y_{d}}-K_{h}\boldsymbol{\rm I}_{h_{k+1}}(\boldsymbol{\rm \lambda}^{k}+\sigma \boldsymbol{\rm I}_{h_{k+1}}\boldsymbol{\rm z}^{k}))\\
  M_{h}\boldsymbol{\rm y}_{r}
\end{array}   
\right].$$ The equivalent linear system (\[equ:linear\]) can be solved inexactly by some Krylov-based method.

Finally, we give a terminal condition of Algorithm \[alg:matrix-vector form\]. Let $\epsilon$ be a given accuracy tolerance, we terminate the algorithm when $\eta<\epsilon$, where the accuracy of a numerical solution is measured by the residual $\eta:={\rm max}\lbrace\eta_{1},\eta_{2},\eta_{3},\eta_{4},\eta_{5}\rbrace,$ where $$\begin{aligned}
\eta_{1}&=\frac{\|K_{h}\boldsymbol{\rm y}-M_{h}\boldsymbol{\rm u}-M_{h}\boldsymbol{\rm y_{r}}\|}{1+\|M_{h}{\boldsymbol{\rm y}}_{r}\|}, \ 
\eta_{2}=\frac{\|M_{h}(\boldsymbol{\rm u}-\boldsymbol{\rm z})\|}{1+\|\boldsymbol{\rm u}\|},\
\eta_{3}=\frac{\|M_{h}(\boldsymbol{\rm y}-{\boldsymbol{\rm y}}_{d})+K_{h}\boldsymbol{\rm p}\|}{1+\|M_{h}\boldsymbol{\rm y}_{d}\|},\\
\eta_{4}&=\frac{\|\alpha M_{h}\boldsymbol{\rm u}-M_{h}\boldsymbol{\rm p}+M_{h}\boldsymbol{\rm \lambda}\|}{1+\|\boldsymbol{\rm u}\|},\
\eta_{5}=\frac{\|\boldsymbol{\rm z}-\Pi_{[a,b]^{N_{h}}}(\boldsymbol{\rm z}+M_{h}\boldsymbol{\rm \lambda})\|}{1+\|\boldsymbol{\rm z}\|}.
\end{aligned}$$

Convergence analysis {#sec:3}
====================

In this section, based on the convergence results of the inexact ADMM in function space in Theorem \[convergence functional space\], we give the convergence analysis and the iteration complexity of the proposed mADMM algorithm.

[([@Song2017Fe], Thm. 2.5)]{} \[convergence functional space\] Suppose that the operator $L$ is uniformly elliptic. Let $(y^{\ast},u^{\ast},z^{\ast},p^{\ast},\lambda^{\ast})$ be the KKT point of (P), the sequence $\lbrace(u^{k},z^{k},\lambda^{k})\rbrace$ is generated by the inexact ADMM algorithm for (\[eqn:reduced problem with linear constraint\]) with the associated state $\lbrace y_{k}\rbrace$ and adjoint state $\lbrace p_{k}\rbrace$, then we have $$\begin{aligned}
&\lim \limits_{k\rightarrow \infty}\lbrace \|u^{k}-u^{\ast}\|_{L^{2}(\Omega)}+\|z^{k}-z^{\ast}\|_{L^{2}(\Omega)}+\|\lambda^{k}-\lambda^{\ast}\|_{L^{2}(\Omega)}\rbrace=0,\\
&\lim \limits_{k\rightarrow \infty}\lbrace \|y^{k}-y^{\ast}\|_{H_{0}^{1}(\Omega)}+\|p^{k}-p^{\ast}\|_{H_{0}^{1}(\Omega)}\rbrace=0.\end{aligned}$$ Moreover, there exists a constant $C$ only depends on the initial point 