A multi-level ADMM algorithm for elliptic PDE-constrained optimization problems

Abstract

In this paper, elliptic PDE-constrained optimization problems with box constraints on the control are considered. To numerically solve the problems, we apply the ‘optimize-discretize-optimize’ strategy. Specifically, the alternating direction method of multipliers (ADMM) algorithm is applied in function space first, and then, the standard piecewise linear finite-element approach is employed to discretize the subproblems in each iteration. Finally, some efficient numerical methods are applied to solve the discretized subproblems based on their structures. Motivated by the idea of the multi-level strategy, instead of fixing the mesh size before the computation process, we propose the strategy of gradually refining the grid. Moreover, the subproblems in each iteration are solved by appropriate inexact methods. Based on the strategies above, an efficient convergent multi-level ADMM (mADMM) algorithm is proposed. We present the convergence analysis and the iteration complexity results o(1/k) for the mADMM algorithm. Some numerical experiments are done and the numerical results show the high efficiency of the mADMM algorithm.

Appendices

Proof of Lemma 2

We employ the mathematical induction to prove the conclusion. While $k=1$, for u-subproblems, $\bar{u}^{1}$ and $\bar{u}_{h_{1}}^{1}$ satisfy the following optimality conditions, respectively:

$$\begin{aligned} \begin{aligned}&S^{*}[S(\bar{u}^{1}+y_{r})-y_{d}]+\alpha \bar{u}^{1}+\lambda ^{0}+\sigma (\bar{u}^{1}-z^{0})=0,\\&S^{*}_{h_{1}}[S_{h_{1}}(\bar{u}_{h_{1}}^{1}+I_{h_{1}}y_{r})-I_{h_{1}}y_{d}]+\alpha \bar{u}^{1}_{h_{1}}+I_{h_{1}}\lambda ^{0}+\sigma (\bar{u}^{1}_{h_{1}}-I_{h_{1}}z^{0})=0. \end{aligned} \end{aligned}$$

By subtracting two equalities above, we have:

$$\begin{aligned} \begin{aligned}&\left[ -(\alpha +\sigma )I-S^{*}_{h_{1}}S_{h_{1}}\right] (\bar{u}^{1}-\bar{u}^{1}_{h_{1}})\\&\quad = S^{*}S\bar{u}^{1}-S^{*}_{h_{1}}S_{h_{1}}\bar{u}^{1}+S^{*}Sy_{r}-S^{*}_{h_{1}}S_{h_{1}}I_{h_{1}}y_{r}-S^{*}y_{d}\\&\qquad +S^{*}_{h_{1}}I_{h_{1}}y_{d}+\lambda ^{0}-I_{h_{1}}\lambda ^{0}+\sigma I_{h_{1}}z^{0}-\sigma z^{0}\\&\quad =E_{1}+E_{2}+E_{3}+E_{4}+E_{5},\\ \end{aligned} \end{aligned}$$

(A.1)

where we define:

$$\begin{aligned} \begin{aligned} E_{1}&:=S^{*}S\bar{u}^{1}-S^{*}_{h_{1}}S_{h_{1}}\bar{u}^{1},\\ E_{2}&:=S^{*}Sy_{r}-S^{*}_{h_{1}}S_{h_{1}}I_{h_{1}}y_{r},\\ E_{3}&:=-S^{*}y_{d}+S^{*}_{h_{1}}I_{h_{1}}y_{d}, \\ E_{4}&:=\lambda ^{0}-I_{h_{1}}\lambda ^{0}, \\ E_{5}&:=\sigma I_{h_{1}}z^{0}-\sigma z^{0}. \end{aligned} \end{aligned}$$

For the term $E_{1}$, we make use of the decomposition:

$$\begin{aligned} \begin{aligned} \Vert E_{1}\Vert _{L^{2}(\varOmega _{h_{1}})}&=\Vert S^{*}S\bar{u}^{1}-S^{*}S_{h_{1}}\bar{u}^{1}+S^{*}S_{h_{1}}\bar{u}^{1}-S^{*}_{h_{1}}S_{h_{1}}\bar{u}^{1}\Vert _{L^{2}(\varOmega _{h_{1}})}\\&\le \Vert S^{*}(S-S_{h_{1}})\bar{u}^{1}\Vert _{L^{2}(\varOmega _{h_{1}})}+\Vert (S^{*}-S^{*}_{h_{1}})S_{h_{1}}\bar{u}^{1}\Vert _{L^{2}(\varOmega _{h_{1}})}\\&\le \Vert S^{*}\Vert \Vert S-S_{h_{1}}\Vert \Vert \bar{u}^{1}\Vert _{L^{2}(\varOmega _{h_{1}})}+\Vert S^{*}-S^{*}_{h_{1}}\Vert \Vert S_{h_{1}}\Vert \Vert \bar{u}^{1}\Vert _{L^{2}(\varOmega _{h_{1}})}. \end{aligned} \end{aligned}$$

Notice that $\Vert S-S_{h_{1}}\Vert \le ch_{1}^{2}$ and $S,S^{*},S_{h_{1}},S^{*}_{h_{1}}$ are bounded linear operators. From Lemma 1 and the triangle inequality, we have $\Vert S_{h_{1}}\Vert _{\mathcal L(L^{2}, L^{2})}\le ch_{1}^{2}+\Vert S\Vert _{\mathcal L(L^{2}, L^{2})}$. With the fact that $h_{1}$ can be bounded above by a non-negative constant, $\Vert S_{h_{1}}\Vert _{\mathcal L(L^{2}, L^{2})}$ can be bounded above by a non-negative constant independent of $h_{1}$. Thus, we have:

$$\begin{aligned} \Vert E_{1}\Vert _{L^{2}(\varOmega _{h_{1}})}\le c_{1}h_{1}^{2}, \end{aligned}$$

where $c_{1}$ is a constant independent of $h_{1}$.

