Robust Iterative Solvers for Algebraic Systems Arising from Elliptic Optimal Control Problems

We consider tracking-type, distributed elliptic optimal control problems with standard \(L_2\) and more general energy regularizations. We propose, analyze, and test new robust preconditioned iterative solvers for systems of linear algebraic equations arising from the finite element discretization of the reduced optimality systems defining the optimal solution in the case of the optimal choice of the regularization parameter. In particular, we study variable regularization parameters adapted to the local behavior of the mesh-size that can heavily change in the case of adaptive mesh refinements as required for discontinuous target functions.