induction to prove the conclusion. The proof is similar to Lemma \[lem:continuous exact and discretized exact\], here we do not talk about it in detail.

To prove the convergence of the mADMM algorithm, let $(\tilde{u}_{h_{k+1}}^{k+1},\tilde{z}^{k+1}_{h_{k+1}})$ represents the exact solutions of the $(k+1)$th iteration of Algorithm \[alg:multi-level ADMM algorithm\]: $$\begin{aligned}
\label{tildeu}
&\tilde{u}_{h_{k+1}}^{k+1}:={\rm argmin}{\hat J}(u_{h_{k+1}})+\langle\lambda_{h_{k+1}}^{k},u_{h_{k+1}}-z_{h_{k+1}}^{k}\rangle+\dfrac{\sigma}{2}\|u_{h_{k+1}}-z_{h_{k+1}}^{k}\|_{L^{2}(\Omega)}^{2},\\
\label{tildez}
&\tilde{z}^{k+1}_{h_{k+1}}:={\rm argmin}\delta_{U_{ad,h}}(z_{h_{k+1}})+\langle{\lambda}^{k}_{h_{k+1}},\tilde{u}^{k+1}_{h_{k+1}}-z_{h_{k+1}}\rangle+\dfrac{\sigma}{2}\|\tilde{u}^{k+1}_{h_{k+1}}-z_{h_{k+1}}\|_{L^{2}(\Omega)}^{2}.\end{aligned}$$ The following lemma gives the gap between $(\tilde{u}_{h_{k+1}}^{k+1}, \tilde{z}_{h_{k+1}}^{k+1})$ and $({u}_{h_{k+1}}^{k+1}, {z}_{h_{k+1}}^{k+1})$.

([@Song2017Fe], Lemma 4.4) \[lmm:tildeu\] For any $k\geqslant 0$, we have $$\begin{aligned}
&\|\tilde{u}_{h_{k+1}}^{k+1}-u_{h_{k+1}}^{k+1}\|_{L^{2}(\Omega)}\leq\rho\|\delta_{u,h_{k+1}}^{k+1}\|_{L^{2}(\Omega)},\\
&\|\tilde{z}_{h_{k+1}}^{k+1}-z_{h_{k+1}}^{k+1}\|_{L^{2}(\Omega)}\leq\rho\|\delta_{u,h_{k+1}}^{k+1}\|_{L^{2}(\Omega)},\end{aligned}$$ where $\rho:=\|[S^{\ast}_{h_{k+1}}S_{h_{k+1}}+(\alpha+\sigma)I]^{-1}\|_{{L^{2}(\Omega)}\rightarrow {L^{2}(\Omega)}}$.

For the convenience of analyzing the non-ergodic iteration complexity, let $(u^{\ast}_{h_{k+1}}, z^{\ast}_{h_{k+1}}, \lambda^{\ast}_{h_{k+1}})$ denotes the KKT point of the discretized reduced problem with the mesh size $h_{k+1}$ $$\label{discretized reduced problem}
\begin{aligned}
\min \limits_{u_{h_{k+1}},z_{h_{k+1}}}
&\frac{1}{2}\|S_{h_{k+1}}(u_{h_{k+1}}+I_{h_{k+1}}y_{r})-I_{h_{k+1}}y_d\|_{L^{2}(\Omega_{h})}^{2}
+\frac{\alpha}{2}\|u_{h_{k+1}}\|_{{L^{2}(\Omega_{h})}}^{2}+\delta_{U_{ad,h_{k+1}}}(z_{h_{k+1}})\\
{\rm s.t.}  \ \   \ \ \ & u_{h_{k+1}}=z_{h_{k+1}}.
\end{aligned} \tag{DRP}$$ Moreover, we provide a lemma and two propositions which are essential for analyzing the iteration complexity of our mADMM.

([@Chen2017], Lemma 6.1) \[lmm:complexity1\] If a sequence $\left\{a_{i}\right\} \in \mathbb{R}$ satisfies the following conditions: $$a_{i} \geq 0 \  for \ any \ i \geq 0 \ \ and \ \ \sum_{i=0}^{\infty} a_{i}=\bar a<\infty .$$ Then we have $\min_{i=1,2, \cdots, k}\left\{a_{i}\right\} \leq \frac{\overline{a}}{k}$ and $\lim_{k \rightarrow \infty}\left\{k \cdot \min _{i=1,2, \cdots, k}\left\{a_{i}\right\}\right\}=0$.

