Appendix A: The proof of Lemma 5
Setting \(v_{M}=u_{M}^{k+1}\), we get
$$\begin{aligned}&{\mathcal {K}}_{1,\alpha ,\delta t}\langle u_{M}^{k+1},u_{M}^{k+1} \rangle _{1,M}+c^{-1}_{\alpha ,\delta t}\gamma _{1}a_{\omega }\langle u_{M}^{k+1},u_{M}^{k+1}\rangle \nonumber \\&\quad = \langle {\mathcal {P}}_{t}^{I,\alpha }u_{M}^{k},u_{M}^{k+1} \rangle _{1,M}\nonumber \\&\qquad + \langle I^{c}_{M}F^{k+1},u_{M}^{k+1}\rangle _{1,M}, \end{aligned}$$
(54)
where
$$\begin{aligned} \langle {\mathcal {P}}_{t}^{I,\alpha }u_{M}^{k},u_{M}^{k+1} \rangle _{1,M}&= {\mathcal {K}}_{2,\alpha ,\delta t}\langle u_{M}^{k}, u_{M}^{k+1} \rangle _{1,M}\\&\quad +\lambda _{2}\sum _{j=1}^{k-1}(d_{\alpha ,j}-d_{\alpha ,j+1}) \langle u_{M}^{k-j},u_{M}^{k+1} \rangle _{1,M}\\&\quad +\lambda _{2} d_{\alpha ,k}\langle u_{M}^{0},u_{M}^{k+1} \rangle _{1,M}. \end{aligned}$$
Using the following inequality
$$\begin{aligned} ab\le \frac{1}{2\varTheta ^2}a^2+\frac{\varTheta ^2}{2}b^2,\quad \forall \varTheta \ne 0, \end{aligned}$$
we have
$$\begin{aligned}&\langle {\mathcal {P}}_{t}^{I,\alpha }u_{M}^{k},u_{M}^{k+1} \rangle _{1,M} \le {\mathcal {K}}_{2,\alpha ,\delta t} (\Vert u_{M}^{k}\Vert ^2_{1,M}\nonumber \\&\quad +\frac{1}{4}\Vert u_{M}^{k+1} \Vert ^{2}_{1,M}) +\lambda _{2}\sum _{j=1}^{k-1}(d_{\alpha ,j}-d_{\alpha ,j+1}) (\Vert u_{M}^{k-j}\Vert ^2_{1,M}\nonumber \\&\quad +\frac{1}{4}\Vert u_{M}^{k+1} \Vert ^{2}_{1,M}) +\lambda _{2} d_{\alpha ,k}(\Vert u_{M}^{0}\Vert ^2_{1,M}\nonumber \\&\quad +\frac{1}{4}\Vert u_{M}^{k+1} \Vert ^{2}_{1,M}) \le \lambda _{1}\beta _{\alpha ,\delta t} \Vert u_{M}^{k}\Vert ^2_{1,M} + \frac{{\mathcal {K}}_{2,\alpha ,\delta t}}{4}\Vert u_{M}^{k+1} \Vert ^{2}_{1,M} \nonumber \\&\quad +\lambda _{2}\sum _{j=0}^{k-1}(d_{\alpha ,j}-d_{\alpha ,j+1})\Vert u_{M}^{k-j}\Vert ^2_{1,M}\nonumber \\&\quad +\frac{\lambda _{2}}{4}(d_{\alpha ,1}-d_{\alpha ,k})\Vert u_{M}^{k+1} \Vert ^{2}_{1,M} \nonumber \\&\quad +\lambda _{2} d_{\alpha ,k}\Vert u_{M}^{0}\Vert ^2_{1,M}+\frac{\lambda _{2} d_{\alpha ,k}}{4}\Vert u_{M}^{k+1} \Vert ^{2}_{1,M}. \end{aligned}$$
(55)
By using Lemma 3 and using (54) and (55), we have
$$\begin{aligned}&{\mathcal {K}}_{1,\alpha ,\delta t}\Vert u_{M}^{k+1}\Vert ^{2}_{1,M}+\frac{c^{-1}_{\alpha ,\delta t}\gamma _{1}}{4}\Vert \partial _{x}u_{M}^{k+1}\Vert ^{2}_{0,\omega }\\&\quad \le \lambda _{1}\beta _{\alpha ,\delta t} \Vert u_{M}^{k}\Vert ^2_{1,M} + \frac{{\mathcal {K}}_{2,\alpha ,\delta t}}{4}\Vert u_{M}^{k+1} \Vert ^{2}_{1,M} \\&\qquad +\lambda _{2}\sum _{j=0}^{k-1}(d_{\alpha ,j}-d_{\alpha ,j+1})\Vert u_{M}^{k-j}\Vert ^2_{1,M}\\&\qquad +\frac{\lambda _{2}}{4}(d_{\alpha ,1}-d_{\alpha ,k})\Vert u_{M}^{k+1} \Vert ^{2}_{1,M} \\&\qquad +\lambda _{2} d_{\alpha ,k}\Vert u_{M}^{0}\Vert ^2_{1,M}+\frac{\lambda _{2} d_{\alpha ,k}}{4}\Vert u_{M}^{k+1} \Vert ^{2}_{1,M}+\langle I^{c}_{M}F^{k+1},u_{M}^{k+1}\rangle _{1,M}. \end{aligned}$$
Using Lemmas 1 and 2 , it follows that
$$\begin{aligned}&\left( {\mathcal {K}}_{1,\alpha ,\delta t}-\frac{{\mathcal {K}}_{2,\alpha , \delta t}}{4}-\frac{\lambda _{2}}{4}(d_{\alpha ,1}-d_{\alpha ,k}) -\frac{\lambda _{2} d_{\alpha ,k}}{4}\right) \Vert u_{M}^{k+1} \Vert ^{2}_{1,M}\nonumber \\&\qquad +\frac{c^{-1}_{\alpha ,\delta t}\gamma _{1}}{4}\Vert \partial _{x}u_{M}^{k+1} \Vert ^{2}_{0,\omega } \nonumber \\&\quad \le \lambda _{1}\beta _{\alpha ,\delta t} \Vert u_{M}^{k}\Vert ^2_{1,M}\nonumber \\&\qquad +\lambda _{2}\sum _{j=0}^{k-1}(d_{\alpha ,j}-d_{\alpha ,j+1})\Vert u_{M}^{k-j} \Vert ^2_{1,M}+\lambda _{2} d_{\alpha ,k}\Vert u_{M}^{0}\Vert ^2_{1,M}\nonumber \\&\qquad +\langle I^{c}_{M}F^{k+1},u_{M}^{k+1}\rangle _{1,M}\le \sqrt{2} \lambda _{1}\beta _{\alpha ,\delta t} \Vert u_{M}^{k}\Vert ^2_{0,\omega }\nonumber \\&\qquad +\lambda _{2}\sum _{j=0}^{k-1}(d_{\alpha ,j}-d_{\alpha ,j+1})\Vert u_{M}^{k-j} \Vert ^2_{1,M}\nonumber \\&\qquad +\lambda _{2} d_{\alpha ,k}\Vert u_{M}^{0}\Vert ^2_{1,M} +\langle I^{c}_{M}F^{k+1},u_{M}^{k+1}\rangle _{1,M}\nonumber \\&\quad \le \sqrt{2}c \lambda _{1}\beta _{\alpha ,\delta t} \Vert \partial _{x}u_{M}^{k}\Vert ^2_{0,\omega } \nonumber \\&\qquad +\lambda _{2}\sum _{j=0}^{k-1}(d_{\alpha ,j}-d_{\alpha ,j+1})\Vert u_{M}^{k-j}\Vert ^2_{1,M}+\lambda _{2} d_{\alpha ,k}\Vert u_{M}^{0}\Vert ^2_{1,M}\nonumber \\&\qquad +\langle I^{c}_{M}F^{k+1},u_{M}^{k+1}\rangle _{1,M}. \end{aligned}$$
