$\lbrace\epsilon_{k}\rbrace_{k=0}^{\infty}\subseteq[0,+\infty)$ and $\sum_{k=0}^{\infty}\epsilon_{k}<\infty$, the iterative scheme of the inexact ADMM in function space for (\[eqn:reduced problem with linear constraint\]) is as follows: $$\left\{\begin{aligned}
\text{Step 1}:\ &\text{Compute an approximation solution $u^{k+1}$ of}\\
&\min_{u} \hat{J}(u)+\langle \lambda^{k},u-z^{k}\rangle_{L^{2}(\Omega)}+\dfrac{\sigma}{2}\|u-z^{k}\|_{L^{2}(\Omega)}^{2}\\
&\text{such that the error vector}\  \delta_{u}^{k+1}:=\nabla \hat{J}(u^{k+1})+\lambda^{k}+\sigma(u^{k+1}-z^{k}) \ \text{satisfies }  \|\delta_{u}^{k+1}\|_{L^{2}(\Omega)}\leq \epsilon_{k}.\\
\text{Step 2}: \ &z^{k+1}={\rm argmin} \delta_{U_{ad}}(z)+\langle \lambda^{k},u^{k+1}-z\rangle_{L^{2}(\Omega)}+\dfrac{\sigma}{2}\|u^{k+1}-z\|_{L^{2}(\Omega)}^{2}.\\
\text{Step 3}: \ &\lambda^{k+1}=\lambda^{k}+\tau \sigma(u^{k+1}-z^{k+1}).
\end{aligned} \right.$$

In [@Song2017Fe], the discretized problem is considered and the level of discretization is fixed. In this paper, instead of considering the discretized problem, the finite element method is employed to discretize the subproblems in each iteration of the inexact ADMM algorithm. This strategy gives the freedom and flexibility to discretize the subproblems by different discretization schemes. The total error of the proposed algorithm for (\[eqn:reduced problem with linear constraint\]) is consisted of two parts: the discretization error and the iteration error resulted from inexactly solving the discretized subproblems. For these two errors, our algorithm can be considered as an approximation of exact ADMM in function space, thus we regard it as an inexact ADMM algorithm in function space. In order to guarantee the convergence behavior of our algorithm, we consider controlling the mesh size and the inexactness of solving the 