This paper presents an efficient numerical framework for solving control-constrained elliptic optimal control problems with \(L^1\) control costs. The sparse regularization term makes the alternating direction method of multipliers (ADMM) a natural and effective solver. However, although each ADMM iteration involves only modest computational effort, the method typically requires many iterations to achieve a high-accuracy solution. We observe that the overall error for this class of problems is often dominated by the inherent discretization error, not the optimization error. This motivates an inexact ADMM scheme that leverages an a priori estimate of the discretization error. The core of our scheme is a novel mesh-dependent stopping criterion, which automatically terminates iterations once the discretization error dominates. The proposed method thus preserves the convergence order of the underlying finite element discretization while drastically reducing computational cost. Numerical experiments confirm the efficiency of our approach, showing substantial savings over standard ADMM. Notably, they also reveal that the early-stopped solution can, in some cases, achieve a lower overall error than the fully converged “exact" solution, providing a counterintuitive yet practically valuable insight.
