\[lmm:complexity2\] Let $\left\{\left(u^{k}_{h_{k}}, z^{k}_{h_{k}}, \lambda^{k}_{h_{k}}\right)\right\}$ be the sequence generated by Algorithm \[alg:multi-level ADMM algorithm\] and $\left\{\left(u^{\ast}_{h_{k}}, z^{\ast}_{h_{k}}, \lambda^{\ast}_{h_{k}}\right)\right\}$ denotes the KKT point of the discretized reduced problem. Then for $k\geq 0$ we have $$\begin{aligned}
&\left\langle\delta^{k+1}_{u,h_{k+1}}, u^{k+1}_{h_{k+1}}-u^{\ast}_{h_{k+1}}\right\rangle+\frac{1}{2\tau\sigma}\|\lambda^{\ast}_{h_{k+1}}-\lambda^{k}_{h_{k+1}}\|_{L^{2}(\Omega)}^{2}+\frac{\sigma}{2}\|z^{\ast}_{h_{k+1}}-z^{k}_{h_{k+1}}\|_{L^{2}(\Omega)}^{2}\\
&-\frac{1}{2\tau\sigma}\|\lambda^{\ast}_{h_{k+1}}-\lambda^{k+1}_{h_{k+1}}\|_{L^{2}(\Omega)}^{2}-\frac{\sigma}{2}\|z^{\ast}_{h_{k+1}}-z^{k+1}_{h_{k+1}}\|_{L^{2}(\Omega)}^{2}\\
\geq &\alpha\|u^{k+1}_{h_{k+1}}-u^{\ast}_{h_{k+1}}\|_{L^{2}(\Omega)}^{2}+\frac{(3-\tau)\sigma}{2}\|u^{k+1}_{h_{k+1}}-z^{k+1}_{h_{k+1}}\|_{L^{2}(\Omega)}^{2}+\frac{\sigma}{2}\|u^{k+1}_{h_{k+1}}-z^{k}_{h_{k+1}}\|_{L^{2}(\Omega)}^{2}.\end{aligned}$$

First, for any $f_{1}, f_{1}^{\prime}, f_{2}, f_{2}^{\prime}\in L^{2}(\Omega)$, we have the following two important equalities hold $$\begin{aligned}
\label{equ1}
&\langle f_{1}, f_{2}\rangle_{L^{2}(\Omega)}=\frac{1}{2}\left(\|f_{1}\|_{L^{2}(\Omega)}^{2}+\|f_{2}\|_{L^{2}(\Omega)}^{2}-\|f_{1}-f_{2}\|_{L^{2}(\Omega)}^{2}\right)
=\frac{1}{2}\left(\|f_{1}+f_{2}\|_{L^{2}(\Omega)}^{2}-\|f_{1}\|_{L^{2}(\Omega)}^{2}-\|f_{2}\|_{L^{2}(\Omega)}^{2}\right),\\
\label{equ2}
&\left\langle f_{1}-f_{1}^{\prime}, f_{2}-f_{2}^{\prime}\right\rangle_{L^{2}(\Omega)}=\frac{1}{2}\left(\|f_{1}+f_{2}\|_{L^{2}(\Omega)}^{2}+\left\|f_{1}^{\prime}+f_{2}^{\prime}\right\|_{L^{2}(\Omega)}^{2}-\left\|f_{1}+f_{2}^{\prime}\right\|_{L^{2}(\Omega)}^{2}-\left\|f_{1}^{\prime}+f_{2}\right\|_{L^{2}(\Omega)}^{2}\right).\end{aligned}$$ The proof of the above two equalities can be easily obtained by the definition of $L^{2}-$norm.