For the term $E_{2}$:

$$\begin{aligned} \begin{aligned} \Vert E_{2}\Vert _{L^{2}(\varOmega _{h_{1}})}&=\Vert S^{*}Sy_{r}-S^{*}_{h_{1}}Sy_{r}+S^{*}_{h_{1}}Sy_{r}-S^{*}_{h_{1}}S_{h_{1}}y_{r}\\&\quad +S^{*}_{h_{1}}S_{h_{1}}y_{r}-S^{*}_{h_{1}}S_{h_{1}}I_{h_{1}}y_{r}\Vert _{L^{2}(\varOmega _{h_{1}})}\\&\le \Vert (S^{*}-S^{*}_{h_{1}})Sy_{r}\Vert _{L^{2}(\varOmega _{h_{1}})}+\Vert S^{*}_{h_{1}}(S-S_{h_{1}})y_{r}\Vert _{L^{2}(\varOmega _{h_{1}})}\\&\quad +\Vert S^{*}_{h_{1}}S_{h_{1}}(y_{r}-I_{h_{1}}y_{r})\Vert _{L^{2}(\varOmega _{h_{1}})}\\&\le \Vert S^{*}-S^{*}_{h_{1}}\Vert \Vert S\Vert \Vert y_{r}\Vert _{L^{2}(\varOmega _{h_{1}})}+\Vert S^{*}_{h_{1}}\Vert \Vert S-S_{h_{1}}\Vert _{L^{2}(\varOmega )}\Vert y_{r}\Vert _{L^{2}(\varOmega _{h_{1}})}\\&\quad +\Vert S^{*}_{h_{1}}\Vert \Vert S_{h_{1}}\Vert \Vert y_{r}-I_{h_{1}}y_{r}\Vert _{L^{2}(\varOmega _{h_{1}})}\\&\le c_{2}h_{1}, \end{aligned} \end{aligned}$$

where $c_{2}$ is a constant independent of $h_{1}$. In the last equality, we used the property that $h_{1}$ can be bounded above by a non-negative constant.

For the term $E_{3}$, similarly:

$$\begin{aligned} \begin{aligned} \Vert E_{3}\Vert _{L^{2}(\varOmega _{h_{1}})}&=\Vert -S^{*}y_{d}+S^{*}I_{h_{1}}y_{d}-S^{*}I_{h_{1}}y_{d}+S^{*}_{h_{1}}I_{h_{1}}y_{d}\Vert _{L^{2}(\varOmega _{h_{1}})}\\&\le \Vert S^{*}\Vert \Vert y_{d}-I_{h_{1}}y_{d}\Vert _{L^{2}(\varOmega _{h_{1}})}+\Vert S^{*}-S^{*}_{h_{1}}\Vert \Vert I_{h_{1}}y_{d}\Vert _{L^{2}(\varOmega _{h_{1}})}\\&\le c_{3}h_{1}, \end{aligned} \end{aligned}$$

where $c_{3}$ is a constant independent of $h_{1}$.

For the term $E_{4}$:

$$\begin{aligned} \begin{aligned} \Vert E_{4}\Vert _{L^{2}(\varOmega _{h_{1}})}&=\Vert \lambda ^{0}-I_{h_{1}}\lambda ^{0}\Vert _{L^{2}(\varOmega _{h_{1}})}\\&\le c_{I}h_{1}\Vert \lambda ^{0}\Vert _{H^{1}(\varOmega _{h_{1}})}\\&\le C_{\lambda , 1}h_{1}, \end{aligned} \end{aligned}$$

where $C_{\lambda , 1}:=c_{I}\Vert \lambda ^{0}\Vert _{H^{1}(\varOmega _{h_{1}})}$ is a constant independent of $h_{1}$.

For the term $E_{5}$, similarly:

$$\begin{aligned} \begin{aligned} \Vert E_{5}\Vert _{L^{2}(\varOmega _{h_{1}})}&=\Vert \sigma I_{h_{1}}z^{0}-\sigma z^{0}\Vert _{L^{2}(\varOmega _{h_{1}})}\\&\le \sigma c_{I}h_{1}\Vert z^{0}\Vert _{H^{1}(\varOmega _{h_{1}})}. \end{aligned} \end{aligned}$$

Then, we know from (A.1) and the estimations of $L^{2}$ norms of $\lbrace E_{i}\rbrace _{i=1}^{5}$ that there exist a constant $\hat{C}_{u,1}$ independent of $h_{1}$, such that:

$$\begin{aligned} \left\| \left[ (\alpha +\sigma )I+S^{*}_{h_{1}}S_{h_{1}}\right] (\bar{u}^{1}-\bar{u}^{1}_{h_{1}})\right\| _{L^{2}(\varOmega _{h_{1}})}\le \hat{C}_{u,1}h_{1}. \end{aligned}$$

