\varepsilon }_{x}}{\partial {u}^{x}}(\widehat{\varvec{U}}_{H})_{\varsigma ,\kappa +1/2}({U}_{h}^{x} -{\tilde{u}}^{x})_{\varsigma ,\kappa +1/2}\nonumber \\&\quad +\dfrac{\partial l^{\varepsilon }_{x}}{\partial {u}^{y}_{1}}(\widehat{\varvec{U}}_{H})_{\varsigma ,\kappa +1/2}({U}_{h}^{y} -{\tilde{u}}^{y})_{\varsigma -1/2,\kappa +1} +\dfrac{\partial l^{\varepsilon }_{x}}{\partial {u}^{y}_{2}}(\widehat{\varvec{U}}_{H})_{\varsigma ,\kappa +1/2}({U}_{h}^{y} -{\tilde{u}}^{y})_{\varsigma -1/2,\kappa }\nonumber \\&\quad +\dfrac{\partial l^{\varepsilon }_{x}}{\partial {u}^{y}_{3}}(\widehat{\varvec{U}}_{H})_{\varsigma ,\kappa +1/2}({U}_{h}^{y} -{\tilde{u}}^{y})_{\varsigma +1/2,\kappa +1}+\dfrac{\partial l^{\varepsilon }_{x}}{\partial {u}^{y}_{4}} (\widehat{\varvec{U}}_{H})_{\varsigma ,\kappa +1/2}({U}_{h}^{y}-{\tilde{u}}^{y})_{\varsigma +1/2,\kappa }\nonumber \\&\quad +\Big [D_{x,h}(e_p+\delta ^{\varvec{{\tilde{u}}}}_{p})\Big ]_{\varsigma ,\kappa +1/2}+O\Big (({\tilde{u}}^x -{\widehat{U}}_{H}^{x})^2_{\varsigma ,\kappa +1/2}+(U_h^x-{\widehat{U}}_{H}^{x})^2_{\varsigma ,\kappa +1/2}\Big )\nonumber \\&=O(h^2). \end{aligned}$$
(A.30)
Multiplying (A.30) by \(e_{\varvec{{\tilde{u}}},\varsigma ,\kappa +1/2}^{x}h^{x}_{\varsigma }h^{y}_{\kappa +1/2}\) and making the summation for \(\varsigma ,\kappa \), we have
$$\begin{aligned}&v\Vert d_{x,h}e^{x}_{\varvec{{\tilde{u}}}}\Vert ^2_{M}+v\Vert D_{y,h}e^{x}_{\varvec{{\tilde{u}}}}\Vert ^2_{T_{y}} +\Big (l_x^\varepsilon (\tilde{\varvec{u}})-l_x(\tilde{\varvec{u}}),e^{x}_{\varvec{{\tilde{u}}}}\Big )_{T,M} +\Big (l_x(\widehat{\varvec{U}}_{H})-l_x^\varepsilon (\widehat{\varvec{U}}_{H}),e^{x}_{\varvec{{\tilde{u}}}}\Big )_{l^2,T,M}\nonumber \\&\quad +\Big (\dfrac{\partial l^{\varepsilon }_{x}}{\partial {u}^{x}}(\widehat{\varvec{U}}_{H}) e^{x}_{\varvec{{\tilde{u}}}},e^{x}_{\varvec{{\tilde{u}}}}\Big )_{T,M} +\Big (\dfrac{\partial l^{\varepsilon }_{x}}{\partial {u}^{y}_1}(\widehat{\varvec{U}}_{H})e^{y}_{\varvec{{\tilde{u}}}}, e^{x}_{\varvec{{\tilde{u}}}}\Big )_{T,M} +\Big (\dfrac{\partial l^{\varepsilon }_{x}}{\partial u^{y}_2}(\widehat{\varvec{U}}_{H})e^{y}_{\varvec{{\tilde{u}}}} ,e^{x}_{\varvec{{\tilde{u}}}}\Big )_{T,M}\nonumber \\&\quad +\Big (\dfrac{\partial l^{\varepsilon }_{x}}{\partial {u}^{y}_3}(\widehat{\varvec{U}}_{H}) e^{y}_{\varvec{{\tilde{u}}}},e^{x}_{\varvec{{\tilde{u}}}}\Big )_{T,M} +\Big (\dfrac{\partial l^{\varepsilon }_{x}}{\partial {u}^{y}_4} (\widehat{\varvec{U}}_{H})e^{y}_{\varvec{{\tilde{u}}}},e^{x}_{\varvec{{\tilde{u}}}}\Big )_{T,M} -(e_{p}+\delta ^{\varvec{{\tilde{u}}}}_{p},d_{x,h}e^{x}_{\varvec{{\tilde{u}}}})_{M}\nonumber \\&\quad +\Big (O\Big (({\tilde{u}}^x-{\widehat{U}}_{H}^{x})^2+(U_h^x-{\widehat{U}}_{H}^{x})^2\Big ), e^{x}_{\varvec{{\tilde{u}}}}\Big )_{T,M}=\Big (O(h^2),e^{x}_{\varvec{{\tilde{u}}}}\Big )_{T,M}. \end{aligned}$$
(A.31)
Similarly, we have that
$$\begin{aligned}&v\Vert d_{y,h}e^{y}_{\varvec{{\tilde{u}}}}\Vert ^2_{M}+v\Vert D_{x,h}e^{y}_{\varvec{{\tilde{u}}}}\Vert ^2_{T_{x}} +\Big (l_y^\varepsilon (\tilde{\varvec{u}})-l_y(\tilde{\varvec{u}}),e^{y}_{\varvec{{\tilde{u}}}}\Big )_{M,T} +\Big (l_y(\widehat{\varvec{U}}_{H})-l_y^\varepsilon (\widehat{\varvec{U}}_{H}),e^{y}_{\varvec{{\tilde{u}}}}\Big )_{M,T}\nonumber \\&\quad +\Big (\dfrac{\partial l^{\varepsilon }_{y}}{\partial {u}^{y}}(\widehat{\varvec{U}}_{H}) e^{y}_{\varvec{{\tilde{u}}}},e^{y}_{\varvec{{\tilde{u}}}}\Big )_{M,T} +\Big (\dfrac{\partial l^{\varepsilon }_{y}}{\partial {u}^{x}_{1}}(\widehat{\varvec{U}}_{H}) e^{x}_{\varvec{{\tilde{u}}}},e^{y}_{\varvec{{\tilde{u}}}}\Big )_{M,T} +\Big (\dfrac{\partial l^{\varepsilon }_{y}}{\partial {u}^{x}_{2}}(\widehat{\varvec{U}}_{H}) e^{x}_{\varvec{{\tilde{u}}}},e^{y}_{\varvec{{\tilde{u}}}}\Big )_{M,T}\nonumber \\&\quad +\Big (\dfrac{\partial l^{\varepsilon }_{y}}{\partial {u}^{x}_{3}}(\widehat{\varvec{U}}_{H}) e^{x}_{\varvec{{\tilde{u}}}},e^{y}_{\varvec{{\tilde{u}}}}\Big )_{M,T}+\Big (\dfrac{\partial l^{\varepsilon }_{y}}{\partial {u}^{x}_{4 }}(\widehat{\varvec{U}}_{H})e^{x}_{\varvec{{\tilde{u}}}},e^{y}_{\varvec{{\tilde{u}}}}\Big )_{M,T} -(e_{p}+\delta ^{\varvec{{\tilde{u}}}}_{p},d_{y,h}e^{y}_{\varvec{{\tilde{u}}}})_{M}\nonumber \\&\quad +\Big (O\Big (({\tilde{u}}^y-{\widehat{U}}_{H}^{y})^2+(U_h^y-{\widehat{U}}_{H}^{y})^2\Big ),e^{y}_{\varvec{{\tilde{u}}}}\Big )_{M,T} =\Big (O(h^2),e^{y}_{\varvec{{\tilde{u}}}}\Big )_{M,T}. \end{aligned}$$
(A.32)
Summing (A.31) and (A.32), we gain
