  We propose, analyze, and test new iterative solvers for large-scale systems
of linear algebraic equations arising from the finite element discretization of
reduced optimality systems defining the finite element approximations to the
solution of elliptic tracking-type distributed optimal control problems with
both the standard $L_2$ and the more general energy regularizations. If we aim
at an approximation of the given desired state $y_d$ by the computed finite
element state $y_h$ that asymptotically differs from $y_d$ in the order of the
best $L_2$ approximation under acceptable costs for the control, then the
optimal choice of the regularization parameter $\varrho$ is linked to the
mesh-size $h$ by the relations $\varrho=h^4$ and $\varrho=h^2$ for the $L_2$
and the energy regularization, respectively. For this setting, we can construct
efficient parallel iterative solvers for the reduced finite element optimality
systems. These results can be generalized to variable regularization parameters
adapted to the local behavior of the mesh-size that can heavily change in case
of adaptive mesh refinement. Similar results can be obtained for the space-time
finite element discretization of the corresponding parabolic and hyperbolic
optimal control problems.