Notice that:

$$\begin{aligned} \begin{aligned}&\left\| \left[ (\alpha +\sigma )I+S^{*}_{h_{1}}S_{h_{1}}\right] (\bar{u}^{1}-\bar{u}^{1}_{h_{1}})\right\| _{L^{2}(\varOmega _{h_{1}})}\\&\quad =\left\langle \left[ (\alpha +\sigma )I+S^{*}_{h_{1}}S_{h_{1}}\right] (\bar{u}^{1}-\bar{u}^{1}_{h_{1}}), \left[ (\alpha +\sigma )I+S^{*}_{h_{1}}S_{h_{1}}\right] (\bar{u}^{1}-\bar{u}^{1}_{h_{1}})\right\rangle _{L^{2}(\varOmega _{h_{1}})}\\&\quad =\left\langle \bar{u}^{1}-\bar{u}^{1}_{h_{1}}, \left[ (\alpha +\sigma )I+S^{*}_{h_{1}}S_{h_{1}}\right] ^{*}\left[ (\alpha +\sigma )I+S^{*}_{h_{1}}S_{h_{1}}\right] (\bar{u}^{1}-\bar{u}^{1}_{h_{1}})\right\rangle _{L^{2}(\varOmega _{h_{1}})}\\&\quad =(\alpha +\sigma )^{2}\Vert \bar{u}^{1}-\bar{u}^{1}_{h_{1}}\Vert _{L^{2}(\varOmega _{h_{1}})}+2(\alpha +\sigma )\Vert S_{h_{1}}(\bar{u}^{1}-\bar{u}^{1}_{h_{1}})\Vert _{L^{2}(\varOmega _{h_{1}})}\\&\qquad +\Vert S_{h_{1}}^{*}S_{h_{1}}(\bar{u}^{1}-\bar{u}^{1}_{h_{1}})\Vert _{L^{2}(\varOmega _{h_{1}})}\\&\quad \ge (\alpha +\sigma )^{2}\Vert \bar{u}^{1}-\bar{u}^{1}_{h_{1}}\Vert _{L^{2}(\varOmega _{h_{1}})}.\\ \end{aligned} \end{aligned}$$

Thus, we have:

$$\begin{aligned} \Vert \bar{u}^{1}-\bar{u}^{1}_{h_{1}}\Vert _{L^{2}(\varOmega _{h_{1}})}\le C_{u,1}h_{1}, \end{aligned}$$

where $C_{u,1}:=\frac{1}{(\alpha +\sigma )^{2}}\hat{C}_{u,1}$ is a constant independent of $h_{1}$.

For z-subproblems, $\bar{z}^{1}$ and $\bar{z}^{1}_{h_{1}}$ satisfy:

$$\begin{aligned} \bar{z}^{1}=\Pi _{U_{ad}}\left( \bar{u}^{1}+\frac{{\lambda }^{0}}{\sigma }\right) ,\ \ \ \bar{z}^{1}_{h_{1}}=\Pi _{U_{ad,h_{1}}}\left( \bar{u}^{1}_{h_{1}}+\frac{I_{h_{1}}{\lambda }^{0}}{\sigma }\right) ; \end{aligned}$$

then, we know from the projection operator $\Pi$ is nonexpansive that:

$$\begin{aligned} \begin{aligned} \Vert \bar{z}^{1}-\bar{z}^{1}_{h_{1}}\Vert _{L^{2}(\varOmega _{h_{1}})}&=\left\| \Pi _{U_{ad}}\left( \bar{u}^{1}+\frac{{\lambda }^{0}}{\sigma }\right) -\Pi _{U_{ad,h_{1}}}\left( \bar{u}^{1}_{h_{1}}+\frac{I_{h_{1}}{\lambda }^{0}}{\sigma }\right) \right\| _{L^{2}(\varOmega _{h_{1}})} \\&\le \left\| \bar{u}^{1}-\bar{u}^{1}_{h_{1}}\right\| _{L^{2}(\varOmega _{h_{1}})}+\frac{1}{\sigma }\left\| {\lambda }^{0}-I_{h_{1}}{\lambda }^{0}\right\| _{L^{2}(\varOmega _{h_{1}})}\\&\le C_{u,1}h_{1}+\frac{1}{\sigma }C_{\lambda , 1}h_{1}\\&=C_{z, 1}h_{1}, \end{aligned} \end{aligned}$$

where $C_{z, 1}:=C_{u,1}+\frac{1}{\sigma }C_{\lambda , 1}$.

Hence, we have completed the proof for the case $k=1$.

While $k>1$, we assume that, for $\forall j\le k,$ we have $\Vert \bar{u}^{j}-\bar{u}^{j}_{h_{j}}\Vert _{L^{2}(\varOmega _{h_{j}})} \le C_{u,j}h_{j}$, $\Vert \bar{z}^{j}-\bar{z}^{j}_{h_{j}}\Vert _{L^{2}(\varOmega _{h_{j}})} \le C_{z,j}h_{j}$ and $\Vert \bar{\lambda }^{j-1}-\bar{\lambda }^{j-1}_{h_{j}}\Vert _{L^{2}(\varOmega _{h_{j}})} \le C_{\lambda ,j}h_{j}$, where $C_{u,j}, C_{z,j}, C_{\lambda ,j}$ are constants independent of $h_{j}$.

For u-subproblems, $\bar{u}^{k+1}$ and $\bar{u}^{k+1}_{h_{k+1}}$ satisfy the following optimality conditions, respectively:

$$\begin{aligned}&S^{*}[S(\bar{u}^{k}+y_{r})-y_{d}]+\alpha \bar{u}^{k+1}+\bar{\lambda }^{k}+\sigma (\bar{u}^{k+1}-\bar{z}^{k})=0,\\&S^{*}_{h_{k+1}}[S_{h_{k+1}}(\bar{u}_{h_{k+1}}^{k+1}+I_{h_{k+1}}y_{r})-I_{h_{k+1}}y_{d}]+\alpha \bar{u}^{k+1}_{h_{k+1}}+\bar{\lambda }^{k}_{h_{k+1}}+\sigma (\bar{u}^{k+1}_{h_{k+1}}-\bar{z}^{k}_{h_{k+1}})=0. \end{aligned}$$