(56)
We know
$$\begin{aligned}&\langle I^{c}_{M}F^{k+1},u_{M}^{k+1}\rangle _{1,M}\le \frac{1}{3(\lambda _{1}\beta _{\alpha ,\delta t}+\lambda _{2})}\Vert I^{c}_{M}F^{k+1}\Vert ^{2}_{1,M}\nonumber \\&\quad +\frac{3}{4}(\lambda _{1}\beta _{\alpha ,\delta t}+\lambda _{2})\Vert u_{M}^{k+1}\Vert ^{2}_{1,M}. \end{aligned}$$
(57)
In view of (56) and (57), we have
$$\begin{aligned}&\gamma _{2}c^{-1}_{\alpha ,\delta t}\Vert u_{M}^{k+1} \Vert ^{2}_{1,M}+\frac{c^{-1}_{\alpha ,\delta t}\gamma _{1}}{4}\Vert \partial _{x}u_{M}^{k+1}\Vert ^{2}_{0,\omega } \le \sqrt{2}c\lambda _{1}\beta _{\alpha ,\delta t} \Vert \partial _{x}u_{M}^{k}\Vert ^2_{0,\omega }\\&\qquad +\lambda _{2}\sum _{j=1}^{k}(d_{\alpha ,k-j}-d_{\alpha ,k-j+1})\Vert u_{M}^{j} \Vert ^2_{1,M}+\lambda _{2} d_{\alpha ,k}\Vert u_{M}^{0}\Vert ^2_{1,M}\\&\qquad +\frac{1}{3(\lambda _{1}\beta _{\alpha ,\delta t}+\lambda _{2})} \Vert I^{c}_{M}F^{k+1}\Vert ^{2}_{1,M}\\&\quad \le \frac{\lambda _{1}\beta _{\alpha ,\delta t}\sqrt{2}c}{1-d_{\alpha ,1}} \sum _{j=1}^{k}(d_{\alpha ,k-j}-d_{\alpha ,k-j+1}) \Vert \partial _{x}u_{M}^{j} \Vert ^2_{0,\omega }\\&\qquad +\lambda _{2}\sum _{j=1}^{k}(d_{\alpha ,k-j}-d_{\alpha ,k-j+1}) \Vert u_{M}^{j}\Vert ^2_{1,M}\\&\qquad +\lambda _{2} d_{\alpha ,k}\Vert u_{M}^{0}\Vert ^2_{1,M}+\frac{1}{3(\lambda _{1}\beta _{\alpha ,\delta t}+\lambda _{2})}\Vert I^{c}_{M}F^{k+1}\Vert ^{2}_{1,M}. \end{aligned}$$
If
$$\begin{aligned}&C_{1,\alpha ,\delta t}=\min \left\{ \gamma _{2}c^{-1}_{\alpha ,\delta t}, \frac{c^{-1}_{\alpha ,\delta t}\gamma _{1}}{4}\right\} ,\\&C_{2,\alpha ,\delta t}=\max \left\{ \lambda _{2},\frac{\lambda _{1}\beta _{\alpha ,\delta t}\sqrt{2}c}{1-d_{\alpha ,1}}\right\} , \end{aligned}$$
we can get the following inequality
$$\begin{aligned}&\Vert u_{M}^{k+1} \Vert ^{2}_{1,M}+\Vert \partial _{x}u_{M}^{k+1} \Vert ^{2}_{0,\omega } \le \sum _{j=1}^{k}\frac{C_{2,\alpha ,\delta t}}{C_{1,\alpha ,\delta t}}(d_{\alpha ,k-j}-d_{\alpha ,k-j+1})\\&\quad (\Vert u_{M}^{j} \Vert ^2_{1,M}+ \Vert \partial _{x}u_{M}^{j}\Vert ^2_{0,\omega }) +C_{1,\alpha ,\delta t}^{-1}\lambda _{2} d_{\alpha ,k}\Vert u_{M}^{0}\Vert ^2_{1,M}\\&\quad +\frac{1}{3C_{1,\alpha ,\delta t}(\lambda _{1}\beta _{\alpha ,\delta t}+\lambda _{2})}\Vert I^{c}_{M}F^{k+1}\Vert ^{2}_{1,M}. \end{aligned}$$
Noting Lemma 4, we have
$$\begin{aligned}&\Vert u_{M}^{k+1} \Vert ^{2}_{1,M}+\Vert \partial _{x}u_{M}^{k+1}\Vert ^{2}_{0,\omega }\\&\quad \le \left( C_{1,\alpha ,\delta t}^{-1}\lambda _{2} d_{\alpha ,k}\Vert u_{M}^{0}\Vert ^2_{1,M}+\frac{\Vert I^{c}_{M}F^{k+1}\Vert ^{2}_{1,M}}{3C_{1,\alpha ,\delta t}(\lambda _{1}\beta _{\alpha ,\delta t}+\lambda _{2})}\right) \\&\quad e^{\frac{C_{2,\alpha ,\delta t}}{C_{1,\alpha ,\delta t}}(d_{\alpha ,0}-d_{\alpha ,k})}. \end{aligned}$$
Therefore
$$\begin{aligned}&\Vert u_{M}^{k+1} \Vert ^{2}_{1,M}+\Vert \partial _{x}u_{M}^{k+1}\Vert ^{2}_{0,\omega }\\&\quad \le {\mathbf {C}}_{1,\alpha ,\delta t}\left( \Vert u_{M}^{0}\Vert ^2_{1,M}+\Vert I^{c}_{M}F^{k+1}\Vert ^{2}_{1,M}\right) e^{{\mathbf {C}}_{2,\alpha ,\delta t}(d_{\alpha ,0}-d_{\alpha ,k})}, \end{aligned}$$
where \({\mathbf {C}}_{1,\alpha ,\delta t}=\max \{C_{1,\alpha ,\delta t}^{-1}\lambda _{2} d_{\alpha ,k},\frac{1}{3C_{1,\alpha ,\delta t}(\lambda _{1}\beta _{\alpha ,\delta t}+\lambda _{2})}\}\) and \({\mathbf {C}}_{2,\alpha ,\delta t}=\frac{C_{2,\alpha ,\delta t}}{C_{1,\alpha ,\delta t}}\).
