$(u^{0}, z^{0}, \lambda^{0})$ and the optimal solution $(u^{\ast}, z^{\ast}, \lambda^{\ast})$ such that for $k\geq 1$, $$\min_{1\leq i\leq k}{R(u^{i}, z^{i}, \lambda^{i})}\leq \frac{C}{k}, \ \ \lim \limits_{k\rightarrow \infty}(k\times \min_{1\leq i\leq k}{R(u^{i}, z^{i}, \lambda^{i})})=0,$$ where the function $R: (u, z, \lambda)\rightarrow [0,\infty)$ defined as $$R(u, z, \lambda):= \|\nabla {\hat J}(u)+\lambda\|^{2}_{L^{2}(\Omega)}+{\rm dist}^{2}(0, -\lambda+\partial \delta_{U_{ad}}(z))+\|u-z\|^{2}_{L^{2}(\Omega)}.$$

For the convenience of proving the convergence results, we give the iterative scheme of the multi-level discretized ADMM for (\[eqn:reduced problem with linear constraint\]) and present a lemma to measure the gap between the solution sequences obtained by the ADMM in function space and the multi-level discretized ADMM.

Given $(u^{0},z^{0},{\lambda}^{0})\in L^{2}(\Omega)\times {\rm dom}(\delta_{U_{ad}}(\cdot))\times L^{2}(\Omega)$, parameters $\sigma>0$, $\tau \in \left(0,\dfrac{1+\sqrt{5}}{2}\right)$. The mesh sizes $\lbrace h_{k} \rbrace_{k=0}^{\infty}$ of each iteration satisfy $\sum_{k=0}^{\infty}h_{k+1}< \infty.$ Then the iterative format of the multi-level discretized ADMM algorithm is as follows: $$\left\{\begin{aligned}
\bar u^{k+1}_{h_{k+1}}&={\rm argmin}\hat J_{h_{k+1}}(u_{h_{k+1}})+\langle\bar{\lambda}_{h_{k+1}}^{k},u_{h_{k+1}}-\bar{z}_{h_{k+1}}^{k}\rangle_{L^{2}(\Omega)}+\dfrac{\sigma}{2}\|u_{h_{k+1}}-\bar{z}_{h_{k+1}}^{k}\|_{L^{2}(\Omega)}^{2},\\
\bar z^{k+1}_{h_{k+1}}&={\rm argmin} \delta_{U_{ad,h_{k+1}}}(z_{h_{k+1}})+\langle {\bar \lambda}_{h_{k+1}}^{k},\bar u^{k+1}_{h_{k+1}}-z_{h_{k+1}}\rangle_{L^{2}(\Omega)}+\dfrac{\sigma}{2}\|\bar u^{k+1}_{h_{k+1}}-z_{h_{k+1}}\|_{L^{2}(\Omega)}^{2},\\
\bar \lambda^{k+1}_{h_{k+1}}&=I_{h_{k+1}}\bar \lambda^{k}_{h_{k}}+\sigma(\bar u^{k+1}_{h_{k+1}}-\bar z^{k+1}_{h_{k+1}}).
\end{aligned} \right.$$

\[lem:continuous exact and discretized exact\] Let the initial point be $(u^{0},z^{0};\lambda^{0})\in L^{2}(\Omega)\times {\rm dom}(\delta_{U_{ad}}(\cdot))\times L^{2}(\Omega)$, then $\|\bar{u}^{k+1}-\bar{u}^{k+1}_{h_{k+1}}\|_{L^{2}(\Omega)}=\|\bar{z}^{k+1}-\bar{z}^{k+1}_{h_{k+1}}\|_{L^{2}(\Omega)}=\|\bar{\lambda}^{k}-\bar{\lambda}^{k}_{h_{k+1}}\|_{L^{2}(\Omega)}=O(h_{k+1})$, $\forall k\geqslant 1,$ and $$\sum_{k=0}^{\infty}\|\bar{u}^{k+1}-\bar{u}^{k+1}_{h_{k+1}}\|_{L^{2}(\Omega)}=O\left(\sum_{k=0}^{\infty}h_{k+1}\right).$$

We employ the mathematical induction to prove the conclusion. While $k=1$, with the definition of $\lambda_{h_{k+1}}^{k}:=I_{h_{k+1}}\lambda_{h_{k}}^{k}$ and the interpolation error estimate in Lemma \[lem:interpolation 