By the optimality conditions of the $u$-subproblem and $z$-subproblem corresponding to $u_{h_{k+1}}^{k+1}$ and $z_{h_{k+1}}^{k+1}$, we have $$\begin{aligned}
\label{optimality1}
&\nabla {\hat J_{h_{k+1}}}(u_{h_{k+1}}^{k+1})=S^{\ast}_{h_{k+1}}[S_{h_{k+1}}(u_{h_{k+1}}^{k+1}+I_{h_{k+1}}y_{r})-I_{h_{k+1}}y_{d}]+\alpha u^{k+1}_{h_{k+1}}=\delta_{u,h_{k+1}}^{k+1}-(\lambda^{k}_{h_{k+1}}+\sigma(u^{k+1}_{h_{k+1}}-z^{k}_{h_{k+1}})),\\
&\lambda^{k}_{h_{k+1}}+\sigma(u^{k+1}_{h_{k+1}}-z^{k+1}_{h_{k+1}}))\in \partial
\delta_{U_{ad,h_{k+1}}}(z_{h_{k+1}}^{k+1}).\end{aligned}$$ Moreover, $\left\{\left(u^{\ast}_{h_{k}}, z^{\ast}_{h_{k}}, \lambda^{\ast}_{h_{k}}\right)\right\}$ denotes the KKT point of the discretized reduced problem, so it satisfies the following KKT system $$\begin{aligned}
\label{optimality2}
&\nabla {\hat J_{h_{k+1}}}(u_{h_{k+1}}^{\ast})=S^{\ast}_{h_{k+1}}[S_{h_{k+1}}(u_{h_{k+1}}^{\ast}+I_{h_{k+1}}y_{r})-I_{h_{k+1}}y_{d}]+\alpha u^{\ast}_{h_{k+1}}=-\lambda^{\ast}_{h_{k+1}},\\
&\lambda^{\ast}_{h_{k+1}}\in \partial
\delta_{U_{ad,h_{k+1}}}(z_{h_{k+1}}^{\ast}),\\
&u^{\ast}_{h_{k+1}}=z^{\ast}_{h_{k+1}}.\end{aligned}$$ Then by combining (\[optimality1\]) and (\[optimality2\]), we obtain $$\begin{aligned}
&\left\langle \nabla {\hat J_{h_{k+1}}}(u_{h_{k+1}}^{k+1})-\nabla {\hat J_{h_{k+1}}}(u_{h_{k+1}}^{\ast}), u_{h_{k+1}}^{k+1}-u_{h_{k+1}}^{\ast} \right\rangle\\
=&\left\langle S^{\ast}_{h_{k+1}}S_{h_{k+1}}(u_{h_{k+1}}^{k+1}-u_{h_{k+1}}^{\ast})+\alpha(u_{h_{k+1}}^{k+1}-u_{h_{k+1}}^{\ast}), u_{h_{k+1}}^{k+1}-u_{h_{k+1}}^{\ast}  \right\rangle\\
=&\|S_{h_{k+1}}(u_{h_{k+1}}^{k+1}-u_{h_{k+1}}^{\ast})\|_{L^{2}(\Omega)}^{2}+\alpha\|u_{h_{k+1}}^{k+1}-u_{h_{k+1}}^{\ast}\|_{L^{2}(\Omega)}^{2}.
\end{aligned}$$ Moreover, the subdifferential operator $\partial
\delta_{U_{ad,h_{k+1}}}(z)$ is a maximal monotone operator, so the following inequality holds, $$\left\langle \partial
\delta_{U_{ad,h_{k+1}}}(z_{h_{k+1}}^{k+1})-\partial
\delta_{U_{ad,h_{k+1}}}(z_{h_{k+1}}^{\ast}), z_{h_{k+1}}^{k+1}-z_{h_{k+1}}^{\ast} \right\rangle\geq 0.$$ For the convenience of analyzing, we define $r_{h_{k+1}}^{k+1}=u_{h_{k+1}}^{k+1}-z_{h_{k+1}}^{k+1}$. Therefore, we can derive that $$\begin{aligned}
&\left\langle \delta_{u,h_{k+1}}^{k+1}-(\tilde \lambda_{h_{k+1}}^{k}+\sigma(z_{h_{k+1}}^{k+1}-z_{h_{k+1}}^{k}))+\lambda_{h_{k+1}}^{\ast},  u_{h_{k+1}}^{k+1}-u_{h_{k+1}}^{\ast}\right\rangle\geq 