Appendix B: The proof of Lemma 10
It is evident that \(a_{\omega }\langle \varPi ^{1,0}_{M}{\mathcal {U}}^{k+1},v_{M} \rangle =a_{\omega }\langle {\mathcal {U}}^{k+1},v_{M} \rangle \), therefore one can write
$$\begin{aligned}&c^{-1}_{\alpha ,\delta t}\gamma _{1}a_{\omega }\langle \varPi ^{1,0}_{M} {\mathcal {U}}^{k+1},v_{M} \rangle =-c^{-1}_{\alpha ,\delta t}\langle (\lambda _{1}\partial _{t} {\mathcal {U}}^{k+1}\nonumber \\&\quad +\lambda _{2}~_{0}^{C} \partial _{t}^{\alpha } {\mathcal {U}}^{k+1}+\gamma _{2} {\mathcal {U}}^{k+1}), v_{M}\rangle _{0,\omega }\nonumber \\&\quad +c^{-1}_{\alpha ,\delta t}\langle f^{k+1},v_{M}\rangle _{0,\omega }, \end{aligned}$$
(58)
Furthermore, we have
$$\begin{aligned}&{\mathcal {K}}_{1,\alpha ,\delta t}\langle \varPi _{M}^{1,0} {\mathcal {U}}^{k+1},v_{M} \rangle _{1,M}-\langle {\mathcal {P}}_{t}^{I, \alpha }\varPi _{M}^{1,0}{\mathcal {U}}^{k},v_{M} \rangle _{1,M}\nonumber \\&\quad =c^{-1}_{\alpha ,\delta t}\langle (\lambda _{1}\partial _{t} \varPi _{M}^{1,0} {\mathcal {U}}^{k+1}\nonumber \\&\qquad +\lambda _{2}~_{0}^{C}\partial _{t}^{\alpha } ~\varPi _{M}^{1,0} {\mathcal {U}}^{k+1}+\gamma _{2} \varPi _{M}^{1,0} {\mathcal {U}}^{k+1}),v_{M}\rangle _{1,M}\nonumber \\&\qquad +c^{-1}_{\alpha ,\delta t}\langle r_{\varPi _{M}^{1,0}{\mathcal {U}}}^{k+1} ,v_{M} \rangle _{0,\omega }, \end{aligned}$$
(59)
where \(r_{\varPi _{M}^{1,0}{\mathcal {U}}}^{k+1}=O(\delta t)\).
Now, from (58) and (59), it easily conclude that
$$\begin{aligned}&{\mathcal {K}}_{1,\alpha ,\delta t}\langle \varPi _{M}^{1,0} {\mathcal {U}}^{k+1},v_{M} \rangle _{1,M}+c^{-1}_{\alpha ,\delta t} \gamma _{1}a_{\omega }\langle \varPi ^{1,0}_{M}{\mathcal {U}}^{k+1},v_{M} \rangle \\&\quad =\langle {\mathcal {P}}_{t}^{I,\alpha }\varPi _{M}^{1,0}{\mathcal {U}}^{k},v_{M} \rangle _{1,M}\\&\qquad -c^{-1}_{\alpha ,\delta t}\langle (\lambda _{1}\partial _{t} {\mathcal {U}}^{k+1}+\lambda _{2}~_{0}^{C}\partial _{t}^{\alpha } {\mathcal {U}}^{k+1}\\&\qquad +\gamma _{2} {\mathcal {U}}^{k+1}),v_{M}\rangle _{0,\omega } +c^{-1}_{\alpha ,\delta t}\langle f^{k+1},v_{M}\rangle _{0,\omega }\\&\qquad +c^{-1}_{\alpha ,\delta t}\langle (\lambda _{1}\partial _{t} \varPi _{M}^{1,0}{\mathcal {U}}^{k+1}\\&\qquad +\lambda _{2}~_{0}^{C}\partial _{t}^{\alpha } ~\varPi _{M}^{1,0} {\mathcal {U}}^{k+1}+\gamma _{2} \varPi _{M}^{1,0} {\mathcal {U}}^{k+1}),v_{M}\rangle _{M}\\&\qquad +c^{-1}_{\alpha ,\delta t}\langle r_{\varPi _{M}^{1,0}{\mathcal {U}}}^{k+1} ,v_{M} \rangle _{M}\\&\quad =\langle {\mathcal {P}}_{t}^{I,\alpha }\varPi _{M}^{1,0} {\mathcal {U}}^{k},v_{M} \rangle _{1,M}-c^{-1}_{\alpha ,\delta t}\langle (\lambda _{1}\partial _{t} {\mathcal {U}}^{k+1}\\&\qquad +\lambda _{2}~_{0}^{C} \partial _{t}^{\alpha } {\mathcal {U}}^{k+1}+\gamma _{2} {\mathcal {U}}^{k+1}), v_{M}\rangle _{0,\omega }\\&\qquad +c^{-1}_{\alpha ,\delta t}\langle (\lambda _{1}\partial _{t} \varPi _{M}^{1,0} {\mathcal {U}}^{k+1}+\lambda _{2}~_{0}^{C}\partial _{t}^{\alpha } ~ \varPi _{M}^{1,0} {\mathcal {U}}^{k+1}\\&\qquad +\gamma _{2} \varPi _{M}^{1,0} {\mathcal {U}}^{k+1}),v_{M}\rangle _{0,\omega }\\&\qquad -c^{-1}_{\alpha ,\delta t}\langle (\lambda _{1}\partial _{t} \varPi _{M}^{1,0} {\mathcal {U}}^{k+1}+\lambda _{2}~_{0}^{C}\partial _{t}^{\alpha } ~ \varPi _{M}^{1,0} {\mathcal {U}}^{k+1}\\&\qquad +\gamma _{2} \varPi _{M}^{1,0} {\mathcal {U}}^{k+1}),v_{M}\rangle _{0,\omega }\\&\qquad +c^{-1}_{\alpha ,\delta t}\langle (\lambda _{1}\partial _{t} \varPi _{M}^{1,0} {\mathcal {U}}^{k+1}+\lambda _{2}~_{0}^{C}\partial _{t}^{\alpha } ~ \varPi _{M}^{1,0} {\mathcal {U}}^{k+1}\\&\qquad +\gamma _{2} \varPi _{M}^{1,0} {\mathcal {U}}^{k+1}),v_{M}\rangle _{1,M}\\&\qquad +c^{-1}_{\alpha ,\delta t}\langle f^{k+1},v_{M}\rangle _{0,\omega }+c^{-1}_{\alpha ,\delta t}\langle r_{\varPi _{M}^{1,0}{\mathcal {U}}}^{k+1} ,v_{M} \rangle _{0,\omega }, \end{aligned}$$
therefore