By subtracting two equalities above, we have:

$$\begin{aligned} \begin{aligned}&-\left[ (\alpha +\sigma )I+S^{*}_{h_{k+1}}S_{h_{k+1}}\right] (\bar{u}^{k+1}-\bar{u}^{k+1}_{h_{k+1}})\\&\quad =S^{*}S\bar{u}^{k+1}-S^{*}_{h_{k+1}}S_{h_{k+1}}\bar{u}^{k+1}+S^{*}Sy_{r}-S^{*}_{h_{k+1}}S_{h_{k+1}}I_{h_{k+1}}y_{r}-S^{*}y_{d}+S^{*}_{h_{k+1}}I_{h_{k+1}}y_{d}\\&\qquad +\bar{\lambda }^{k}-\bar{\lambda }^{k}_{h_{k+1}}-\sigma (\bar{z}^{k}-\bar{z}^{k}_{h_{k+1}})\\&\quad =\hat{E}_{1}+\hat{E}_{2}+\hat{E}_{3}+\hat{E}_{4}+\hat{E}_{5}, \end{aligned} \end{aligned}$$

(A.2)

where we define:

$$\begin{aligned} \begin{aligned} \hat{E}_{1}&:=S^{*}S\bar{u}^{k+1}-S^{*}_{h_{k+1}}S_{h_{k+1}}\bar{u}^{k+1},\\ \hat{E}_{2}&:=S^{*}Sy_{r}-S^{*}_{h_{k+1}}S_{h_{k+1}}I_{h_{k+1}}y_{r},\\ \hat{E}_{3}&:=-S^{*}y_{d}+S^{*}_{h_{k+1}}I_{h_{k+1}}y_{d},\\ \hat{E}_{4}&:=\bar{\lambda }^{k}-\bar{\lambda }^{k}_{h_{k+1}},\\ \hat{E}_{5}&:=-\sigma (\bar{z}^{k}-\bar{z}^{k}_{h_{k+1}}). \end{aligned} \end{aligned}$$

For the term $\hat{E}_{1}$, we make use of the decomposition:

$$\begin{aligned} \begin{aligned} \Vert \hat{E}_{1}\Vert _{L^{2}(\varOmega _{h_{k+1}})}&=\Vert S^{*}S\bar{u}^{k+1}-S^{*}S_{h_{k+1}}\bar{u}^{k+1}+S^{*}S_{h_{k+1}}\bar{u}^{k+1}-S^{*}_{h_{k+1}}S_{h_{k+1}}\bar{u}^{k+1}\Vert _{L^{2}(\varOmega _{h_{k+1}})} \\&\le \Vert S^{*}(S-S_{h_{k+1}})\bar{u}^{k+1}\Vert _{L^{2}(\varOmega _{h_{k+1}})}+\Vert (S^{*}-S^{*}_{h_{k+1}})S_{h_{k+1}}\bar{u}^{k+1}\Vert _{L^{2}(\varOmega _{h_{k+1}})}\\&\le \Vert S^{*}\Vert \Vert S-S_{h_{k+1}}\Vert \Vert \bar{u}^{k+1}\Vert _{L^{2}(\varOmega _{h_{k+1}})}+\Vert S^{*}-S^{*}_{h_{k+1}}\Vert \Vert S_{h_{k+1}}\Vert \Vert \bar{u}^{k+1}\Vert _{L^{2}(\varOmega _{h_{k+1}})}. \end{aligned} \end{aligned}$$

Notice that $\Vert S-S_{h_{k+1}}\Vert \le ch_{k+1}^{2}$ and $S,S^{*},S_{h_{k+1}},S^{*}_{h_{k+1}}$ are bounded linear operators. Moreover, we know from Lemma 1 and the triangle inequality that $\Vert S_{h_{k+1}}\Vert _{\mathcal L(L^{2}, L^{2})}\le ch_{k+1}^{2}+\Vert S\Vert _{\mathcal L(L^{2}, L^{2})}$. With the fact that $h_{k+1}$ can be bounded above by a non-negative constant, $\Vert S_{h_{k+1}}\Vert _{\mathcal L(L^{2}, L^{2})}$ can be bounded above by a non-negative constant independent of $h_{k+1}$. Thus, we have:

$$\begin{aligned} \Vert \hat{E}_{1}\Vert _{L^{2}(\varOmega _{h_{k+1}})}\le \hat{c}_{1}h^{2}_{k+1}, \end{aligned}$$

where $\hat{c}_{1}$ is a constant independent of $h_{k+1}$ and it can be bounded above by a non-negative constant.

Similarly, for the term $\hat{E}_{2}$ and $\hat{E}_{3}$, we have:

$$\begin{aligned} \begin{aligned} \Vert \hat{E}_{2}\Vert _{L^{2}(\varOmega _{h_{k+1}})}&=\Vert S^{*}Sy_{r}-S^{*}_{h_{k+1}}Sy_{r}+S^{*}_{h_{k+1}}Sy_{r}-S^{*}_{h_{k+1}}S_{h_{k+1}}y_{r}\\&\qquad +S^{*}_{h_{k+1}}S_{h_{k+1}}y_{r}-S^{*}_{h_{k+1}}S_{h_{k+1}}I_{h_{k+1}}y_{r}\Vert _{L^{2}} \\&\le \Vert (S^{*}-S^{*}_{h_{k+1}})Sy_{r}\Vert _{L^{2}} +\Vert S^{*}_{h_{k+1}}(S-S_{h_{k+1}})y_{r}\Vert _{L^{2}}\\&\qquad +\Vert S^{*}_{h_{k+1}}S_{h_{k+1}}(y_{r}-I_{h_{k+1}}y_{r})\Vert _{L^{2}} \\&\le \Vert S^{*}-S^{*}_{h_{k+1}}\Vert \Vert S\Vert \Vert y_{r}\Vert _{L^{2}}+\Vert S^{*}_{h_{k+1}}\Vert \Vert S-S_{h_{k+1}}\Vert \Vert y_{r}\Vert _{L^{2}}\\&\qquad +\Vert S^{*}_{h_{k+1}}\Vert \Vert S_{h_{k+1}}\Vert \Vert y_{r}-I_{h_{k+1}}y_{r}\Vert _{L^{2}}\\&\le \hat{c}_{2}h_{k+1}, \end{aligned} \end{aligned}$$

