$u$-subproblem, we have $$S^{\ast}[S(u_{h_{k+1}}^{k+1}+y_{r})-y_{d}]+\alpha u_{h_{k+1}}^{k+1}+\lambda^{k}+\sigma(u_{h_{k+1}}^{k+1}-z^{k})=\delta_{u}^{k+1}.$$ Moreover, by the optimality conditions of the $u-$subproblem corresponding to $u_{h_{k+1}}^{k+1}$ and $\bar u_{h_{k+1}}^{k+1}$ in Algorithm \[alg:multi-level ADMM algorithm\] and multi-level discretized ADMM, we have $$\begin{aligned}
&S^{\ast}_{h_{k+1}}[S_{h_{k+1}}(u_{h_{k+1}}^{k+1}+I_{h_{k+1}}y_{r})-I_{h_{k+1}}y_{d}]+\alpha u_{h_{k+1}}^{k+1}+\lambda^{k}_{h_{k+1}}+\sigma(u_{h_{k+1}}^{k+1}-z^{k}_{h_{k+1}})=\delta_{u,h_{k+1}}^{k+1},\\
&S^{\ast}_{h_{k+1}}[S_{h_{k+1}}(\bar u_{h_{k+1}}^{k+1}+I_{h_{k+1}}y_{r})-I_{h_{k+1}}y_{d}]+\alpha \bar u_{h_{k+1}}^{k+1}+\bar \lambda^{k}_{h_{k+1}}+\sigma(\bar u_{h_{k+1}}^{k+1}-\bar z^{k}_{h_{k+1}})=0.\end{aligned}$$

Then we know from the four equalities above that $$\begin{aligned}
\delta_{u}^{k+1}=&\delta_{u}^{k+1}-\delta_{u,h_{k+1}}^{k+1}+\delta_{u,h_{k+1}}^{k+1}\\
=&\delta_{u,h_{k+1}}^{k+1}+S^{\ast}S(u_{h_{k+1}}^{k+1}-\bar u^{k+1})+(\alpha+\sigma)(u_{h_{k+1}}^{k+1}-\bar u^{k+1})+(\lambda^{k}-\bar \lambda^{k})-\sigma(z^{k}-\bar z^{k})\\
&+S^{\ast}_{h_{k+1}}S_{h_{k+1}}(\bar u_{h_{k+1}}^{k+1}-u_{h_{k+1}}^{k+1})+(\alpha+\sigma)(\bar u_{h_{k+1}}^{k+1}-u_{h_{k+1}}^{k+1})+(\bar \lambda^{k}_{h_{k+1}}-\lambda^{k}_{h_{k+1}})-\sigma(\bar z^{k}_{h_{k+1}}-z^{k}_{h_{k+1}})\\
=&\delta_{u,h_{k+1}}^{k+1}+(\alpha+\sigma)(\bar u_{h_{k+1}}^{k+1}-\bar u^{k+1})+
(\lambda^{k}-\lambda^{k}_{h_{k+1}})-\sigma(z^{k}-z^{k}_{h_{k+1}})+(\bar \lambda^{k}_{h_{k+1}}-\bar \lambda^{k})-\sigma(\bar z^{k}_{h_{k+1}}-\bar z^{k})\\
&+(S^{\ast}S-S^{\ast}_{h_{k+1}}S_{h_{k+1}})(u_{h_{k+1}}^{k+1}-\bar u_{h_{k+1}}^{k+1})+S^{\ast}S(\bar u_{h_{k+1}}^{k+1}-\bar u^{k+1}).
\end{aligned}$$ Moreover, we have the estimate $$\begin{aligned}
\label{equ:estimate}
&\|(S^{\ast}S-S^{\ast}_{h_{k+1}}S_{h_{k+1}})(u_{h_{k+1}}^{k+1}-\bar u_{h_{k+1}}^{k+1})\|_{L^{2}(\Omega)}\\
=&\|(S^{\ast}S-S^{\ast}S_{h_{k+1}})(u_{h_{k+1}}^{k+1}-\bar u_{h_{k+1}}^{k+1})\|_{L^{2}(\Omega)}
+\|(S^{\ast}S_{h_{k+1}}-S^{\ast}_{h_{k+1}}S_{h_{k+1}})(u_{h_{k+1}}^{k+1}-\bar u_{h_{k+1}}^{k+1})\|_{L^{2}(\Omega)}\\
\leq &\|S^{\ast}\|\|S-S_{h_{k+1}}\|\|u_{h_{k+1}}^{k+1}-\bar u_{h_{k+1}}^{k+1}\|_{L^{2}(\Omega)}+\|S^{\ast}-S^{\ast}_{h_{k+1}}\|\|S_{h_{k+1}}\|\|u_{h_{k+1}}^{k+1}-\bar u_{h_{k+1}}^{k+1}\|_{L^{2}(\Omega)}\\
= &O(h_{k+1}^{2}).
\end{aligned}$$

Then we know from (\[equ:estimate\]), Lemma \[lem:continuous exact and discretized exact\] and Lemma \[lem:continuous inexact and discretized inexact\] that there exists constant C such that $$\label{equ: delta}
\begin{aligned}
\sum_{k=0}^{\infty}\|\delta_{u}^{k+1}\|_{L^{2}(\Omega)}
\leq \sum_{k=0}^{\infty}\|\delta_{u,h_{k+1}}^{k+1}\|_{L^{2}(\Omega)}+C\sum_{k=0}^{\infty}\left(h_{k+1}+\|\delta_{u,h_{k+1}}^{k+1}\|_{L^{2}(\Omega)}\right).
\end{aligned}$$ We know from Algorithm \[alg:multi-level ADMM algorithm\] that the mesh sizes $\lbrace h_{k} \rbrace_{k=0}^{\infty}$ of each iteration satisfy $\sum_{k=0}^{\infty}h_{k+1}<\infty.$ The error vector of the multi-level ADMM satisfy $\sum_{k=0}^{\infty}\|\delta_{u,h_{k+1}}^{k+1}\|_{L^{2}(\Omega)}\leq\sum_{k=0}^{\infty}\xi_{k+1}<\infty,$ where $\xi_{k+1}$ is the upper bound of $\delta_{u,h_{k+1}}^{k+1}$, i.e. $\|\delta_{u,h_{k+1}}^{k+1}\|_{L^{2}(\Omega)}\leq \xi_{k+1}$. Thus we have $$\sum_{k=0}^{\infty}\|\delta_{u}^{k+1}\|_{L^{2}(\Omega)}
< \infty.$$ For the discretization error and the iteration error, Algorithm \[alg:multi-level ADMM algorithm\] can be considered as an inexact ADMM algorithm in function space, then we know from Theorem \[convergence 