E^{n+1})_{i,j} - \rho_\alpha({\mathcal{B}}_{i,j}(U^{n},M^{n+1})-{\mathcal{B}}_{i,j}({\tilde U}^{n},{\tilde M}^{n+1})) =\sum_{k=0}^n c^n_k E^k+{\rho_\alpha}b^{n+1}.$$ Multiplying the previous equation by $E^{n+1}_{i,j}$, summing over $i$ and $j$ and using the identities and , we get $$\label{eq:conv_es1}
\begin{split}
&\|(E^{n+1},[D_hE^{n+1}])\|_2^2-{\rho_\alpha}h^2 \sum_{i,j} E^{n+1}_{i,j} [D_hE^{n+1}]_{i,j}\cdot g_q(x_{i,j}, [D U^n]_{i,j})\\
&=h^2\sum_{k=0}^n c^n_k (E^k,E^{n+1})+{\rho_\alpha}h^2(b^{n+1}, E^{n+1})\\
&+{\rho_\alpha}h^2  \sum_{i,j}{\tilde M}_{i,j}^{n+1}[DE^{n+1}]_{i,j}\cdot(g_q(x_{i,j}, [D_hU^n]_{i,j})-g_q(x_{i,j}, [D_h{\tilde U}^n]_{i,j})).
\end{split}$$ Taking into account , we estimate for ${{\Delta t}}$ sufficiently small $$\left|{\rho_\alpha}h^2 \sum_{i,j} E^{n+1}_{i,j} [D_hE^{n+1}]_{i,j}\cdot g_q(x_{i,j}, [D U^n]_{i,j})\right|\le 
\frac 1 2\|(E^{n+1},[D_hE^{n+1}])\|_2^2.$$ Using again and since $(u,m)$ is a classical solution to , we have for all $\eta>0$ (see[@acc Equation (4.5)]) $$\begin{aligned}
&\left|{\tilde M}_{i,j}^{n+1}[DE^{n+1}]_{i,j}\cdot(g_q(x_{i,j}, [D_hU^n]_{i,j})-g_q(x_{i,j}, [D_h{\tilde U}^n]_{i,j}))\right|\\
&\le  C \Big(\frac{1}{\eta} |[D_hE^{n+1}]_{i,j}|^2+\eta |[D_hU^n]_{i,j}-[D_h{\tilde U}^n]_{i,j}|^\beta\Big)\end{aligned}$$ Plugging the previous estimates in and dividing by $\|(E^{n+1},[D_hE^{n+1}])\|_2^2$, we get $$\label{eq:conv_es2}
\|(E^{n+1},[D_hE^{n+1}])\|_2\le C\left(\sum_{k=0}^n c^n_k \|E^k\|_2+{\rho_\alpha}\varepsilon^{n+1}\right)$$ where $$\varepsilon^{n+1}=\|b^{n+1}\|_\infty+  h^2  \sum_{i,j}|[D_hU^n]_{i,j}-[D_h{\tilde U}^n]_{i,j}|^\beta.$$ and the constant $C$ is independent of $h,{{\Delta t}}$.\
Following [@lx Theorem 3.2], we rewrite as $$\label{eq:conv_es3}
\|(E^{n+1},[D_hE^{n+1}])\|_2\le C\Big((1-B_1)\|E^n\|_2+\sum_{k=1}^{n-1}(B_k-B_{k+1})\|E^{n-k}\|_2
+B_n\|E^0\|_2+{\rho_\alpha}\varepsilon^{n+1}\Big),$$ where $B_k=(k+1)^{1-\alpha}-k^{1-\alpha}$, $k=0,\dots,n$. Recall that, since $M^0={\tilde M}^0$, then $E^0\equiv 0$. We claim that for $n=1,\dots,N$, $$\label{eq:conv_es4}
\|(E^{n},[D_hE^{n}])\|_2\le C{\rho_\alpha}B_{n-1}\max_{0\le k\le N}\varepsilon^k.$$ For $n=1$, by we have $$\|(E^{1},[D_hE^{1}])\|_2\le C{\rho_\alpha}\varepsilon^1\le C {\rho_\alpha}B_0^{-1}\max_{0\le k\le N}\varepsilon^k,$$ since $B_0=1$. Assuming that is true for $n$, by and the fact that $B_k^{-1}/B_{k+1}^{-1}<1$, we obtain $$\begin{aligned}
&\|(E^{n+1},[D_hE^{n+1}])\|_2\le \left[ (1-B_1)B_k^{-1}+\sum_{k=1}^{n-1}(B_k-B_{k+1})B^{-1}_{n-k-1}\right]C{\rho_\alpha}\max_{0\le k\le N}\varepsilon^k\\
&+C{\rho_\alpha}\max_{0\le k\le N}\varepsilon^k
\le \left[ (1-B_1) 