and

$$\begin{aligned} \begin{aligned} \Vert \hat{E}_{3}\Vert _{L^{2}(\varOmega _{h_{k+1}})}&=\Vert -S^{*}y_{d}+S^{*}I_{h_{k+1}}y_{d}-S^{*}I_{h_{k+1}}y_{d}+S^{*}_{h_{k+1}}I_{h_{k+1}}y_{d}\Vert _{L^{2}(\varOmega _{h_{k+1}})} \\&\le \Vert S^{*}\Vert \Vert y_{d}-I_{h_{k+1}}y_{d}\Vert _{L^{2}(\varOmega _{h_{k+1}})} +\Vert S^{*}-S^{*}_{h_{k+1}}\Vert \Vert I_{h_{k+1}}y_{d}\Vert _{L^{2}(\varOmega _{h_{k+1}})}\\&\le \hat{c}_{3}h_{k+1}, \end{aligned} \end{aligned}$$

where $\hat{c}_{2}$ and $\hat{c}_{3}$ are constants independent of $h_{k+1}$, and both of them can be bounded above by a non-negative constant.

For the term $\hat{E}_{4}$:

$$\begin{aligned} \begin{aligned} \Vert \hat{E}_{4}\Vert _{L^{2}(\varOmega _{h_{k+1}})}&=\Vert \bar{\lambda }^{k}-\bar{\lambda }^{k}_{h_{k}}+\bar{\lambda }^{k}_{h_{k}}-I_{h_{k+1}}\bar{\lambda }^{k}_{h_{k}}\Vert _{L^{2}(\varOmega _{h_{k+1}})}\\&\le \Vert \bar{\lambda }^{k}-\bar{\lambda }^{k}_{h_{k}}\Vert _{L^{2}(\varOmega _{h_{k}})}+\Vert \bar{\lambda }^{k}_{h_{k}}-I_{h_{k+1}}\bar{\lambda }^{k}_{h_{k}}\Vert _{L^{2}(\varOmega _{h_{k+1}})}\\&\le \Vert \bar{\lambda }^{k-1}-\bar{\lambda }^{k-1}_{h_{k}}+\tau \sigma (\bar{u}^{k}-\bar{u}^{k}_{h_{k}})-\tau \sigma (\bar{z}^{k}-\bar{z}^{k}_{h_{k}})\Vert _{L^{2}(\varOmega _{h_{k}})}\\&\qquad +\Vert \bar{\lambda }^{k}_{h_{k}}-I_{h_{k+1}}\bar{\lambda }^{k}_{h_{k}}\Vert _{L^{2}(\varOmega _{h_{k+1}})} \\&\le \Vert \bar{\lambda }^{k-1}-\bar{\lambda }^{k-1}_{h_{k}}\Vert _{L^{2}(\varOmega _{h_{k}})} + \tau \sigma \left( \Vert \bar{u}^{k}-\bar{u}^{k}_{h_{k}}\Vert _{L^{2}(\varOmega _{h_{k}})} +\Vert \bar{z}^{k}-\bar{z}^{k}_{h_{k}}\Vert _{L^{2}(\varOmega _{h_{k}})}\right) \\&+\Vert \bar{\lambda }^{k}_{h_{k}}-I_{h_{k+1}}\bar{\lambda }^{k}_{h_{k}}\Vert _{L^{2}(\varOmega _{h_{k+1}})}\\&\le (C_{\lambda ,k}+\tau \sigma C_{k}+\tau \sigma C_{z,k})h_{k}+c_{I}h_{k+1}\Vert \bar{\lambda }^{k}_{h_{k}}\Vert _{H^{1}(\varOmega _{h_{k+1}})}\\&\le C_{\lambda ,k+1}h_{k+1}, \end{aligned} \end{aligned}$$

where $C_{\lambda ,k+1}$ is a constant independent of $h_{k+1}$, and it can be bounded above by a non-negative constant. In the last equality, we use the property $\sum _{k=0}^{\infty }h_{k}< \infty$, and thus, there exists a constant $C_{k+1}$, such that $h_{k}<C_{k+1}h_{k+1}$.

For the term $\hat{E}_{5}$, similarly:

$$\begin{aligned} \begin{aligned} \Vert \hat{E}_{5}\Vert _{L^{2}(\varOmega _{h_{k+1}})}&=\sigma \Vert \bar{z}^{k}-\bar{z}^{k}_{h_{k}}+\bar{z}^{k}_{h_{k}}-\bar{z}^{k}_{h_{k+1}}\Vert _{L^{2}(\varOmega _{h_{k+1}})}\\&\le \sigma \Vert \bar{z}^{k}-\bar{z}^{k}_{h_{k}}\Vert _{L^{2}(\varOmega _{h_{k}})}+\sigma \Vert \bar{z}^{k}_{h_{k}}-\bar{z}^{k}_{h_{k+1}}\Vert _{L^{2}(\varOmega _{h_{k+1}})}\\&\le \hat{c}_{5}h_{k+1}, \end{aligned} \end{aligned}$$

where $\hat{c}_{5}$ is a constant independent of $h_{k+1}$ and it can be bounded above by a non-negative constant.