$$\begin{aligned}&{\mathcal {K}}_{1,\alpha ,\delta t}\langle \varPi _{M}^{1,0} {\mathcal {U}}^{k+1},v_{M} \rangle _{1,M}+c^{-1}_{\alpha ,\delta t} \gamma _{1}a_{\omega }\langle \varPi ^{1,0}_{M}{\mathcal {U}}^{k+1},v_{M} \rangle \nonumber \\&\quad =\langle {\mathcal {P}}_{t}^{I,\alpha }\varPi _{M}^{1,0}{\mathcal {U}}^{k}, v_{M} \rangle _{1,M}\nonumber \\&\qquad -c^{-1}_{\alpha ,\delta t}\langle (I_{d}-\varPi _{M}^{1,0}) (\lambda _{1}\partial _{t} {\mathcal {U}}^{k+1}+\lambda _{2}~_{0}^{C} \partial _{t}^{\alpha } {\mathcal {U}}^{k+1}\nonumber \\&\qquad +\gamma _{2} {\mathcal {U}}^{k+1}), v_{M}\rangle _{0,\omega }\nonumber \\&\qquad +c^{-1}_{\alpha ,\delta t}\langle E(\lambda _{1}\partial _{t} \varPi _{M}^{1,0}{\mathcal {U}}^{k+1}+\lambda _{2}~_{0}^{C}\partial _{t}^{\alpha } ~\varPi _{M}^{1,0} {\mathcal {U}}^{k+1}\nonumber \\&\qquad +\gamma _{2} \varPi _{M}^{1,0} {\mathcal {U}}^{k+1}), v_{M}\rangle _{0,\omega }\nonumber \\&\qquad +c^{-1}_{\alpha ,\delta t}\langle f^{k+1},v_{M}\rangle _{0,\omega }+c^{-1}_{\alpha ,\delta t}\langle r_{\varPi _{M}^{1,0}{\mathcal {U}}}^{k+1} ,v_{M} \rangle _{0,\omega }. \end{aligned}$$
(60)
Let \(e_{M}^{k+1}=u_{M}^{k+1}-\varPi ^{1,0}_{M}{\mathcal {U}}^{k+1}\). From (11) and (60), we obtain
$$\begin{aligned}&{\mathcal {K}}_{1,\alpha ,\delta t}\langle e_{M}^{k+1},v_{M} \rangle _{1,M}+c^{-1}_{\alpha ,\delta t}\gamma _{1}a_{\omega }\langle e_{M}^{k+1},v_{M} \rangle \nonumber \\&\quad =\langle {\mathcal {P}}_{t}^{I,\alpha }e_{M}^{k}, v_{M} \rangle _{1,M}\nonumber \\&\qquad +c^{-1}_{\alpha ,\delta t}\langle (I_{d}-\varPi _{M}^{1,0}) (\lambda _{1}\partial _{t} {\mathcal {U}}^{k+1}+\lambda _{2}~_{0}^{C} \partial _{t}^{\alpha } {\mathcal {U}}^{k+1}\nonumber \\&\qquad +\gamma _{2} {\mathcal {U}}^{k+1}), v_{M}\rangle _{0,\omega }\nonumber \\&\qquad -c^{-1}_{\alpha ,\delta t}\langle E(\lambda _{1}\partial _{t} \varPi _{M}^{1,0} {\mathcal {U}}^{k+1}+\lambda _{2}~_{0}^{C}\partial _{t}^{\alpha } ~\varPi _{M}^{1,0} {\mathcal {U}}^{k+1}\nonumber \\&\qquad +\gamma _{2} \varPi _{M}^{1,0} {\mathcal {U}}^{k+1}),v_{M}\rangle _{0,\omega }\nonumber \\&\qquad +c^{-1}_{\alpha ,\delta t}\langle I_{N}^{c}f^{k+1},v_{M}\rangle _{N,\omega }-c^{-1}_{\alpha ,\delta t}\langle f^{k+1},v_{M}\rangle _{0,\omega }\nonumber \\&\qquad -c^{-1}_{\alpha ,\delta t}\langle r_{\varPi _{M}^{1,0}{\mathcal {U}}}^{k+1} ,v_{M} \rangle _{1,M}, \end{aligned}$$
(61)
therefore we can write
$$\begin{aligned}&{\mathcal {K}}_{1,\alpha ,\delta t}\langle e_{M}^{k+1},v_{M} \rangle _{1,M}+c^{-1}_{\alpha ,\delta t}\gamma _{1}a_{\omega }\langle e_{M}^{k+1},v_{M} \rangle \\&\quad = \langle {\mathcal {P}}_{t}^{I,\alpha }e_{M}^{k},v_{M} \rangle _{1,M}+ \langle \delta ^{k+1},v_{M} \rangle _{0,\omega }, \end{aligned}$$
where
$$\begin{aligned} \langle \delta ^{k+1},v_{M}\rangle _{0,\omega }&=\langle \delta _{1}^{k+1},v_{M}\rangle _{0,\omega }+\langle \delta _{2}^{k+1},v_{M} \rangle _{0,\omega }\\&\quad +\langle \delta _{3}^{k+1},v_{M}\rangle _{0,\omega }, \end{aligned}$$
in which
$$\begin{aligned}&\langle \delta _{1}^{k+1},v_{M}\rangle _{0,\omega }=c^{-1}_{\alpha ,\delta t}\langle (I_{d}-\varPi _{M}^{1,0})(\lambda _{1}\partial _{t} {\mathcal {U}}^{k+1}\\&\quad +\lambda _{2}~_{0}^{C}\partial _{t}^{\alpha } {\mathcal {U}}^{k+1}+\gamma _{2} {\mathcal {U}}^{k+1}),v_{M}\rangle _{0,\omega }, \\&\langle \delta _{2}^{k+1},v_{M}\rangle _{0,\omega }=-c^{-1}_{\alpha ,\delta t}\langle E(\lambda _{1}\partial _{t} \varPi _{M}^{1,0}{\mathcal {U}}^{k+1}\\&\quad +\lambda _{2}~_{0}^{C}\partial _{t}^{\alpha } ~\varPi _{M}^{1,0} {\mathcal {U}}^{k+1}+\gamma _{2} \varPi _{M}^{1,0} {\mathcal {U}}^{k+1}),v_{M}\rangle _{0,\omega }, \end{aligned}$$
and
$$\begin{aligned} \langle \delta _{3}^{k+1},v_{M}\rangle _{0,\omega }&=c^{-1}_{\alpha ,\delta t}\langle I_{N}^{c}f^{k+1},v_{M}\rangle _{N,\omega }-c^{-1}_{\alpha ,\delta t}\langle f^{k+1},v_{M}\rangle _{0,\omega }\\&\quad -c^{-1}_{\alpha ,\delta t}\langle r_{\varPi _{M}^{1,0}{\mathcal {U}}}^{k+1} ,v_{M} \rangle _{0,\omega }. \end{aligned}$$
It easily conclude that
$$\begin{aligned}&{\mathcal {K}}_{1,\alpha ,\delta t}\langle e_{M}^{k+1},v_{M} \rangle _{1,M}+c^{-1}_{\alpha ,\delta t}\gamma _{1}a_{\omega }\langle e_{M}^{k+1},v_{M} \rangle \nonumber \\&\quad = {\mathcal {K}}_{2,\alpha ,\delta t}\langle e_{M}^{k} ,v_{M}\rangle _{1,M} \nonumber \\&\qquad +\lambda _{2}\sum _{j=1}^{k-1}(d_{\alpha ,j}-d_{\alpha ,j+1})\langle e_{M}^{k-j},v_{M}\rangle _{1,M}\nonumber \\&\qquad +\lambda _{2} d_{\alpha ,k}\langle e_{M}^{0},v_{M}\rangle _{1,M}\nonumber \\&\qquad +\langle \delta ^{k+1},v_{M}\rangle _{0,\omega },\quad \forall v_{M} \in {\mathbb {P}}^{0}_{M}. \end{aligned}$$
(62)
This completes the proof of Lemma 10.
