error estimate\], we have $$\|\bar{\lambda}^{0}-\bar{\lambda}^{0}_{h_{1}}\|_{L^{2}(\Omega)}=\|\lambda^{0}-I_{h_{1}}\lambda^{0}\|_{L^{2}(\Omega)}=O(h_{1}),$$ Then we can easily obtain that $\|\bar{u}^{1}-\bar{u}^{1}_{h_{1}}\|_{L^{2}(\Omega)}=O(h_{1})$. The proof is similar to the case $k>1$, here we omit it.

While $k>1$, we assume for $\forall j\leq k,$ we have $\|\bar{u}^{j}-\bar{u}^{j}_{h_{j}}\|_{L^{2}(\Omega)}=\|\bar{\lambda}^{j-1}-\bar{\lambda}^{j-1}_{h_{j}}\|_{L^{2}(\Omega)}=O(h_{j})$. Then for $z$-subproblems in exact ADMM in function space and the multi-level discretized ADMM, we know $\bar{z}^{k}$ and $\bar{z}^{k}_{h_{k}}$ satisfy the following optimality conditions: $$\bar{z}^{k}=\Pi_{[a,b]}\left(\bar{u}^{k}+\frac{\bar{\lambda}^{k-1}}{\sigma}\right),\
\bar{z}^{k}_{h_{k}}=\Pi_{[a,b]}\left(\bar{u}^{k}_{h_{k}}+\frac{\bar{\lambda}^{k-1}_{h_{k}}}{\sigma}\right).$$ Then subtracting the above two equalities we obtain $$\begin{aligned}
\|\bar{z}^{k}-\bar{z}^{k}_{h_{k}}\|_{L^{2}(\Omega)}
&=\left\|\Pi_{[a,b]}\left(\bar{u}^{k}+\dfrac{\bar{\lambda}^{k-1}}{\sigma}\right)-\Pi_{[a,b]}\left(\bar{u}^{k}_{h_{k}}+\dfrac{\bar{\lambda}^{k-1}_{h_{k}}}{\sigma}\right)\right\|_{L^{2}(\Omega)}
\\
&\leq \left\|\bar{u}^{k}+\dfrac{\bar{\lambda}^{k-1}}{\sigma}-\bar{u}^{k}_{h_{k}}-\dfrac{\bar{\lambda}^{k-1}_{h_{k}}}{\sigma}\right\|_{L^{2}(\Omega)}\\
&\leq \|\bar{u}^{k}-\bar{u}^{k}_{h_{k}}\|_{L^{2}(\Omega)}+\dfrac{1}{\sigma}\|\bar{\lambda}^{k-1}-\bar{\lambda}^{k-1}_{h_{k}}\|_{L^{2}(\Omega)}\\
&=O(h_{k}).
\end{aligned}$$ For the multiplier $\bar{\lambda}^{k}$ and $\bar{\lambda}^{k}_{h_{k}}$, $$\bar{\lambda}^{k}=\bar{\lambda}^{k-1}+\sigma(\bar{u}^{k}-\bar{z}^{k}), \
\bar{\lambda}^{k}_{h_{k}}=\bar{\lambda}^{k-1}_{h_{k}}+\sigma(\bar{u}^{k}_{h_{k}}-\bar{z}^{k}_{h_{k}}),$$ we can get the estimate $$\label{equ:lambda}
\begin{aligned}
\|\bar{\lambda}^{k}-\bar{\lambda}^{k}_{h_{k}}\|_{L^{2}(\Omega)}
&=\|\bar{\lambda}^{k-1}-\bar{\lambda}^{k-1}_{h_{k}}+\sigma(\bar{u}^{k}-\bar{u}^{k}_{h_{k}})-\sigma(\bar{z}^{k}-\bar{z}^{k}_{h_{k}})\|_{L^{2}(\Omega)}\\
&\leq \|\bar{\lambda}^{k-1}-\bar{\lambda}^{k-1}_{h_{k}}\|_{L^{2}(\Omega)}+ \sigma\|\bar{u}^{k}-\bar{u}^{k}_{h_{k}}\|_{L^{2}(\Omega)}+\sigma\|\bar{z}^{k}-\bar{z}^{k}_{h_{k}}\|_{L^{2}(\Omega)}\\
&=O(h_{k}).
