\alpha\|u_{h_{k+1}}^{k+1}-u_{h_{k+1}}^{\ast}\|_{L^{2}(\Omega)}^{2},\\
&\left\langle \lambda_{h_{k+1}}^{k}+\sigma r_{h_{k+1}}^{k+1}, z_{h_{k+1}}^{k+1}-z_{h_{k+1}}^{\ast} \right\rangle\geq 0.\end{aligned}$$ Then adding the above two equalities we obtain $$\label{estimate}
\left\langle \delta_{u,h_{k+1}}^{k+1},  u_{h_{k+1}}^{k+1}-u_{h_{k+1}}^{\ast}\right\rangle-\left\langle \lambda_{h_{k+1}}^{k}+\sigma r_{h_{k+1}}^{k+1}-\lambda_{h_{k+1}}^{\ast}, r_{h_{k+1}}^{k+1}\right\rangle-\sigma\left\langle z_{h_{k+1}}^{k+1}-z_{h_{k+1}}^{k}, u_{h_{k+1}}^{k+1}-u_{h_{k+1}}^{\ast} \right\rangle \geq \alpha\|u_{h_{k+1}}^{k+1}-u_{h_{k+1}}^{\ast}\|_{L^{2}(\Omega)}^{2}.$$ Next, we estimate the last two terms on the left side separately, $$\begin{aligned}
\label{estimate1}
&\left\langle \lambda_{h_{k+1}}^{\ast}-(\lambda_{h_{k+1}}^{k}+\sigma r_{h_{k+1}}^{k+1}), r_{h_{k+1}}^{k+1}\right\rangle
\\
=&\frac{1}{\tau\sigma}\left\langle \lambda_{h_{k+1}}^{\ast}-\lambda_{h_{k+1}}^{k}, \lambda_{h_{k+1}}^{k+1}-\lambda_{h_{k+1}}^{k} \right\rangle-\sigma\|r_{h_{k+1}}^{k+1}\|_{L^{2}(\Omega)}^{2}\\
=&\frac{1}{2\tau\sigma}(\|\lambda_{h_{k+1}}^{\ast}-\lambda_{h_{k+1}}^{k}\|_{L^{2}(\Omega)}^{2}+\|\lambda_{h_{k+1}}^{k+1}-\lambda_{h_{k+1}}^{k}\|_{L^{2}(\Omega)}^{2}-\|\lambda_{h_{k+1}}^{\ast}-\lambda_{h_{k+1}}^{k+1}\|_{L^{2}(\Omega)}^{2})-\sigma\|r_{h_{k+1}}^{k+1}\|_{L^{2}(\Omega)}^{2}\\
=&\frac{1}{2\tau\sigma}(\|\lambda_{h_{k+1}}^{\ast}-\lambda_{h_{k+1}}^{k}\|_{L^{2}(\Omega)}^{2}-\|\lambda_{h_{k+1}}^{\ast}-\lambda_{h_{k+1}}^{k+1}\|_{L^{2}(\Omega)}^{2})+\frac{(\tau-2)\sigma}{2}\|r_{h_{k+1}}^{k+1}\|_{L^{2}(\Omega)}^{2},
\end{aligned}$$ where we used the equality (\[equ1\]).

Moreover, by employing the equality (\[equ2\]) and using $u^{\ast}_{h_{k+1}}=z^{\ast}_{h_{k+1}}$, we have $$\begin{aligned}
\label{estimate2}
&\sigma\left\langle z_{h_{k+1}}^{k+1}-z_{h_{k+1}}^{k}, u_{h_{k+1}}^{\ast}-u_{h_{k+1}}^{k+1} \right\rangle \\
=&\sigma\left\langle z_{h_{k+1}}^{k+1}-z_{h_{k+1}}^{k}, -u_{h_{k+1}}^{k+1}-(-z_{h_{k+1}}^{\ast}) \right\rangle \\
=&\frac{\sigma}{2}(\|r_{h_{k+1}}^{k+1}\|_{L^{2}(\Omega)}^{2}+\|z_{h_{k+1}}^{k}-z_{h_{k+1}}^{\ast}\|_{L^{2}(\Omega)}^{2}-\|z_{h_{k+1}}^{k+1}-z_{h_{k+1}}^{\ast}\|_{L^{2}(\Omega)}^{2}
-\|z_{h_{k+1}}^{k}-u_{h_{k+1}}^{k+1}\|_{L^{2}(\Omega)}^{2}).
\end{aligned}$$ Then, substituting (\[estimate1\]), (\[estimate2\]) into (\[estimate\]), we can get the assertion of Proposition \[lmm:complexity2\].

\[lmm:complexity3\] Let $\left\{\left(u^{k}_{h_{k}}, z^{k}_{h_{k}}, \lambda^{k}_{h_{k}}\right)\right\}$ be the sequence generated by Algorithm \[alg:multi-level ADMM algorithm\], $\left\{\left(u^{\ast}_{h_{k}}, z^{\ast}_{h_{k}}, \lambda^{\ast}_{h_{k}}\right)\right\}$ denotes the KKT point of the discretized reduced problem and $\tilde{u}_{h_{k+1}}^{k+1}$, $\tilde{z}^{k+1}_{h_{k+1}}$ defined in (\[tildeu\]), (\[tildez\]), respectively. Then for $k\geq 0$ we have $$\begin{aligned}
&\frac{1}{2\tau\sigma}\|\lambda^{\ast}_{h_{k+1}}-\lambda^{k}_{h_{k+1}}\|_{L^{2}(\Omega)}^{2}+\frac{\sigma}{2}\|z^{\ast}_{h_{k+1}}-z^{k}_{h_{k+1}}\|_{L^{2}(\Omega)}^{2}-\frac{1}{2\tau\sigma}\|\lambda^{\ast}_{h_{k+1}}-\tilde \lambda^{k+1}_{h_{k+1}}\|_{L^{2}(\Omega)}^{2}-\frac{\sigma}{2}\|z^{\ast}_{h_{k+1}}-\tilde z^{k+1}_{h_{k+1}}\|_{L^{2}(\Omega)}^{2}\\
\geq & \alpha\|\tilde u^{k+1}_{h_{k+1}}-u^{\ast}_{h_{k+1}}\|_{L^{2}(\Omega)}^{2}+\frac{(3-\tau)\sigma}{2}\|\tilde u^{k+1}_{h_{k+1}}-\tilde z^{k+1}_{h_{k+1}}\|_{L^{2}(\Omega)}^{2}+\frac{\sigma}{2}\|\tilde u^{k+1}_{h_{k+1}}-z^{k}_{h_{k+1}}\|_{L^{2}(\Omega)}^{2}.\end{aligned}$$

For the proof of Proposition \[lmm:complexity3\], by substituting $\tilde u^{k+1}_{h_{k+1}}$ and $\tilde z^{k+1}_{h_{k+1}}$ for $u^{k+1}_{h_{k+1}}$ and $z^{k+1}_{h_{k+1}}$ in the proof of Proposition \[lmm:complexity2\], we can get the assertion.

Finally, based on the above results, the convergence results of Algorithm \[alg:multi-level ADMM algorithm\] is given by the following theorem.

\[convergence theorem\] 