Appendix C: The proof of Theorem 5
We have
$$\begin{aligned}&\langle {\mathcal {P}}_{t}^{I,\alpha }e_{M}^{k},e_{M}^{k+1} \rangle _{1,M} \le {\mathcal {K}}_{2,\alpha ,\delta t} (\Vert e_{M}^{k}\Vert ^2_{1,M}\nonumber \\&\quad +\frac{1}{4}\Vert e_{M}^{k+1} \Vert ^{2}_{1,M}) +\lambda _{2}\sum _{j=1}^{k-1}(d_{\alpha ,j}-d_{\alpha ,j+1})(\Vert e_{M}^{k-j} \Vert ^2_{1,M}\nonumber \\&\quad +\frac{1}{4}\Vert e_{M}^{k+1} \Vert ^{2}_{1,M}) +\lambda _{2} d_{\alpha ,k}(\Vert u_{M}^{0}\Vert ^2_{1,M}\nonumber \\&\quad +\frac{1}{4}\Vert e_{M}^{k+1} \Vert ^{2}_{1,M}) \le \lambda _{1}\beta _{\alpha ,\delta t} \Vert e_{M}^{k}\Vert ^2_{1,M} + \frac{{\mathcal {K}}_{2,\alpha ,\delta t}}{4}\Vert e_{M}^{k+1} \Vert ^{2}_{1,M} \nonumber \\&\quad +\lambda _{2}\sum _{j=0}^{k-1}(d_{\alpha ,j}-d_{\alpha ,j+1})\Vert e_{M}^{k-j}\Vert ^2_{1,M}\nonumber \\&\quad +\frac{\lambda _{2}}{4}(d_{\alpha ,1}-d_{\alpha ,k})\Vert e_{M}^{k+1} \Vert ^{2}_{1,M} \nonumber \\&\quad +\lambda _{2} d_{\alpha ,k}\Vert e_{M}^{0}\Vert ^2_{1,M}+\frac{\lambda _{2} d_{\alpha ,k}}{4}\Vert e_{M}^{k+1} \Vert ^{2}_{1,M}.~ \end{aligned}$$
(63)
By Lemmas 1 and 3 and using (63), we have
$$\begin{aligned}&{\mathcal {K}}_{1,\alpha ,\delta t}\Vert e_{M}^{k+1}\Vert ^{2}_{1,M}+\frac{c^{-1}_{\alpha ,\delta t}\gamma _{1}}{8}\Vert \partial _{x}e_{M}^{k+1}\Vert ^{2}_{1,M}\\&\quad \le \lambda _{1}\beta _{\alpha ,\delta t} \Vert e_{M}^{k}\Vert ^2_{1,M}\\&\qquad + \frac{{\mathcal {K}}_{2,\alpha ,\delta t}}{4}\Vert e_{M}^{k+1} \Vert ^{2}_{1,M}+\lambda _{2}\sum _{j=0}^{k-1}(d_{\alpha ,j}-d_{\alpha ,j+1}) \Vert e_{M}^{k-j}\Vert ^2_{1,M}\\&\qquad +\frac{\lambda _{2}}{4}(d_{\alpha ,1}-d_{\alpha ,k})\Vert e_{M}^{k+1} \Vert ^{2}_{1,M} +\lambda _{2} d_{\alpha ,k}\Vert e_{M}^{0}\Vert ^2_{1,M}\\&\qquad +\frac{\lambda _{2} d_{\alpha ,k}}{4}\Vert e_{M}^{k+1} \Vert ^{2}_{1,M}+\langle \delta ^{k+1},e_{M}^{k+1}\rangle _{0,\omega }. \end{aligned}$$
It follows that
$$\begin{aligned}&\left( \frac{3}{4}(\lambda _{1}\beta _{\alpha ,\delta t}+\lambda _{2}) +\gamma _{2}c^{-1}_{\alpha ,\delta t}\right) \Vert e_{M}^{k+1} \Vert ^{2}_{1,M}\nonumber \\&\qquad +\frac{c^{-1}_{\alpha ,\delta t}\gamma _{1}}{8}\Vert \partial _{x}e_{M}^{k+1} \Vert ^{2}_{1,M} \nonumber \\&\qquad \le \lambda _{1}\beta _{\alpha ,\delta t} \Vert e_{M}^{k}\Vert ^2_{1,M} +\lambda _{2}\sum _{j=0}^{k-1}(d_{\alpha ,j}-d_{\alpha ,j+1})\Vert e_{M}^{k-j} \Vert ^2_{1,M}\nonumber \\&\quad +\lambda _{2} d_{\alpha ,k}\Vert e_{M}^{0}\Vert ^2_{1,M}+\langle \delta ^{k+1},e_{M}^{k+1}\rangle _{0,\omega }. \end{aligned}$$
(64)
Using Lemmas 8 and 9 , we obtain
$$\begin{aligned}&|\langle \delta _{1}^{k+1},e^{k+1}_{M}\rangle _{0,\omega }| =c^{-1}_{\alpha ,\delta t}|\langle (I_{d}-\varPi _{M}^{1,0}) ((\lambda _{1}\partial _{t} {\mathcal {U}}^{k+1}\nonumber \\&\qquad +\lambda _{2}~_{0}^{C} \partial _{t}^{\alpha } {\mathcal {U}}^{k+1}+\gamma _{2} {\mathcal {U}}^{k+1})), e^{k+1}_{M}\rangle _{0,\omega }| \nonumber \\&\quad \le D_{1,\alpha }\delta t^{\alpha } M^{-2r}+\frac{1}{4}(\lambda _{1}\beta _{\alpha ,\delta t}+\lambda _{2}) \Vert e_{M}^{k+1}\Vert ^{2}_{1,M} , \end{aligned}$$
(65)
$$\begin{aligned}&|\langle \delta _{2}^{k+1},e^{k+1}_{M}\rangle _{0,\omega }=c^{-1}_{\alpha , \delta t}|\langle E(\lambda _{1}\partial _{t} \varPi _{M}^{1,0}{\mathcal {U}}^{k+1}\nonumber \\&\qquad +\lambda _{2}~_{0}^{C}\partial _{t}^{\alpha } ~\varPi _{M}^{1,0} {\mathcal {U}}^{k+1} +\gamma _{2} \varPi _{M}^{1,0} {\mathcal {U}}^{k+1}), e^{k+1}_{M}\rangle _{0,\omega }|\nonumber \\&\quad \le D_{2,\alpha } \delta t^{\alpha } M^{-2r}+\frac{1}{4}(\lambda _{1}\beta _{\alpha ,\delta t}+\lambda _{2}) \Vert e_{M}^{k+1}\Vert ^{2}_{1,M}, \end{aligned}$$
and
