guarantee the sequence $\lbrace\xi_{k+1}\rbrace_{k=0}^{\infty}\subseteq[0,+\infty)$ satisfies $\sum_{k=0}^{\infty}\xi_{k+1}<\infty$. In the numerical experiment, we can choose $\xi_{k+1}=\frac{1}{(k+1)^{2}}$ as an example.

In order to guarantee the mesh sizes $\lbrace h_{k} \rbrace_{k=0}^{\infty}$ of each iteration satisfy $\sum_{k=0}^{\infty}h_{k+1}< \infty.$ In the numerical experiment, we choose $h_{k}=2^{-(k+4)}$ in Example \[ex1\], $h_{k}=\sqrt{2}/2^{(k+3)}$ in Example \[ex2\].

Numerical computation of the subproblems in Algorithm \[alg:multi-level ADMM algorithm\]
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For $u=\sum_{i=1}^{N_{h}}u_{i}\phi_{i}\in U_{h}$, $y=\sum_{i=1}^{N_{h}}y_{i}\phi_{i}\in Y_{h}$, let $\boldsymbol{\rm u}=(u_{1},...,u_{N_{h}})$, $\boldsymbol{\rm y}=(y_{1},...,y_{N_{h}})$ be the relative coefficient vectors respectively. Let the $L^{2}$-projections of $y_{r}$ and $y_{d}$ onto $Y_{h}$ be $y_{r,h}:=\sum_{i=1}^{N_{h}}y_{r}^{i}\phi_{i}(x)$ and $y_{d,h}:=\sum_{i=1}^{N_{h}}y_{d}^{i}\phi_{i}(x)$, respectively. Similarly, $\boldsymbol{\rm y}_{r}=(y_{r}^{1},y_{r}^{2}, ...,y_{r}^{N_{h}})$ and $\boldsymbol{\rm y}_{d}=(y_{d}^{1},y_{d}^{2}, ...,y_{d}^{N_{h}})$ denote their coefficient vectors. The stiffness matrix and the mass matrix are defined as: $$K_{h}:=(a(\phi_{i},\phi_{j}))_{i,j=1}^{N_{h}}, \ M_{h}:=\left( \int_{\Omega_{h}}\phi_{i}\phi_{j}dx \right)_{i,j=1}^{N_{h}}.$$ Furthermore, we define $$\begin{aligned}
f(\boldsymbol{\rm u})&:=\dfrac{1}{2}\|K_{h}^{-1}M_{h}(\boldsymbol{\rm u}+\boldsymbol{\rm y}_{r})-\boldsymbol{\rm y}_{d}\|_{M_{h}}^{2}+\dfrac{\alpha}{2}\|\boldsymbol{\rm u}\|_{M_{h}}^{2}, \\
g(\boldsymbol{\rm z})&:=\delta_{[a,b]^{N_{h}}}(\boldsymbol{\rm z}).\end{aligned}$$ Then we can obtain the matrix-vector form of Algorithm \[alg:multi-level ADMM algorithm\] in Algorithm \[alg:matrix-vector form\].

$(\boldsymbol{\rm u}^{0},\boldsymbol{\rm z}^{0},\boldsymbol{\rm \lambda}^{0})\in \mathbb{R}^{N_{h}}\times{[a,b]^{N_{h}}}\times\mathbb{R}^{N_{h}}$, 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 $O(\sum_{k=0}^{\infty}h_{k+1})\leq \infty.$ Set $k=0$. $\boldsymbol{\rm u}^{k},\boldsymbol{\rm z}^{k},\boldsymbol{\rm \lambda}^{k}$. Compute an approximation solution of $$\min_{\boldsymbol{\rm u}} f(\boldsymbol{\rm u})+\langle \boldsymbol{\rm I}_{h_{k+1}}\boldsymbol{\rm \lambda}_{h_{k}}^{k},M_{h_{k+1}}(\boldsymbol{\rm u}-\boldsymbol{\rm I}_{h_{k+1}}\boldsymbol{\rm z}^{k}_{h_{k}})\rangle+\dfrac{\sigma}{2}\|\boldsymbol{\rm u}-\boldsymbol{\rm I}_{h_{k+1}}\boldsymbol{\rm z}^{k}_{h_{k}}\|_{M_{h_{k+1}}}^{2},$$      such that 