  We consider standard tracking-type, distributed elliptic optimal control
problems with $L^2$ regularization, and their finite element discretization. We
are investigating the $L^2$ error between the finite element approximation
$u_{\varrho h}$ of the state $u_\varrho$ and the desired state (target)
$\bar{u}$ in terms of the regularization parameter $\varrho$ and the mesh size
$h$ that leads to the optimal choice $\varrho = h^4$. It turns out that, for
this choice of the regularization parameter, we can devise simple Jacobi-like
preconditioned MINRES or Bramble-Pasciak CG methods that allow us to solve the
reduced discrete optimality system in asymptotically optimal complexity with
respect to the arithmetical operations and memory demand. The theoretical
results are confirmed by several benchmark problems with targets of various
regularities including discontinuous targets.