$$\begin{aligned} |\langle \delta _{3}^{k+1},e^{k+1}_{M}\rangle _{0,\omega }|&\le c^{-1}_{\alpha ,\delta t}|\langle I_{N}^{c}f^{k+1},v_{M} \rangle _{N,\omega }\nonumber \\&\quad -\langle f^{k+1},v_{M}\rangle _{0,\omega }| +c^{-1}_{\alpha ,\delta t}|\langle r_{\varPi _{M}^{1,0}{\mathcal {U}}}^{k+1} , e^{k+1}_{M} \rangle _{0,\omega }|\nonumber \\&\le D_{3,\alpha } \delta t^{\alpha } M^{-2r}+D_{4,\alpha } \delta t^{2+\alpha } \nonumber \\&\quad +\frac{1}{4}(\lambda _{1}\beta _{\alpha ,\delta t}+\lambda _{2})\Vert e_{M}^{k+1}\Vert ^{2}_{1,M}. \end{aligned}$$
(66)
In view of (64), (65) and (66), we have
$$\begin{aligned}&\gamma _{2}\Vert e_{M}^{k+1} \Vert ^{2}_{1,M}+\frac{\gamma _{1}}{8}\Vert \partial _{x}e_{M}^{k+1}\Vert ^{2}_{1,M}\nonumber \\&\quad \le \lambda _{1}c_{\alpha , \delta t}\beta _{\alpha ,\delta t} \Vert e_{M}^{k}\Vert ^2_{1,M} +\lambda _{2}c_{\alpha ,\delta t}\sum _{j=0}^{k-1}(d_{\alpha ,j} -d_{\alpha ,j+1})\Vert e_{M}^{k-j}\Vert ^2_{1,M}\nonumber \\&\qquad +\lambda _{2}c_{\alpha ,\delta t} d_{\alpha ,k}\Vert e_{M}^{0}\Vert ^2_{1,M} +{\widetilde{D}}_{1,\alpha } M^{-2r} +{\widetilde{D}}_{2,\alpha } \delta t^{2}. \end{aligned}$$
(67)
If
$$\begin{aligned}&B_{1}=\min \left\{ \gamma _{2},\frac{\gamma _{1}}{8}\right\} ,\\&B_{2,\alpha ,\delta t}=\max \left\{ \lambda _{2}c_{\alpha ,\delta t},\frac{\lambda _{1}c_{\alpha ,\delta t}\beta _{\alpha ,\delta t}}{1-d_{\alpha ,1}}\right\} . \end{aligned}$$
Now, we can get the following inequality
$$\begin{aligned}&\Vert e_{M}^{k+1} \Vert ^{2}_{1,M}+\Vert \partial _{x}e_{M}^{k+1}\Vert ^{2}_{1,M}\\&\quad \le \sum _{j=1}^{k}\frac{B_{2,\alpha ,\delta t}}{B_{1}}(d_{\alpha ,k-j} -d_{\alpha ,k-j+1})(\Vert e_{M}^{j}\Vert ^2_{1,M} + \Vert \partial _{x}e_{M}^{j}\Vert ^2_{1,M})\\&\qquad +B_{1}^{-1}\lambda _{2}(1-\alpha ) \frac{t_{k}^{-\alpha }}{\varGamma (2-\alpha )} \Vert u_{M}^{0}\Vert ^2_{1,M}+{\widetilde{D}}_{1,\alpha } M^{-2r}+{\widetilde{D}}_{2,\alpha } \delta t^{2}. \end{aligned}$$
By Lemma 4, we have
$$\begin{aligned}&\Vert u_{M}^{k+1} \Vert ^{2}_{1,M}\le \Vert u_{M}^{k+1} \Vert ^{2}_{1,M} +\Vert \partial _{x}u_{M}^{k+1}\Vert ^{2}_{0,\omega }\nonumber \\&\quad \le \left( \frac{B_{1}^{-1}\lambda _{2}(1-\alpha )t_{k}^{-\alpha }}{\varGamma (2-\alpha )} \Vert u_{M}^{0}\Vert ^2_{1,M}+{\widetilde{D}}_{1,\alpha } M^{-2r}+{\widetilde{D}}_{2,\alpha } \delta t^{2}\right) \nonumber \\&\qquad e^{\frac{B_{2,\alpha ,\delta t}}{B_{1}}(d_{\alpha ,0}-d_{\alpha ,k})}. \end{aligned}$$
(68)
Appendix D: The proof of Lemma 15
Setting \(v_{M}=u_{M}^{k+1}\), we get
$$\begin{aligned}&{\mathcal {A}}_{1,\alpha ,\delta t}\langle u_{M}^{k+1},u_{M}^{k+1} \rangle _{1,M}+c^{-1}_{\alpha ,\delta t}\gamma a_{\omega }\langle u_{M}^{k+1},u_{M}^{k+1}\rangle =\langle {\mathcal {P}}_{t}^{I,\alpha }u_{M}^{k},u_{M}^{k+1} \rangle _{1,M}\nonumber \\&\quad + \left\{ {\begin{array}{ll} \langle I^{c}_{M}{\mathcal {Q}}(u_{M}^{0}),u_{M}^{1} \rangle _{1,M} + \langle I^{c}_{M}F^{1},u_{M}^{1}\rangle _{1,M},&{}k=0,\\ \langle 2I^{c}_{M}{\mathcal {Q}}(u_{M}^{k})-I^{c}_{M}{\mathcal {Q}}(u_{M}^{k-1}), u_{M}^{k+1} \rangle _{1,M}+ \langle I^{c}_{M}F^{k+1},u_{M}^{k+1}\rangle _{1,M},&{}k\ge 1. \end{array} } \right. \end{aligned}$$
(69)
We know
$$\begin{aligned}&({\mathcal {A}}_{2,\alpha ,\delta t}+\lambda _{2} d_{\alpha ,1}) \langle u_{M}^{0},u_{M}^{1} \rangle _{1,M}\le (\lambda _{1}\beta _{\alpha ,\delta t}+\lambda _{2}) \Vert u_{M}^{0}\Vert ^2_{1,M}\nonumber \\&\quad + \frac{(\lambda _{1}\beta _{\alpha ,\delta t}+\lambda _{2})}{4}\Vert u_{M}^{1} \Vert ^{2}_{1,M}, \end{aligned}$$
(70)
and