Then, we know from (A.2) and the estimations of $L^{2}$ norms of $\lbrace \hat{E}_{i}\rbrace _{i=1}^{5}$ that there exists a constant $\hat{C}_{u, k+1}$ independent of $h_{k+1}$, such that:

$$\begin{aligned} \left\| \left[ (\alpha +\sigma )I+S^{*}_{h_{k+1}}S_{h_{k+1}}\right] (\bar{u}^{k+1}-\bar{u}^{k+1}_{h_{k+1}})\right\| _{L^{2}(\varOmega _{h_{1}})}\le \hat{C}_{u,k+1}h_{k+1}. \end{aligned}$$

Notice that:

$$\begin{aligned} \begin{aligned}&\left\| \left[ (\alpha +\sigma )I+S^{*}_{h_{k+1}}S_{h_{k+1}}\right] (\bar{u}^{k+1}-\bar{u}^{k+1}_{h_{k+1}})\right\| _{L^{2}(\varOmega _{h_{k+1}})}\\&\quad =\left\langle \left[ (\alpha +\sigma )I+S^{*}_{h_{k+1}}S_{h_{k+1}}\right] (\bar{u}^{k+1}-\bar{u}^{k+1}_{h_{k+1}}),\right. \\&\qquad \left. \left[ (\alpha +\sigma )I+S^{*}_{h_{k+1}}S_{h_{k+1}}\right] (\bar{u}^{k+1}-\bar{u}^{k+1}_{h_{k+1}})\right\rangle _{L^{2}(\varOmega _{h_{k+1}})}\\&\quad =\left\langle \bar{u}^{k+1}-\bar{u}^{k+1}_{h_{k+1}}, \left[ (\alpha +\sigma )I+S^{*}_{h_{k+1}}S_{h_{k+1}}\right] ^{*}\right. \\&\qquad \left. \left[ (\alpha +\sigma )I+S^{*}_{h_{k+1}}S_{h_{k+1}}\right] (\bar{u}^{k+1}-\bar{u}^{k+1}_{h_{k+1}})\right\rangle _{L^{2}(\varOmega _{h_{k+1}})}\\&\quad =(\alpha +\sigma )^{2}\Vert \bar{u}^{k+1}-\bar{u}^{k+1}_{h_{k+1}}\Vert _{L^{2}(\varOmega _{h_{k+1}})}+2(\alpha +\sigma )\Vert S_{h_{k+1}}(\bar{u}^{k+1}-\bar{u}^{k+1}_{h_{k+1}})\Vert _{L^{2}(\varOmega _{h_{k+1}})}\\&\qquad +\Vert S_{h_{k+1}}^{*}S_{h_{k+1}}(\bar{u}^{k+1}-\bar{u}^{k+1}_{h_{k+1}})\Vert _{L^{2}(\varOmega _{h_{k+1}})}\\&\quad \ge (\alpha +\sigma )^{2}\Vert \bar{u}^{k+1}-\bar{u}^{k+1}_{h_{k+1}}\Vert _{L^{2}(\varOmega _{h_{k+1}})}.\\ \end{aligned} \end{aligned}$$

Thus, we have:

$$\begin{aligned} \Vert \bar{u}^{k+1}-\bar{u}^{k+1}_{h_{k+1}}\Vert _{L^{2}(\varOmega _{h_{k+1}})}\le C_{u,k+1}h_{k+1}, \end{aligned}$$

where $C_{u,k+1}:=\frac{1}{(\alpha +\sigma )^{2}}\hat{C}_{u,k+1}$ is a constant independent of $h_{k+1}$.

Hence, the conclusion holds for the case $k+1$, and we can get the assertion:

$$\begin{aligned} \sum _{k=0}^{\infty }\Vert \bar{u}^{k+1}-\bar{u}^{k+1}_{h_{k+1}}\Vert _{L^{2}(\varOmega _{h_{k+1}})}\le C\sum _{k=0}^{\infty }h_{k+1}< \infty . \end{aligned}$$

Moreover, for z-subproblems, we have:

$$\begin{aligned} \begin{aligned} \Vert \bar{z}^{k+1}-\bar{z}^{k+1}_{h_{k+1}}\Vert _{L^{2}(\varOmega _{h_{k+1}})}&=\left\| \Pi _{U_{ad}}\left( \bar{u}^{k+1}+\dfrac{\bar{\lambda }^{k}}{\sigma }\right) -\Pi _{U_{ad,h_{k+1}}}\left( \bar{u}^{k+1}_{h_{k+1}}+\dfrac{\bar{\lambda }^{k}_{h_{k+1}}}{\sigma }\right) \right\| _{L^{2}(\varOmega _{h_{k+1}})} \\&\le \left\| \bar{u}^{k+1}+\dfrac{\bar{\lambda }^{k}}{\sigma }-\bar{u}^{k+1}_{h_{k+1}}-\dfrac{\bar{\lambda }^{k}_{h_{k+1}}}{\sigma }\right\| _{L^{2}(\varOmega _{h_{k+1}})}\\&\le \Vert \bar{u}^{k+1}-\bar{u}^{k+1}_{h_{k+1}}\Vert _{L^{2}(\varOmega _{h_{k+1}})}+\dfrac{1}{\sigma }\Vert \bar{\lambda }^{k}-\bar{\lambda }^{k}_{h_{k+1}}\Vert _{L^{2}(\varOmega _{h_{k+1}})}\\&\le C_{u,k+1}h_{k+1}+\dfrac{1}{\sigma }C_{\lambda ,k+1}h_{k+1}\\&= C_{z,k+1}h_{k+1}, \end{aligned} \end{aligned}$$

where $C_{z,k+1}:=C_{u,k+1}+\dfrac{1}{\sigma }C_{\lambda ,k+1}$ is a constant independent of $h_{k+1}$, and it can be bounded above by a non-negative constant. Hence, we have completed the proof.

