to the finer mesh. Finally, we apply appropriate numerical methods to solve the subproblems in the finer mesh and obtain a more precise solution, so on and so forth.

Let the mesh size of the $k$th iterate denotes by $h_{k}, k\in \mathbb{Z}, k\geq 1$, then the discretized reduced cost function is defined as follows, $${\hat J_{h_{k+1}}}(u_{h_{k+1}}):=\dfrac{1}{2}\|S_{h_{k+1}}(u_{h_{k+1}}+I_{h_{k+1}}y_{r})-I_{h_{k+1}}y_{d}\|^{2}_{L^{2}(\Omega)}+\dfrac{\alpha}{2}\|u_{h_{k+1}}\|^{2}_{L^{2}(\Omega)}.$$ Let $U_{ad, h_{k+1}}$ denotes the discretized feasible set, $$U_{a d, h_{k+1}} :=U_{h_{k+1}} \cap U_{a d}=\left\{z_{h_{k+1}}=\sum_{i=1}^{N_{h_{k+1}}} z_{i} \phi_{i}(x) | a \leq z_{i} \leq b, \forall i=1, \cdots, N_{h_{k+1}}\right\} \subset U_{a d}.$$ Moreover, we define $\lambda_{h_{k+1}}^{k}:=I_{h_{k+1}}\lambda_{h_{k}}^{k}$ and $z_{h_{k+1}}^{k}:=I_{h_{k+1}}z_{h_{k}}^{k}$, where $I_{h_{k+1}}$ denotes the node interpolation operator. Based on the above representations, we present the iterative scheme of the multi-level ADMM algorithm (mADMM) in Algorithm \[alg:multi-level ADMM algorithm\].

$(u^{0},z^{0},{\lambda}^{0})\in L^{2}(\Omega)\times {\rm dom}(\delta_{U_{ad}}(\cdot))\times L^{2}(\Omega)$, parameters $\sigma>0$, $\tau \in \left(0,\dfrac{1+\sqrt{5}}{2}\right)$. Let$\lbrace\xi_{k+1}\rbrace_{k=0}^{\infty}$ be a sequence satisfying $\lbrace\xi_{k+1}\rbrace_{k=0}^{\infty}\subseteq[0,+\infty)$ and $\sum_{k=0}^{\infty}\xi_{k+1}<\infty$, mesh sizes $\lbrace h_{k} \rbrace_{k=0}^{\infty}$ of each iteration satisfy $\sum_{k=0}^{\infty}h_{k+1}< \infty.$ Set $k=0$. $u^{k}_{h_{k}},z^{k}_{h_{k}},{\lambda}^{k}_{h_{k}}$. Compute an approximation solution of

\_[u\_[h\_[k+1]{}]{}]{} [J\_[h\_[k+1]{}]{}]{}(u\_[h\_[k+1]{}]{})+\_[h\_[k+1]{}]{}\^[k]{},u\_[h\_[k+1]{}]{}-[z]{}\_[h\_[k+1]{}]{}\^[k]{}\_[L\^[2]{}()]{} +u\_[h\_[k+1]{}]{}-[z]{}\_[h\_[k+1]{}]{}\^[k]{}\^[2]{}\_[L\^[2]{}()]{}

    such that the error vector $\delta_{u,h_{k+1}}^{k+1}:=\nabla {\hat J_{h_{k+1}}}(u_{h_{k+1}})+{\lambda}_{h_{k+1}}^{k}+\sigma(u_{h_{k+1}}^{k+1}-{z}_{h_{k+1}}^{k})$ satisfies\
     $\|\delta_{u,h_{k+1}}^{k+1}\|_{L^{2}(\Omega)}\leq \xi_{k+1}.$ Compute $z^{k+1}_{h_{k+1}}$ as follows: $$z^{k+1}_{h_{k+1}}={\rm argmin} \delta_{U_{ad,h_{k+1}}}(z_{h_{k+1}})+\langle {\lambda}_{h_{k+1}}^{k},u^{k+1}_{h_{k+1}}-z_{h_{k+1}}\rangle_{L^{2}(\Omega)}+\dfrac{\sigma}{2}\|u^{k+1}_{h_{k+1}}-z_{h_{k+1}}\|^{2}_{L^{2}(\Omega)}.$$ Compute $$\lambda^{k+1}_{h_{k+1}}=I_{h_{k+1}}\lambda^{k}_{h_{k}}+\tau \sigma(u^{k+1}_{h_{k+1}}-z^{k+1}_{h_{k+1}}).$$ If a termination criterion is met, stop; else, set $k:=k+1$ and go to Step 1.

In order to 