$$\begin{aligned}&v\Vert d_{x,h}e^{x}_{\varvec{{\tilde{u}}}}\Vert ^2_{M}+v\Vert D_{y,h}e^{x}_{\varvec{{\tilde{u}}}}\Vert ^2_{T_{y}} +v\Vert d_{y,h}e^{y}_{\varvec{{\tilde{u}}}}\Vert ^2_{M}+v\Vert D_{x,h}e^{y}_{\varvec{{\tilde{u}}}}\Vert ^2_{T_{x}} +\Big (l_x^\varepsilon (\tilde{\varvec{u}})-l_x(\tilde{\varvec{u}}),e^{x}_{\varvec{{\tilde{u}}}}\Big )_{T,M}\nonumber \\&\quad +\Big (l_x(\widehat{\varvec{U}}_{H})-l_x^\varepsilon (\widehat{\varvec{U}}_{H}),e^{x}_{\varvec{{\tilde{u}}}}\Big )_{T,M} +\Big (l_y^\varepsilon (\tilde{\varvec{u}})-l_y(\tilde{\varvec{u}}),e^{y}_{\varvec{{\tilde{u}}}}\Big )_{M,T} +\Big (l_y(\widehat{\varvec{U}}_{H})-l_y^\varepsilon (\widehat{\varvec{U}}_{H}),e^{y}_{\varvec{{\tilde{u}}}}\Big )_{M,T}\nonumber \\&\quad +\Big (\dfrac{\partial l^{\varepsilon }_{x}}{\partial {u}^{x}}(\widehat{\varvec{U}}_{H}) e^{x}_{\varvec{{\tilde{u}}}},e^{x}_{\varvec{{\tilde{u}}}}\Big )_{T,M} +\Big (\dfrac{\partial l^{\varepsilon }_{y}}{\partial {u}^{y}}(\widehat{\varvec{U}}_{H}) e^{y}_{\varvec{{\tilde{u}}}},e^{y}_{\varvec{{\tilde{u}}}}\Big )_{M,T} +\Big (\dfrac{\partial l^{\varepsilon }_{x}}{\partial {u}^{y}_1}(\widehat{\varvec{U}}_{H}) e^{y}_{\varvec{{\tilde{u}}}},e^{x}_{\varvec{{\tilde{u}}}}\Big )_{T,M}\nonumber \\&\quad +\Big (\dfrac{\partial l^{\varepsilon }_{x}}{\partial u^{y}_2}(\widehat{\varvec{U}}_{H}) e^{y}_{\varvec{{\tilde{u}}}},e^{x}_{\varvec{{\tilde{u}}}}\Big )_{T,M} +\Big (\dfrac{\partial l^{\varepsilon }_{x}}{\partial {u}^{y}_3}(\widehat{\varvec{U}}_{H}) e^{y}_{\varvec{{\tilde{u}}}},e^{x}_{\varvec{{\tilde{u}}}}\Big )_{T,M} +\Big (\dfrac{\partial l^{\varepsilon }_{x}}{\partial {u}^{y}_4}(\widehat{\varvec{U}}_{H}) e^{y}_{\varvec{{\tilde{u}}}},e^{x}_{\varvec{{\tilde{u}}}}\Big )_{T,M}\nonumber \\&\quad +\Big (\dfrac{\partial l^{\varepsilon }_{y}}{\partial {u}^{x}_{1}}(\widehat{\varvec{U}}_{H}) e^{x}_{\varvec{{\tilde{u}}}},e^{y}_{\varvec{{\tilde{u}}}}\Big )_{M,T} +\Big (\dfrac{\partial l^{\varepsilon }_{y}}{\partial {u}^{x}_{2}}(\widehat{\varvec{U}}_{H}) e^{x}_{\varvec{{\tilde{u}}}},e^{y}_{\varvec{u}}\Big )_{M,T} +\Big (\dfrac{\partial l^{\varepsilon }_{y}}{\partial {u}^{x}_{3}}(\widehat{\varvec{U}}_{H}) e^{x}_{\varvec{{\tilde{u}}}},e^{y}_{\varvec{u}}\Big )_{M,T}\nonumber \\&\quad +\Big (\dfrac{\partial l^{\varepsilon }_{y}}{\partial {u}^{x}_{4}}(\widehat{\varvec{U}}_{H})e^{x}_{\varvec{{\tilde{u}}}},e^{y}_{\varvec{{\tilde{u}}}}\Big )_{M,T} +\Big (O\Big (({\tilde{u}}^x-{\widehat{U}}_{H}^{x})^2+(U_h^x-{\widehat{U}}_{H}^{x})^2\Big ),e^{x}_{\varvec{{\tilde{u}}}}\Big )_{T,M}\nonumber \\&\quad +\Big (O\Big (({\tilde{u}}^y-{\widehat{U}}_{H}^{y})^2+(U_h^y-{\widehat{U}}_{H}^{y})^2\Big ),e^{y}_{\varvec{{\tilde{u}}}}\Big )_{M,T} -(e_{p}+\delta ^{\varvec{{\tilde{u}}}}_{p},d_{x,h}e^{x}_{\varvec{{\tilde{u}}}}+d_{y,h}e^{y}_{\varvec{{\tilde{u}}}})_{M}\nonumber \\&=\Big (O(h^2),e^{x}_{\varvec{{\tilde{u}}}}\Big )_{T,M}+\Big (O(h^2),e^{y}_{\varvec{{\tilde{u}}}}\Big )_{M,T}. \end{aligned}$$
