$$\|E_{1}\|_{L^{2}(\Omega)}\leq \hat Ch_{k+1}^{2}+\|S^{\ast}_{h_{k+1}}S_{h_{k+1}}\|_{L^{2}(\Omega)\rightarrow L^{2}(\Omega)}\|\bar{u}^{k+1}-\bar{u}^{k+1}_{h_{k+1}}\|_{L^{2}(\Omega)}.$$

Similarly, based on the property of the projection operator $\|y_{r}-I_{h_{k+1}}y_{r}\|_{L^{2}(\Omega)}=\|y_{d}-I_{h_{k+1}}y_{d}\|_{L^{2}(\Omega)}=O(h_{k+1})$ in Lemma \[lem:interpolation error estimate\], for the term $E_{2}$, we have $$\begin{aligned}
\|E_{2}\|_{L^{2}(\Omega)}=&\|S^{\ast}Sy_{r}-S^{\ast}_{h_{k+1}}Sy_{r}+S^{\ast}_{h_{k+1}}Sy_{r}-S^{\ast}_{h_{k+1}}S_{h_{k+1}}y_{r}+S^{\ast}_{h_{k+1}}S_{h_{k+1}}y_{r}-S^{\ast}_{h_{k+1}}S_{h_{k+1}}I_{h_{k+1}}y_{r}\|_{L^{2}(\Omega)}\\
\leq& \|(S^{\ast}-S^{\ast}_{h_{k+1}})Sy_{r}\|_{L^{2}(\Omega)}+\|S^{\ast}_{h_{k+1}}(S-S_{h_{k+1}})y_{r}\|_{L^{2}(\Omega)}+\|S^{\ast}_{h_{k+1}}S_{h_{k+1}}(y_{r}-I_{h_{k+1}}y_{r})\|_{L^{2}(\Omega)}\\
\leq & \|S^{\ast}-S^{\ast}_{h_{k+1}}\|_{L^{2}(\Omega)\rightarrow L^{2}(\Omega)}\|Sy_{r}\|_{L^{2}(\Omega)}+\|S^{\ast}_{h_{k+1}}\|_{L^{2}(\Omega)\rightarrow L^{2}(\Omega)}\|S-S_{h_{k+1}}\|_{L^{2}(\Omega)\rightarrow L^{2}(\Omega)}\|y_{r}\|_{L^{2}(\Omega)}\\
&+\|S^{\ast}_{h_{k+1}}S_{h_{k+1}}\|_{L^{2}(\Omega)\rightarrow L^{2}(\Omega)}\|y_{r}-I_{h_{k+1}}y_{r}\|_{L^{2}(\Omega)}\\
=&O(h_{k+1}),
\end{aligned}$$ and $$\begin{aligned}
\|E_{3}\|_{L^{2}(\Omega)}&=\|-S^{\ast}y_{d}+S^{\ast}I_{h_{k+1}}y_{d}-S^{\ast}I_{h_{k+1}}y_{d}+S^{\ast}_{h_{k+1}}I_{h_{k+1}}y_{d}\|_{L^{2}(\Omega)}\\
&\leq \|S^{\ast}(y_{d}-I_{h_{k+1}}y_{d})\|_{L^{2}(\Omega)}+\|(S^{\ast}-S^{\ast}_{h_{k+1}})I_{h_{k+1}}y_{d}\|_{L^{2}(\Omega)}\\
&=O(h_{k+1}).
\end{aligned}$$

