PDE constraints numerically. One is *First discretize, then optimize*, another is *First optimize, then discretize* [@Hinze2009Optimization]. In the first approach, the discretization is applied to the original PDE-constrained optimization problem, while in the second one, the discretization is applied to the KKT system of the PDE-constrained optimization problem. Different from the strategies mentioned above, instead of applying discretized concept to problem (\[eqn:orginal problems\]) or its KKT system directly, *‘optimize-discretize-optimize’* strategy was proposed by Song in [@Song2018]. First the optimization algorithm is given in the sense of continuous function space, then the subproblems in each iteration are discretized by the standard piecewise linear finite element approach. Finally, numerical optimization methods are applied to solve the discretized subproblems numerically. The advantage of this method is that from the optimization algorithm in function space, we can have a better knowledge of the structure of the PDE-constrained optimization problem, which is important for choosing an appropriate discretization format to discretize the subproblems. Moreover, this strategy gives the freedom to discretize the subproblems differently, which makes the subproblems can be solved more effectively.

In this paper, we focus on the alternating direction method of multipliers (ADMM) method, which was originally proposed by Glowinski et al. 