(A.33)
Then using \(\Vert \varvec{w}\Vert \le \Vert d_{x,h}w^{x}\Vert _{M}+\Vert d_{y,h}w^{y}\Vert _{M}\), Lemma 4.2 and Cauchy-Schwartz inequality, we have
$$\begin{aligned}&-(e_{p}+\delta ^{\varvec{{\tilde{u}}}}_{p},d_{x,h}e^{x}_{\varvec{{\tilde{u}}}}+d_{y,h}e^{y}_{\varvec{{\tilde{u}}}})_{M}\nonumber \\&\quad \le v\Vert D_he_{\varvec{{\tilde{u}}}}\Vert ^2+ \Big (\Vert O(h^2)\Vert _{T,M}+\Vert l_x^\varepsilon (\tilde{\varvec{u}}) -l_x(\tilde{\varvec{u}})\Vert _{T,M}+\Vert l_x(\widehat{\varvec{U}}_{H})-l_x^\varepsilon (\widehat{\varvec{U}}_{H})\Vert _{T,M}\nonumber \\&\quad \quad +\Vert \dfrac{\partial l^{\varepsilon }_{x}}{\partial {u}^{x}}(\widehat{\varvec{U}}_{H})e^{x}_{\varvec{{\tilde{u}}}}\Vert _{T,M} +\Vert \dfrac{\partial l^{\varepsilon }_{x}}{\partial {u}^{y}_1}(\widehat{\varvec{U}}_{H})e^{y}_{\varvec{{\tilde{u}}}}\Vert _{T,M} +\Vert \dfrac{\partial l^{\varepsilon }_{x}}{\partial {u}^{y}_2}(\widehat{\varvec{U}}_{H})e^{y}_{\varvec{{\tilde{u}}}}\Vert _{T,M} +\Vert \dfrac{\partial l^{\varepsilon }_{x}}{\partial {u}^{y}_3}(\widehat{\varvec{U}}_{H})e^{y}_{\varvec{{\tilde{u}}}}\Vert _{T,M}\nonumber \\&\quad \quad +\Vert \dfrac{\partial l^{\varepsilon }_{x}}{\partial {u}^{y}_4}(\widehat{\varvec{U}}_{H})e^{y}_{\varvec{{\tilde{u}}}}\Vert _{M,T}+\Vert O\Big (({\tilde{u}}^x -{\widehat{U}}_{H}^{x})^2+(U_h^x-{\widehat{U}}_{H}^{x})^2\Big )\Vert _{T,M}\Big )\Vert e^{x}_{\varvec{{\tilde{u}}}}\Vert _{ T,M}+\Big (\Vert O(h^2)\Vert _{M,T}\nonumber \\&\quad \quad +\Vert l_y^\varepsilon (\tilde{\varvec{u}})-l_y(\tilde{\varvec{u}})\Vert _{M,T} +\Vert l_y(\widehat{\varvec{U}}_{H})-l_y^\varepsilon (\widehat{\varvec{U}}_{H})\Vert _{M,T} +\Vert \dfrac{\partial l^{\varepsilon }_{y}}{\partial {u}^{y}}(\widehat{\varvec{U}}_{H})e^{y}_{\varvec{{\tilde{u}}}}\Vert _{M,T}\nonumber \\&\quad \quad +\Vert \dfrac{\partial l^{\varepsilon }_{y}}{\partial {u}^{x}_{1}}(\widehat{\varvec{U}}_{H})e^{x}_{\varvec{{\tilde{u}}}}\Vert _{M,T} +\Vert \dfrac{\partial l^{\varepsilon }_{y}}{\partial {u}^{x}_{2}}(\widehat{\varvec{U}}_{H})e^{x}_{\varvec{{\tilde{u}}}}\Vert _{M,T} +\Vert \dfrac{\partial l^{\varepsilon }_{y}}{\partial {u}^{x}_{3}}(\widehat{\varvec{U}}_{H})e^{x}_{\varvec{{\tilde{u}}}}\Vert _{M,T} +\Vert \dfrac{\partial l^{\varepsilon }_{y}}{\partial {u}^{x}_{4}}(\widehat{\varvec{U}}_{H})e^{x}_{\varvec{{\tilde{u}}}}\Vert _{M,T}\nonumber \\&\quad \quad +\Vert O\Big (({\tilde{u}}^y-{\widehat{U}}_{H}^{y})^2+(U_h^y-{\widehat{U}}_{H}^{y})^2\Big )\Vert _{M,T}\Big )\Vert e^{y}_{\varvec{{\tilde{u}}}}\Vert _{M,T}\nonumber \\&\quad \le C\Big (\Vert D_he_{\varvec{{\tilde{u}}}}\Vert +O(\varepsilon +H^4+h^2)\Big )\Vert D_he_{\varvec{{\tilde{u}}}}\Vert . \end{aligned}$$
(A.34)
Then, making several steps of proof which are similar to Theorem 4.1, we can get
$$\begin{aligned} \beta \parallel e_{p}+\delta ^{\varvec{{\tilde{u}}}}_{p}\parallel _{M} \le C\Big (\Vert D_he_{\varvec{{\tilde{u}}}}\Vert +O(\varepsilon +H^4+h^2)\Big ) \end{aligned}$$
(A.35)
and
$$\begin{aligned} (e_{p}+\delta ^{\varvec{{\tilde{u}}}}_{p},d_{x,h}e^{x}_{\varvec{{\tilde{u}}}}+d_{y,h}e^{y}_{\varvec{{\tilde{u}}}})_{M}\le \frac{v}{4}\Vert D_he_{\varvec{{\tilde{u}}}}\Vert ^{2}+C(\varepsilon ^2+H^8+h^4). \end{aligned}$$
(A.36)
By Lemmas 4.11-4.14, we get that
$$\begin{aligned}&\Big (O(h^2),e^{x}_{\varvec{{\tilde{u}}}}\Big )_{T,M}+\Big (O(h^2),e^{y}_{\varvec{{\tilde{u}}}}\Big )_{M,T} -\Big (l_x^\varepsilon ({\tilde{u}})-l_x({\tilde{u}}),e^{x}_{\varvec{{\tilde{u}}}}\Big )_{T,M} -\Big (l_y^\varepsilon ({\tilde{u}})-l_y({\tilde{u}}),e^{y}_{\varvec{{\tilde{u}}}}\Big )_{M,T}\nonumber \\&\quad \quad -\Big (l_x(\widehat{\varvec{U}}_{H})-l_x^\varepsilon (\widehat{\varvec{U}}_{H}),e^{x}_{\varvec{{\tilde{u}}}}\Big )_{T,M} -\Big (l_y(\widehat{\varvec{U}}_{H})-l_y^\varepsilon (\widehat{\varvec{U}}_{H}),e^{y}_{\varvec{{\tilde{u}}}}\Big )_{M,T} -\Big (\dfrac{\partial l^{\varepsilon }_{x}}{\partial {u}^{x}}(\widehat{\varvec{U}}_{H})e^{x}_{\varvec{{\tilde{u}}}},e^{x}_{\varvec{{\tilde{u}}}}\Big )_{T,M}\nonumber \\&\quad \quad -\Big (\dfrac{\partial l^{\varepsilon }_{y}}{\partial {u}^{y}}(\widehat{\varvec{U}}_{H})e^{y}_{\varvec{{\tilde{u}}}},e^{y}_{\varvec{{\tilde{u}}}}\Big )_{M,T} -\Big (\dfrac{\partial l^{\varepsilon }_{x}}{\partial {u}^{y}_1}(\widehat{\varvec{U}}_{H})e^{y}_{\varvec{{\tilde{u}}}},e^{x}_{\varvec{{\tilde{u}}}}\Big )_{T,M}-\Big (\dfrac{\partial l^{\varepsilon }_{x}}{\partial {u}^{y}_2}(\widehat{\varvec{U}}_{H})e^{y}_{\varvec{{\tilde{u}}}},e^{x}_{\varvec{{\tilde{u}}}}\Big )_{T,M}\nonumber \\&\quad \quad -\Big (\dfrac{\partial l^{\varepsilon }_{x}}{\partial {u}^{y}_3}(\widehat{\varvec{U}}_{H})e^{y}_{\varvec{{\tilde{u}}}},e^{x}_{\varvec{{\tilde{u}}}}\Big )_{T,M} -\Big (\dfrac{\partial l^{\varepsilon }_{x}}{\partial {u}^{y}_4}(\widehat{\varvec{U}}_{H})e^{y}_{\varvec{{\tilde{u}}}},e^{x}_{\varvec{{\tilde{u}}}}\Big )_{T,M}-\Big (\dfrac{\partial l^{\varepsilon }_{y}}{\partial {u}^{x}_{1}}(\widehat{\varvec{U}}_{H})e^{x}_{\varvec{{\tilde{u}}}},e^{y}_{\varvec{{\tilde{u}}}}\Big )_{M,T}\nonumber \\&\quad \quad -\Big (\dfrac{\partial l^{\varepsilon }_{y}}{\partial {u}^{x}_{2}}(\widehat{\varvec{U}}_{H})e^{x}_{\varvec{{\tilde{u}}}},e^{y}_{\varvec{{\tilde{u}}}}\Big )_{M,T} -\Big (\dfrac{\partial l^{\varepsilon }_{y}}{\partial {u}^{x}_{3}}(\widehat{\varvec{U}}_{H})e^{x}_{\varvec{{\tilde{u}}}},e^{y}_{\varvec{{\tilde{u}}}}\Big )_{M,T}-\Big (\dfrac{\partial l^{\varepsilon }_{y}}{\partial {u}^{x}_{4 }}(\widehat{\varvec{U}}_{H})e^{x}_{\varvec{{\tilde{u}}}},e^{y}_{\varvec{{\tilde{u}}}}\Big )_{M,T}\nonumber \\&\quad \quad +\Big (O\Big (({\tilde{u}}^x-{\widehat{U}}_{H}^{x})^2+(U_h^x-{\widehat{U}}_{H}^{x})^2\Big ),e^{x}_ {\varvec{{\tilde{u}}}}\Big )_{T,M}+\Big (O\Big (({\tilde{u}}^y-{\widehat{U}}_{H}^{y})^2+(U_h^y-{\widehat{U}}_{H}^{y})^2\Big ), e^{y}_{\varvec{{\tilde{u}}}}\Big )_{M,T}\nonumber \\&\quad \le \dfrac{v}{4}\Big (\parallel d_{x,h}e^{x}_{\varvec{{\tilde{u}}}}\parallel _{M}^2 +\parallel D_{y,h}e^{x}_{\varvec{{\tilde{u}}}}\parallel _{T_{y}}^2 +\parallel d_{y,h}e^{y}_{\varvec{{\tilde{u}}}}\parallel _{M}^2 +\parallel D_{x,h}e^{y}_{\varvec{{\tilde{u}}}}\parallel _{T_{x}}^2\Big )\nonumber \\&\quad \quad +C(\varepsilon ^2+H^8+h^4). \end{aligned}$$
(A.37)
Combining this with (A.32) and (A.36), we have
$$\begin{aligned}&\parallel D_he_{\varvec{{\tilde{u}}}}\parallel ^{2}\nonumber \\&\quad =\parallel d_{x,h}e^{x}_{\varvec{{\tilde{u}}}}\parallel _{M}^2 +\parallel D_{y,h}e^{x}_{\varvec{{\tilde{u}}}}\parallel _{T_{y}}^2 +\parallel d_{y,h}e^{y}_{\varvec{{\tilde{u}}}}\parallel _{M}^2 +\parallel D_{x,h}e^{y}_{\varvec{{\tilde{u}}}}\parallel _{T_{x}}^2\nonumber \\&\quad \le C(\varepsilon ^2+H^8+h^4). \end{aligned}$$
(A.38)
Therefore, the Lemma can be established. \(\square \)
