strategy and propose an efficient convergent multi-level ADMM (mADMM) algorithm in Algorithm \[alg:multi-level ADMM algorithm\]. The strategies of gradually refining the grid and inexactly solving the subproblems are given to guarantee the convergence and efficiency of the mADMM algorithm.

The mADMM algorithm {#sec:2.1}
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Based on the inexact ADMM in function space, to numerically solve problem (\[eqn:reduced problem with linear constraint\]), we consider the finite element method. Specifically, piecewise linear functions are utilized to discretize the variables in the related subproblems appearing in each iteration of the inexact ADMM in function space. We first consider a family of regular and quasi-uniform triangulations $\lbrace \mathcal{T}_{h}\rbrace$ of $\bar \Omega$, i.e. $\bar \Omega=\bigcup_{T\in \mathcal{T}_{h}}\bar T$. With each element $T\in  \mathcal{T}_{h}$, we define the diameter of the set $T$ by $\rho_{T}:={\rm diam} \ T$ and let $\sigma_{T}$ denotes the diameter of the largest ball contained in $T$. The mesh size of the grid is defined by $h:={\rm max}_{T\in \mathcal{T}_{h}}\ \rho_{T}$. We suppose the following standard assumption in the context of error estimates holds (see[@Hinze2010Variational; @Hinze2009Optimization]).

(Regular and quasi-uniform triangulations) There exist two positive constants $\kappa$ and $\tau$ such that $$\frac{\rho_{T}}{\sigma_{T}}\leq \kappa, \quad \frac{h}{\rho_{T}}\leq \tau$$ hold for all $T\in \mathcal{T}_{h}$ and all $h>0$. Moreover, let 