\end{aligned}$$ For $u$-subproblems, $\bar{u}^{k+1}$ and $\bar{u}^{k+1}_{h_{k+1}}$ satisfy the following optimality conditions respectively, $$\begin{aligned}
&S^{\ast}[S(\bar{u}^{k}+y_{r})-y_{d}]+\alpha\bar{u}^{k+1}+\bar{\lambda}^{k}+\sigma(\bar{u}^{k+1}-\bar{z}^{k})=0,\\
&S^{\ast}_{h_{k+1}}[S_{h_{k+1}}(\bar{u}_{h_{k+1}}^{k+1}+I_{h_{k+1}}y_{r})-I_{h_{k+1}}y_{d}]+\alpha\bar{u}^{k+1}_{h_{k+1}}+\bar{\lambda}^{k}_{h_{k+1}}+\sigma(\bar{u}^{k+1}_{h_{k+1}}-\bar{z}^{k}_{h_{k+1}})=0.\end{aligned}$$ Then we know from the above two equalities that $$\begin{aligned}
0=&
S^{\ast}S\bar{u}^{k+1}-S^{\ast}_{h_{k+1}}S_{h_{k+1}}\bar{u}^{k+1}_{h_{k+1}}+S^{\ast}Sy_{r}-S^{\ast}_{h_{k+1}}S_{h_{k+1}}I_{h_{k+1}}y_{r}\\&-S^{\ast}y_{d}+S^{\ast}_{h_{k+1}}I_{h_{k+1}}y_{d}+(\alpha+\sigma)(\bar{u}^{k+1}-\bar{u}^{k+1}_{h_{k+1}})+\bar{\lambda}^{k}-\bar{\lambda}^{k}_{h_{k+1}}-\sigma(\bar{z}^{k}-\bar{z}^{k}_{h_{k+1}}),
\end{aligned}$$ so $$\begin{aligned}
\label{equality}
(\alpha+\sigma)(\bar{u}^{k+1}-\bar{u}^{k+1}_{h_{k+1}})=&
-(S^{\ast}S\bar{u}^{k+1}-S^{\ast}_{h_{k+1}}S_{h_{k+1}}\bar{u}^{k+1}_{h_{k+1}}+S^{\ast}Sy_{r}-S^{\ast}_{h_{k+1}}S_{h_{k+1}}I_{h_{k+1}}y_{r}
\\&-S^{\ast}y_{d}+S^{\ast}_{h_{k+1}}I_{h_{k+1}}y_{d}+\bar{\lambda}^{k}-\bar{\lambda}^{k}_{h_{k+1}}-\sigma(\bar{z}^{k}-\bar{z}^{k}_{h_{k+1}}))\\
=&-(E_{1}+E_{2}+E_{3}+E_{4}+E_{5}),
\end{aligned}$$ where we define $$\begin{aligned}
E_{1}&:=S^{\ast}S\bar{u}^{k+1}-S^{\ast}_{h_{k+1}}S_{h_{k+1}}\bar{u}^{k+1}_{h_{k+1}},\\
E_{2}&:=S^{\ast}Sy_{r}-S^{\ast}_{h_{k+1}}S_{h_{k+1}}I_{h_{k+1}}y_{r},\\
E_{3}&:=-S^{\ast}y_{d}+S^{\ast}_{h_{k+1}}I_{h_{k+1}}y_{d},\\
E_{4}&:=\bar{\lambda}^{k}-\bar{\lambda}^{k}_{h_{k+1}},\\
E_{5}&:=-\sigma(\bar{z}^{k}-\bar{z}^{k}_{h_{k+1}}).
\end{aligned}$$

For the term $E_{1}$, we make use of the decomposition, $$\begin{aligned}
\|E_{1}\|_{L^{2}(\Omega)}&=\|S^{\ast}S\bar{u}^{k+1}-S^{\ast}S_{h_{k+1}}\bar{u}^{k+1}+S^{\ast}S_{h_{k+1}}\bar{u}^{k+1}-S^{\ast}_{h_{k+1}}S_{h_{k+1}}\bar{u}^{k+1}+S^{\ast}_{h_{k+1}}S_{h_{k+1}}\bar{u}^{k+1}-S^{\ast}_{h_{k+1}}S_{h_{k+1}}\bar{u}^{k+1}_{h_{k+1}}\|_{L^{2}(\Omega)}\\
&\leq\|S^{\ast}(S-S_{h_{k+1}})\bar{u}^{k+1}\|_{L^{2}(\Omega)}+\|(S^{\ast}-S^{\ast}_{h_{k+1}})S_{h_{k+1}}\bar{u}^{k+1}\|_{L^{2}(\Omega)}+\|S^{\ast}_{h_{k+1}}S_{h_{k+1}}(\bar{u}^{k+1}-\bar{u}^{k+1}_{h_{k+1}})\|_{L^{2}(\Omega)}.
\end{aligned}$$ From the well known error estimate $\|S-S_{h}\|_{L^{2}(\Omega)\rightarrow L^{2}(\Omega)}=O(h^{2})$ in lemma \[lem:error estimates\] and the property that $S^{\ast},S_{h_{k+1}}$ are bounded linear operators, we have $$\begin{aligned}
&\|S^{\ast}(S-S_{h_{k+1}})\bar{u}^{k+1}\|_{L^{2}(\Omega)}\leq
\|S^{\ast}\|_{L^{2}(\Omega)\rightarrow L^{2}(\Omega)}\|S-S_{h_{k+1}}\|_{L^{2}(\Omega)\rightarrow L^{2}(\Omega)}\|\bar{u}^{k+1}\|_{L^{2}(\Omega)}=O(h_{k+1}^{2}),\\
&\|(S^{\ast}-S^{\ast}_{h_{k+1}})S_{h_{k+1}}\bar{u}^{k+1}\|_{L^{2}(\Omega)}\leq\|S^{\ast}-S^{\ast}_{h_{k+1}}\|_{L^{2}(\Omega)\rightarrow L^{2}(\Omega)}\|S_{h_{k+1}}\bar{u}^{k+1}\|_{L^{2}(\Omega)}=O(h_{k+1}^{2}).\end{aligned}$$ Hence, there exists a constant $\hat C$ such that 