$$\begin{aligned}&\langle {\mathcal {P}}_{t}^{I,\alpha }u_{M}^{k},u_{M}^{k+1} \rangle _{1,M}\le \lambda _{1}\beta _{\alpha ,\delta t} \Vert u_{M}^{k} \Vert ^2_{1,M}+ \frac{{\mathcal {A}}_{2,\alpha ,\delta t}}{4}\Vert u_{M}^{k+1} \Vert ^{2}_{1,M}\nonumber \\&\quad +\lambda _{2}\sum _{j=0}^{k-1}(d_{\alpha ,j}-d_{\alpha ,j+1}) \Vert u_{M}^{k-j}\Vert ^2_{1,M}\nonumber \\&\quad +\frac{\lambda _{2}}{4}(d_{\alpha ,1}-d_{\alpha ,k})\Vert u_{M}^{k+1} \Vert ^{2}_{1,M} +\lambda _{2} d_{\alpha ,k}\Vert u_{M}^{0}\Vert ^2_{1,M}\nonumber \\&\quad +\frac{\lambda _{2} d_{\alpha ,k}}{4}\Vert u_{M}^{k+1} \Vert ^{2}_{1,M},\quad k\ge 1. \end{aligned}$$
(71)
Using Lemmas 1, 2 and 3 and applying Eqs. (69), (70) and (71), we obtain
$$\begin{aligned}&\frac{3(\lambda _{1}\beta _{\alpha ,\delta t}+\lambda _{2})}{4}\Vert u_{M}^{1} \Vert ^{2}_{1,M}+\frac{c^{-1}_{\alpha ,\delta t}\gamma }{4}\Vert \partial _{x}u_{M}^{1}\Vert ^{2}_{0,\omega }\nonumber \\&\quad \le (\lambda _{1}\beta _{\alpha ,\delta t} +\lambda _{2}) \Vert u_{M}^{0}\Vert ^2_{1,M}\nonumber \\&\qquad +\langle I^{c}_{M}{\mathcal {Q}}(u_{M}^{0}),u_{M}^{1} \rangle _{1,M}+ \langle I^{c}_{M}F^{1},u_{M}^{1}\rangle _{1,M} \nonumber \\&\quad \le \sqrt{2}c_{1}(\lambda _{1}\beta _{\alpha ,\delta t} +\lambda _{2}) \Vert \partial _{x}u_{M}^{0}\Vert ^2_{0,\omega } \nonumber \\&\qquad +\langle I^{c}_{M} {\mathcal {Q}}(u_{M}^{0}),u_{M}^{1} \rangle _{1,M}+ \langle I^{c}_{M}F^{1},u_{M}^{1}\rangle _{1,M}\nonumber \\&\quad \le \sqrt{2}c_{1}(\lambda _{1}\beta _{\alpha ,\delta t} +\lambda _{2}) \Vert \partial _{x}u_{M}^{0}\Vert ^2_{0,\omega }\nonumber \\&\qquad + \frac{1}{(\lambda _{1}\beta _{\alpha ,\delta t}+\lambda _{2})}(\Vert I^{c}_{M}F^{1}\Vert ^{2}_{1,M}+\Vert I^{c}_{M}{\mathcal {Q}}(u_{M}^{0}) \Vert ^{2}_{1,M})\nonumber \\&\qquad +\frac{(\lambda _{1}\beta _{\alpha ,\delta t}+\lambda _{2})}{2}\Vert u_{M}^{1}\Vert ^{2}_{1,M},\quad k=0. \end{aligned}$$
(72)
We know that there exists a positive constants \(c_{2}\) such that \(|{\mathcal {Q}}({\mathcal {U}})|\le c_{2} |{\mathcal {U}}|\). Then, we conclude that
$$\begin{aligned}&\frac{(\lambda _{1}\beta _{\alpha ,\delta t}+\lambda _{2})}{4} \Vert u_{M}^{1} \Vert ^{2}_{1,M}+\frac{c^{-1}_{\alpha ,\delta t}\gamma }{4} \Vert \partial _{x}u_{M}^{1}\Vert ^{2}_{0,\omega } \\&\quad \le \frac{1}{(\lambda _{1}\beta _{\alpha ,\delta t}+\lambda _{2})}\Vert I^{c}_{M}F^{1} \Vert ^{2}_{1,M}\\&\qquad +\frac{c_{2}^{2}}{(\lambda _{1}\beta _{\alpha ,\delta t}+\lambda _{2})} \Vert u_{M}^{0}\Vert ^2_{1,M}+ \sqrt{2}c_{1}(\lambda _{1}\beta _{\alpha ,\delta t} \\&\qquad +\lambda _{2}) \Vert \partial _{x}u_{M}^{0}\Vert ^2_{0,\omega },~k=0. \end{aligned}$$
If
$$\begin{aligned}&C_{1,\alpha ,\delta t}=\min \left\{ \frac{\lambda _{1}\beta _{\alpha ,\delta t} +\lambda _{2})}{4},\frac{c^{-1}_{\alpha ,\delta t}\gamma }{4}\right\} ,\\&C_{2,\alpha ,\delta t}=\max \left\{ \frac{c_{2}^{2}}{(\lambda _{1}\beta _{\alpha ,\delta t}+\lambda _{2})},\sqrt{2}c_{1}(\lambda _{1}\beta _{\alpha ,\delta t}+\lambda _{2})\right\} , \end{aligned}$$
we can get the following inequality
$$\begin{aligned}&\Vert u_{M}^{1} \Vert ^{2}_{1,M}+\Vert \partial _{x}u_{M}^{1}\Vert ^{2}_{0,\omega } \le \frac{1}{C_{1,\alpha ,\delta t}(\lambda _{1}\beta _{\alpha ,\delta t}+\lambda _{2})}\Vert I^{c}_{M}F^{1}\Vert ^{2}_{1,M}\\&\quad +\frac{C_{2,\alpha ,\delta t}}{C_{1,\alpha ,\delta t}} (\Vert u_{M}^{0}\Vert ^2_{1,M}+ \Vert \partial _{x}u_{M}^{0}\Vert ^2_{0,\omega }). \end{aligned}$$
Now, noting Lemma 4, we obtain
$$\begin{aligned} \Vert u_{M}^{1} \Vert ^{2}_{1,M}+\Vert \partial _{x}u_{M}^{1}\Vert ^{2}_{0,\omega } \le {\mathbf {C}}_{1,\alpha ,\delta t}\Vert I^{c}_{M}F^{1}\Vert ^{2}_{1,M}, \end{aligned}$$
where \({\mathbf {C}}_{1,\alpha ,\delta t}=\frac{e^{\frac{C_{2,\alpha ,\delta t}}{C_{1}}}}{C_{1,\alpha ,\delta t}(\lambda _{1}\beta _{\alpha ,\delta t}+\lambda _{2})}\).