Proof of Proposition 1

First, for any $f_{1}, f_{1}^{\prime }, f_{2}, f_{2}^{\prime }\in L^{2}(\varOmega )$, we have the following two important equalities:

$$\begin{aligned} \langle f_{1}, f_{2}\rangle _{L^{2}(\varOmega )}&=\frac{1}{2}\left( \Vert f_{1}\Vert _{L^{2}(\varOmega )}^{2}+\Vert f_{2}\Vert _{L^{2}(\varOmega )}^{2}-\Vert f_{1}-f_{2}\Vert _{L^{2}(\varOmega )}^{2}\right) \nonumber \\&=\frac{1}{2}\left( \Vert f_{1}+f_{2}\Vert _{L^{2}(\varOmega )}^{2}-\Vert f_{1}\Vert _{L^{2}(\varOmega )}^{2}-\Vert f_{2}\Vert _{L^{2}(\varOmega )}^{2}\right) , \end{aligned}$$

(B.1)

$$\begin{aligned} \left\langle f_{1}-f_{1}^{\prime }, f_{2}-f_{2}^{\prime }\right\rangle _{L^{2}(\varOmega )}&=\frac{1}{2}\left( \Vert f_{1}+f_{2}\Vert _{L^{2}(\varOmega )}^{2}+\left\| f_{1}^{\prime }+f_{2}^{\prime }\right\| _{L^{2}(\varOmega )}^{2}\right. \nonumber \\&\quad \left. -\left\| f_{1}+f_{2}^{\prime }\right\| _{L^{2}(\varOmega )}^{2}-\left\| f_{1}^{\prime }+f_{2}\right\| _{L^{2}(\varOmega )}^{2}\right) . \end{aligned}$$

(B.2)

The proof of the above two equalities can be easily obtained by the definition of $L^{2}-$norm.

By the optimality condition at point $(u^{k}_{h_{k}},z^{k}_{h_{k}})$, we have:

$$\begin{aligned}&\nabla {\hat{J}_{h_{k}}}(u_{h_{k}}^{k})=S^{*}_{h_{k}}[S_{h_{k}}(u_{h_{k}}^{k}+I_{h_{k}}y_{r})-I_{h_{k}}y_{d}]+\alpha u^{k}_{h_{k}}=\delta _{u,h_{k}}^{k}-(\lambda ^{k-1}_{h_{k}}+\sigma (u^{k}_{h_{k}}-z^{k-1}_{h_{k}})), \end{aligned}$$

(B.3)

$$\begin{aligned}&\lambda ^{k-1}_{h_{k}}+\sigma (u^{k}_{h_{k}}-z^{k}_{h_{k}})\in \partial \delta _{U_{ad,h_{k}}}(z_{h_{k}}^{k}). \end{aligned}$$

(B.4)

Moreover, the KKT point $\left( u^{*}_{h_{k}}, z^{*}_{h_{k}}, \lambda ^{*}_{h_{k}}\right)$ of (DRP) satisfies:

$$\begin{aligned}&\nabla {\hat{J}_{h_{k}}}(u_{h_{k}}^{*})=S^{*}_{h_{k}}[S_{h_{k}}(u_{h_{k}}^{*}+I_{h_{k}}y_{r})-I_{h_{k}}y_{d}]+\alpha u^{*}_{h_{k}}=-\lambda ^{*}_{h_{k}}, \end{aligned}$$

(B.5)

$$\begin{aligned}&\lambda ^{*}_{h_{k}}\in \partial \delta _{U_{ad,h_{k}}}(z_{h_{k}}^{*}), \end{aligned}$$

(B.6)

$$\begin{aligned}&u^{*}_{h_{k}}=z^{*}_{h_{k}}. \end{aligned}$$

(B.7)

Then, we have:

$$\begin{aligned} \begin{aligned}&\left\langle \nabla {\hat{J}_{h_{k}}}(u_{h_{k}}^{k})-\nabla {\hat{J}_{h_{k}}}(u_{h_{k}}^{*}), u_{h_{k}}^{k}-u_{h_{k}}^{*} \right\rangle _{L^{2}(\varOmega _{h_{k}})}\\&\quad =\left\langle S^{*}_{h_{k}}S_{h_{k}}(u_{h_{k}}^{k}-u_{h_{k}}^{*})+\alpha (u_{h_{k}}^{k}-u_{h_{k}}^{*}), u_{h_{k}}^{k}-u_{h_{k}}^{*} \right\rangle _{L^{2}(\varOmega _{h_{k}})}\\&\quad =\Vert S_{h_{k}}(u_{h_{k}}^{k}-u_{h_{k}}^{*})\Vert _{L^{2}(\varOmega _{h_{k}})}^{2}+\alpha \Vert u_{h_{k}}^{k}-u_{h_{k}}^{*}\Vert _{L^{2}(\varOmega _{h_{k}})}^{2}. \end{aligned} \end{aligned}$$

(B.8)

Moreover, from the definition of the subdifferential operator, we have the following inequality:

$$\begin{aligned} \left\langle \lambda _{h_{k}}^{k}-\lambda _{h_{k}}^{*}, z_{h_{k}}^{k}-z_{h_{k}}^{*} \right\rangle _{L^{2}(\varOmega _{h_{k}})}\ge 0, \ \ \forall \lambda _{h_{k}}^{k}\in \partial \delta _{U_{ad,h_{k}}}(z_{h_{k}}^{k}), \ \lambda _{h_{k}}^{*}\in \partial \delta _{U_{ad,h_{k}}}(z_{h_{k}}^{*}). \end{aligned}$$

(B.9)

