subproblems.

In the classical finite element based ADMM-type algorithm to solve the PDE constrained optimization problem, the subproblems are always discretized on a fixed mesh size, which results in large scale optimization problems. Thus it is important to consider reducing the computation cost. Multi-grid method is a modern field of research starting with the works of Brandt [@Brandt1977Multi] and Hackbusch [@Hackbusch1985; @Hackbusch1978On]. It is well known that multi-grid method solves elliptic problems with optimal computational complexity. Motivated by the idea of applying multi-grid method to tackle infinite dimension problems by Newton method in [@Deuflhard2011newton; @Hackbusch1985], we apply the multi-level strategy to our algorithm. It is important to point out that in the initial stage of the algorithm, the iteration precision is required to be relatively low, which means using coarse mesh is sufficient. While as the iteration process proceeds, the iteration precision is supposed to be higher and higher. In this case, using finer mesh is necessary. It is obvious that the strategy of gradually refining the grid can strongly reduce the computation cost and make the algorithm faster than computing the problem on a fixed mesh size.

The main contribution of this paper is that we give the reasonable 