For \(k\ge 1\), it can also be shown
$$\begin{aligned}&\frac{3(\lambda _{1}\beta _{\alpha ,\delta t}+\lambda _{2})}{4} \Vert u_{M}^{k+1}\Vert ^{2}_{1,M}+\frac{c^{-1}_{\alpha ,\delta t}\gamma }{4}\Vert \partial _{x}u_{M}^{k+1}\Vert ^{2}_{0,\omega } \\&\quad \le \lambda _{1}\beta _{\alpha ,\delta t} \Vert u_{M}^{k}\Vert ^2_{1,M} \\&\qquad + \lambda _{2}\sum _{j=0}^{k-1}(d_{\alpha ,j}-d_{\alpha ,j+1}) \Vert u_{M}^{k-j}\Vert ^2_{1,M}+\lambda _{2} d_{\alpha ,k}\Vert u_{M}^{0}\Vert ^2_{1,M}\\&\qquad +\langle 2I^{c}_{M}{\mathcal {Q}}(u_{M}^{k})-I^{c}_{M} {\mathcal {Q}}(u_{M}^{k-1}),u_{M}^{k+1} \rangle _{1,M}\\&\qquad +\langle I^{c}_{M}F^{k+1},u_{M}^{k+1}\rangle _{1,M}\le \sqrt{2} \lambda _{1}\beta _{\alpha ,\delta t} \Vert u_{M}^{k}\Vert ^2_{0,\omega }\\&\qquad +\lambda _{2}\sum _{j=0}^{k-1}(d_{\alpha ,j}-d_{\alpha ,j+1})\Vert u_{M}^{k-j} \Vert ^2_{1,M}\\&\qquad +\lambda _{2} d_{\alpha ,k}\Vert u_{M}^{0}\Vert ^2_{1,M}+\langle 2I^{c}_{M} {\mathcal {Q}}(u_{M}^{k})-I^{c}_{M}{\mathcal {Q}}(u_{M}^{k-1}),u_{M}^{k+1} \rangle _{1,M}\\&\qquad +\langle I^{c}_{M}F^{k+1},u_{M}^{k+1}\rangle _{1,M}\\&\quad \le \sqrt{2}c_{1}\lambda _{1}\beta _{\alpha ,\delta t} \Vert \partial _{x}u_{M}^{k}\Vert ^2_{0,\omega }\\&\qquad +\lambda _{2}\sum _{j=0}^{k-1} (d_{\alpha ,j}-d_{\alpha ,j+1})\Vert u_{M}^{k-j}\Vert ^2_{1,M} +\lambda _{2} d_{\alpha ,k}\Vert u_{M}^{0}\Vert ^2_{1,M}\\&\qquad +\frac{1}{(\lambda _{1}\beta _{\alpha ,\delta t}+\lambda _{2})} (\Vert I^{c}_{M}F^{k+1}\Vert ^{2}_{1,M}+\Vert 2I^{c}_{M}{\mathcal {Q}}(u_{M}^{k})\\&\qquad -I^{c}_{M}{\mathcal {Q}}(u_{M}^{k-1})\Vert ^{2}_{1,M}) +\frac{(\lambda _{1}\beta _{\alpha ,\delta t}+\lambda _{2})}{2}\Vert u_{M}^{k+1} \Vert ^2_{1,M}\\&\quad \le \sqrt{2}c \lambda _{1}\beta _{\alpha ,\delta t} \Vert \partial _{x}u_{M}^{k}\Vert ^2_{0,\omega }+\lambda _{2}\sum _{j=1}^{k} (d_{\alpha ,k-j}-d_{\alpha ,k-j+1})\Vert u_{M}^{j}\Vert ^2_{1,M}\\&\qquad +\lambda _{2} d_{\alpha ,k}\Vert u_{M}^{0}\Vert ^2_{1,M}\\&\qquad +\frac{1}{(\lambda _{1}\beta _{\alpha ,\delta t}+\lambda _{2})}(\Vert I^{c}_{M}F^{k+1}\Vert ^{2}_{1,M}+ 8c_{2}^{2}\Vert u_{M}^{k}\Vert ^2_{1,M}\\&\qquad +2c_{2}^{2}\Vert u_{M}^{k-1}\Vert ^2_{1,M})+\frac{(\lambda _{1}\beta _{\alpha ,\delta t}+\lambda _{2})}{2}\Vert u_{M}^{k+1}\Vert ^2_{1,M}, \end{aligned}$$
and therefore
$$\begin{aligned}&\frac{(\lambda _{1}\beta _{\alpha ,\delta t}+\lambda _{2})}{4} \Vert u_{M}^{k+1}\Vert ^{2}_{1,M}+\frac{c^{-1}_{\alpha ,\delta t}\gamma }{4}\Vert \partial _{x}u_{M}^{k+1}\Vert ^{2}_{0,\omega }\nonumber \\&\quad \le \left( \lambda _{2}+\frac{1}{(\lambda _{1}\beta _{\alpha ,\delta t}+\lambda _{2})} \left( \frac{8c_{2}^{2}}{(1-d_{\alpha ,1})}+\frac{2c_{2}^{2}}{(d_{\alpha ,1} -d_{\alpha ,2})}\right) \right) \nonumber \\&\qquad \sum _{j=1}^{k}(d_{\alpha ,k-j}-d_{\alpha ,k-j+1}) \Vert u_{M}^{j}\Vert ^2_{1,M}\nonumber \\&\qquad +\frac{\sqrt{2}c_{1}\lambda _{1}\beta _{\alpha ,\delta t}}{1-d_{\alpha ,1}} \sum _{j=1}^{k}(d_{\alpha ,k-j}-d_{\alpha ,k-j+1}) \Vert \partial _{x}u_{M}^{j} \Vert ^2_{0,\omega }\nonumber \\&\qquad +\lambda _{2} d_{\alpha ,k}\Vert u_{M}^{0}\Vert ^2_{1,M} +\frac{1}{(\lambda _{1}\beta _{\alpha ,\delta t}+\lambda _{2})}\Vert I^{c}_{M}F^{k+1}\Vert ^{2}_{1,M}. \end{aligned}$$
(73)
If
$$\begin{aligned} C_{1,\alpha ,\delta t}&=\min \left\{ \frac{(\lambda _{1}\beta _{\alpha ,\delta t} +\lambda _{2})}{4},\frac{c^{-1}_{\alpha ,\delta t}\gamma }{4}\right\} ,\\ C_{2,\alpha ,\delta t}&=\max \left\{ \left( \lambda _{2}+\frac{1}{(\lambda _{1}\beta _{\alpha ,\delta t}+\lambda _{2})}\right. \right. \\&\quad \left. \left. \times \left( \frac{8c_{2}^{2}}{(1-d_{\alpha ,1})} +\frac{2c_{2}^{2}}{(d_{\alpha ,1}-d_{\alpha ,2})}\right) \right) ,\right. \\&\quad \left. \frac{\sqrt{2}c_{1}\lambda _{1}\beta _{\alpha ,\delta t}}{1-d_{\alpha ,1}}\right\} , \end{aligned}$$
we can get the following inequality
$$\begin{aligned}&\Vert u_{M}^{k+1} \Vert ^{2}_{1,M}+\Vert \partial _{x}u_{M}^{k+1}\Vert ^{2}_{0,\omega } \le \sum _{j=1}^{k}\frac{C_{2,\alpha ,\delta t}}{C_{1,\alpha ,\delta t}} (d_{\alpha ,k-j}-d_{\alpha ,k-j+1})\\&\quad (\Vert u_{M}^{j}\Vert ^2_{1,M} + \Vert \partial _{x}u_{M}^{j}\Vert ^2_{0,\omega })\\&\quad +C_{1,\alpha ,\delta t}^{-1}\lambda _{2} d_{\alpha ,k}\Vert u_{M}^{0}\Vert ^2_{1,M}\\&\quad +\frac{1}{C_{1,\alpha ,\delta t}(\lambda _{1}\beta _{\alpha ,\delta t} +\lambda _{2})}\Vert I^{c}_{M}F^{k+1}\Vert ^{2}_{1,M}. \end{aligned}$$
Noting Lemma 4, we have
$$\begin{aligned}&\Vert u_{M}^{k+1} \Vert ^{2}_{1,M}+\Vert \partial _{x}u_{M}^{k+1}\Vert ^{2}_{0,\omega }\\&\quad \le {\mathbf {C}}_{2,\alpha ,\delta t} \left( \Vert u_{M}^{0}\Vert ^2_{1,M}+\Vert I^{c}_{M}F^{k+1}\Vert ^{2}_{1,M}\right) \\&\qquad e^{{\mathbf {C}}_{3,\alpha ,\delta t}(d_{\alpha ,0}-d_{\alpha ,k})}, \end{aligned}$$
where \({\mathbf {C}}_{2,\alpha ,\delta t}=\max \{C_{1,\alpha ,\delta t}^{-1}\lambda _{2} d_{\alpha ,k},\frac{1}{C_{1,\alpha ,\delta t}(\lambda _{1}\beta _{\alpha ,\delta t}+\lambda _{2})}\}\) and \({\mathbf {C}}_{3,\alpha ,\delta t}=\frac{C_{2,\alpha ,\delta t}}{C_{1,\alpha ,\delta t}}\).
