Moreau–Yosida Regularization in State Constrained Elliptic Control Problems: Error Estimates and Parameter Adjustment

An adjustment scheme for the regularization parameter of a Moreau–Yosida-based regularization, or relaxation, approach to the numerical solution of pointwise state constrained elliptic optimal control problems is introduced. The method utilizes error estimates of an associated finite element discretization of the regularized problems for the optimal selection of the regularization parameter in dependence on the mesh size of discretization and error estimates for the approximation error due to regularization. The theoretical results are verified numerically.