Suppose that the operator $L$ is uniformly elliptic. Let $(y^{\ast},u^{\ast},z^{\ast},p^{\ast},\lambda^{\ast})$ be the KKT point of [(\[eqn:orginal problems\])]{}, $(u^{k}_{h_{k}},z^{k}_{h_{k}},\lambda^{k}_{h_{k}})$ is obtained in the $k$th iterate of Algorithm \[alg:multi-level ADMM algorithm\], where we suppose the mesh sizes $\lbrace h_{k} \rbrace_{k=0}^{\infty}$ of each iteration satisfy $\sum_{k=0}^{\infty}h_{k+1}<\infty,$ and the error vector $\delta_{u,h_{k+1}}^{k+1}$ satisfies $\|\delta_{u,h_{k+1}}^{k+1}\|_{L^{2}(\Omega)}\leq \xi_{k+1}$, $\sum_{k=0}^{\infty}\xi_{k+1}<\infty.$ Then we have $$\begin{aligned}
&\lim \limits_{k\rightarrow \infty}\lbrace \|u^{k}_{h_{k}}-u^{\ast}\|_{L^{2}(\Omega)}+\|z^{k}_{h_{k}}-z^{\ast}\|_{L^{2}(\Omega}+\|\lambda^{k}_{h_{k}}-\lambda^{\ast}\|_{L^{2}(\Omega)}\rbrace=0,\\
&\lim \limits_{k\rightarrow \infty}\lbrace \|y^{k}_{h_{k}}-y^{\ast}\|_{H_{0}^{1}(\Omega)}+\|p^{k}_{h_{k}}-p^{\ast}\|_{H_{0}^{1}(\Omega)}\rbrace=0.\end{aligned}$$ Moreover, there exists a constant $\tilde C$ only depends on the initial point $(u^{0}, z^{0}, \lambda^{0})$ and the optimal solution $(u^{\ast}, z^{\ast}, \lambda^{\ast})$ such that for $k\geq 1$, $$\min_{1\leq i\leq k}{R_{h_{i}}}(u^{i}_{h_{i}}, z^{i}_{h_{i}}, \lambda^{i}_{h_{i}})\leq \frac{\tilde C}{k}, \ \ \lim \limits_{k\rightarrow \infty}(k\times \min_{1\leq i\leq k}{R_{h_{i}}}(u^{i}_{h_{i}}, z^{i}_{h_{i}}, \lambda^{i}_{h_{i}}))=0,$$ where $R_{h_{i}}: (u^{i}_{h_{i}}, z^{i}_{h_{i}}, \lambda^{i}_{h_{i}})\rightarrow [0,\infty)$ is defined as $$R_{h_{i}}(u^{i}_{h_{i}}, z^{i}_{h_{i}}, \lambda^{i}_{h_{i}}):=\|\nabla {\hat J_{h_{i}}}(u^{i}_{h_{i}})+\lambda_{h_{i}}^{i-1}\|^{2}_{L^{2}(\Omega)}+{\rm dist}^{2}(0,-\lambda_{h_{i}}^{i-1}+\partial \delta_{U_{ad,h_{i}}}(z^{i}_{h_{i}}))+\|u^{i}_{h_{i}}-z^{i}_{h_{i}}\|^{2}_{L^{2}(\Omega)}.$$

By the optimality condition of the $u$-subproblem in the ADMM in function space, we have $$\begin{aligned}
S^{\ast}[S(\bar{u}^{k+1}+y_{r})-y_{d}]+\alpha\bar{u}^{k+1}+\bar \lambda^{k}+\sigma(\bar{u}^{k+1}-\bar z^{k})=0.\end{aligned}$$ As we know, the error between the inexact solution and exact solution contains two parts, error from gradually refining the grid and error from the inexactly solving the subproblems. We take them into consideration together as a total error, let $u_{h_{k+1}}^{k+1}$ represents the inexact solution of the $(k+1)$th iteration, then from the optimality condition of the 