  In this paper, the elliptic PDE-constrained optimization problem with box
constraints on the control is studied. To numerically solve the problem, we
apply the 'optimize-discretize-optimize' strategy. Specifically, the
alternating direction method of multipliers (ADMM) algorithm is applied in
function space first, then the standard piecewise linear finite element
approach is employed to discretize the subproblems in each iteration. Finally,
some efficient numerical methods are applied to solve the discretized
subproblems based on their structures. Motivated by the idea of the multi-level
strategy, instead of fixing the mesh size before the computation process, we
propose the strategy of gradually refining the grid. Moreover, the subproblems
in each iteration are solved inexactly. Based on the strategies above, an
efficient convergent multi-level ADMM (mADMM) algorithm is proposed. We present
the convergence analysis and the iteration complexity results o(1/k) of the
proposed algorithm for the PDE-constrained optimization problems. Numerical
results show the high efficiency of the mADMM algorithm.