For the term $E_{4}$, $$\begin{aligned}
\|E_{4}\|_{L^{2}(\Omega)}
&=\|\bar{\lambda}^{k}-\bar{\lambda}^{k}_{h_{k}}+\bar{\lambda}^{k}_{h_{k}}-I_{h_{k+1}}\bar{\lambda}^{k}_{h_{k}}\|_{L^{2}(\Omega)}\\
&\leq \|\bar{\lambda}^{k}-\bar{\lambda}^{k}_{h_{k}}\|_{L^{2}(\Omega)}+\|\bar{\lambda}^{k}_{h_{k}}-I_{h_{k+1}}\bar{\lambda}^{k}_{h_{k}}\|_{L^{2}(\Omega)}\\
&=O(h_{k}+h_{k+1}), 
\end{aligned}$$ where we used (\[equ:lambda\]), the property of the projection operator $\|\bar{\lambda}^{k}_{h_{k}}-I_{h_{k+1}}\bar{\lambda}^{k}_{h_{k}}\|_{L^{2}(\Omega)}=O(h_{k+1})$ and the property of the mesh size $h_{k}>h_{k+1}$. Moreover, as the mesh sizes satisfy $\sum_{k=0}^{\infty}h_{k+1}< \infty,$ there exists a constant $C_{k+1}$ such that $h_{k}<C_{k+1}h_{k+1}$, then we have $$\|E_{4}\|_{L^{2}(\Omega)}=\|\bar{\lambda}^{k}-\bar{\lambda}^{k}_{h_{k+1}}\|_{L^{2}(\Omega)}=O(h_{k+1}).$$

Similarly, $$\begin{aligned}
\|E_{5}\|_{L^{2}(\Omega)}
&=\sigma\|\bar{z}^{k}-\bar{z}^{k}_{h_{k}}+\bar{z}^{k}_{h_{k}}-\bar{z}^{k}_{h_{k+1}}\|_{L^{2}(\Omega)}\\
&\leq \sigma\|\bar{z}^{k}-\bar{z}^{k}_{h_{k}}\|_{L^{2}(\Omega)}+\sigma\|\bar{z}^{k}_{h_{k}}-\bar{z}^{k}_{h_{k+1}}\|_{L^{2}(\Omega)}\\
&=O(h_{k+1}).
\end{aligned}$$

Then with the fact that operators $S^{\ast}_{h_{k+1}}, S_{h_{k+1}}$ are bounded linear operators, we know from the equality (\[equality\]) and the estimations of $L^{2}$ norms of $\lbrace E_{i}\rbrace_{i=1}^{5}$ above that $$\|\bar{u}^{k+1}-\bar{u}^{k+1}_{h_{k+1}}\|_{L^{2}(\Omega)}=O(h_{k+1}).$$ Moreover, we have $$\begin{aligned}
\|\bar{z}^{k+1}-\bar{z}^{k+1}_{h_{k+1}}\|_{L^{2}(\Omega)}
&=\left\|\Pi_{[a,b]}\left(\bar{u}^{k+1}+\dfrac{\bar{\lambda}^{k}}{\sigma}\right)-\Pi_{[a,b]}\left(\bar{u}^{k+1}_{h_{k+1}}+\dfrac{\bar{\lambda}^{k}_{h_{k+1}}}{\sigma}\right)\right\|_{L^{2}(\Omega)}
\\
&\leq \|\bar{u}^{k+1}-\bar{u}^{k+1}_{h_{k+1}}\|_{L^{2}(\Omega)}+\dfrac{1}{\sigma}\|(\bar{\lambda}^{k}-\bar{\lambda}^{k}_{h_{k+1}})\|_{L^{2}(\Omega)}\\
&=O(h_{k+1}).
\end{aligned}$$ Hence the conclusion holds for the case $k+1$ and we can get the assertion $$\sum_{k=0}^{\infty}\|\bar{u}^{k+1}-\bar{u}^{k+1}_{h_{k+1}}\|_{L^{2}(\Omega)}=O\left(\sum_{k=0}^{\infty}h_{k+1}\right).$$

Similar to the Lemma \[lem:continuous exact and discretized exact\], we have the following lemma.

\[lem:continuous inexact and discretized inexact\] 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 $\|u^{k+1}-{u}^{k+1}_{h_{k+1}}\|_{L^{2}(\Omega)}=\|z^{k+1}-{z}^{k+1}_{h_{k+1}}\|_{L^{2}(\Omega)}=\|{\lambda}^{k}-{\lambda}^{k}_{h_{k+1}}\|_{L^{2}(\Omega)}=O(h_{k+1}+\delta_{u,h_{k+1}}^{k+1})$, $\forall k\geqslant 1,$ and $$\sum_{k=0}^{\infty}\|{u}^{k+1}-{u}^{k+1}_{h_{k+1}}\|_{L^{2}(\Omega)}=O\left(\sum_{k=0}^{\infty}\left(h_{k+1}+\delta_{u,h_{k+1}}^{k+1}\right)\right).$$

We employ the mathematical 