For the convenience of analyzing, we define $r_{h_{k}}^{k}=u_{h_{k}}^{k}-z_{h_{k}}^{k}$. By substituting (B.3)–(B.7) into (B.8) and (B.9), we have:

$$\begin{aligned}&\left\langle \delta _{u,h_{k}}^{k}- \lambda _{h_{k}}^{k-1}-\sigma r_{h_{k}}^{k}-\sigma (z_{h_{k}}^{k}-z_{h_{k}}^{k-1})+\lambda _{h_{k}}^{*}, u_{h_{k}}^{k}-u_{h_{k}}^{*}\right\rangle _{L^{2}(\varOmega _{h_{k}})}\ge \alpha \Vert u_{h_{k}}^{k}-u_{h_{k}}^{*}\Vert _{L^{2}(\varOmega _{h_{k}})},\\&\left\langle \lambda _{h_{k}}^{k-1}+\sigma r_{h_{k}}^{k}-\lambda _{h_{k}}^{*}, z_{h_{k}}^{k}-z_{h_{k}}^{*} \right\rangle _{L^{2}(\varOmega _{h_{k}})}\ge 0. \end{aligned}$$

By adding the above two equalities, we obtain:

$$\begin{aligned} \begin{aligned}&\left\langle \delta _{u,h_{k}}^{k}, u_{h_{k}}^{k}-u_{h_{k}}^{*}\right\rangle _{L^{2}(\varOmega _{h_{k}})}-\left\langle \lambda _{h_{k}}^{k-1}+\sigma r_{h_{k}}^{k}-\lambda _{h_{k}}^{*}, r_{h_{k}}^{k}\right\rangle _{L^{2}(\varOmega _{h_{k}})}\\&\qquad -\sigma \left\langle z_{h_{k}}^{k}-z_{h_{k}}^{k-1}, u_{h_{k}}^{k}-u_{h_{k}}^{*} \right\rangle _{L^{2}(\varOmega _{h_{k}})}\\&\quad \ge \alpha \Vert u_{h_{k}}^{k}-u_{h_{k}}^{*}\Vert _{L^{2}(\varOmega _{h_{k}})}. \end{aligned} \end{aligned}$$

(B.10)

Next, we estimate the last two terms on the left side separately:

$$\begin{aligned} \begin{aligned}&\left\langle \lambda _{h_{k}}^{*}-(\lambda _{h_{k}}^{k-1}+\sigma r_{h_{k}}^{k}), r_{h_{k}}^{k}\right\rangle _{L^{2}(\varOmega _{h_{k}})} \\&\quad =\frac{1}{\tau \sigma }\left\langle \lambda _{h_{k}}^{*}-\lambda _{h_{k}}^{k-1}, \lambda _{h_{k}}^{k}-\lambda _{h_{k}}^{k-1} \right\rangle _{L^{2}(\varOmega _{h_{k}})}-\sigma \Vert r_{h_{k}}^{k}\Vert _{L^{2}(\varOmega _{h_{k}})}^{2}\\&\quad =\frac{1}{2\tau \sigma }(\Vert \lambda _{h_{k}}^{*}-\lambda _{h_{k}}^{k-1}\Vert _{L^{2}(\varOmega _{h_{k}})}^{2}+\Vert \lambda _{h_{k}}^{k}-\lambda _{h_{k}}^{k-1}\Vert _{L^{2}(\varOmega _{h_{k}})}^{2}\\&\qquad -\Vert \lambda _{h_{k}}^{*}-\lambda _{h_{k}}^{k}\Vert _{L^{2}(\varOmega _{h_{k}})}^{2})-\sigma \Vert r_{h_{k}}^{k}\Vert _{L^{2}(\varOmega _{h_{k}})}^{2}\\&\quad =\frac{1}{2\tau \sigma }(\Vert \lambda _{h_{k}}^{*}-\lambda _{h_{k}}^{k-1}\Vert _{L^{2}(\varOmega _{h_{k}})}^{2}-\Vert \lambda _{h_{k}}^{*}-\lambda _{h_{k}}^{k}\Vert _{L^{2}(\varOmega _{h_{k}})}^{2})+\frac{(\tau -2)\sigma }{2}\Vert r_{h_{k}}^{k}\Vert _{L^{2}(\varOmega _{h_{k}})}^{2}, \end{aligned} \end{aligned}$$

(B.11)

where we used (B.1).

Moreover, by employing (B.2) and using $u^{*}_{h_{k}}=z^{*}_{h_{k}}$, we have:

$$\begin{aligned} \begin{aligned}&\sigma \left\langle z_{h_{k}}^{k}-z_{h_{k}}^{k-1}, u_{h_{k}}^{*}-u_{h_{k}}^{k} \right\rangle _{L^{2}(\varOmega _{h_{k}})} \\&\quad =\sigma \left\langle z_{h_{k}}^{k}-z_{h_{k}}^{k-1}, -u_{h_{k}}^{k}-(-z_{h_{k}}^{*}) \right\rangle _{L^{2}(\varOmega _{h_{k}})} \\&\quad =\frac{\sigma }{2}(\Vert r_{h_{k}}^{k}\Vert _{L^{2}(\varOmega _{h_{k}})}^{2}+\Vert z_{h_{k}}^{k-1}-z_{h_{k}}^{*}\Vert _{L^{2}(\varOmega _{h_{k}})}^{2}\\&\qquad -\Vert z_{h_{k}}^{k}-z_{h_{k}}^{*}\Vert _{L^{2}(\varOmega _{h_{k}})}^{2} -\Vert z_{h_{k}}^{k-1}-u_{h_{k}}^{k}\Vert _{L^{2}(\varOmega _{h_{k}})}^{2}). \end{aligned} \end{aligned}$$

(B.12)

Then, substituting (B.11) and (B.12) into (B.10), we can get the assertion of Proposition 1.