  We analyze the finite element discretization of distributed elliptic optimal
control problems with variable energy regularization, where the usual
$L^2(\Omega)$ norm regularization term with a constant regularization parameter
$\varrho$ is replaced by a suitable representation of the energy norm in
$H^{-1}(\Omega)$ involving a variable, mesh-dependent regularization parameter
$\varrho(x)$. It turns out that the error between the computed finite element
state $\widetilde{u}_{\varrho h}$ and the desired state $\overline{u}$ (target)
is optimal in the $L^2(\Omega)$ norm provided that $\varrho(x)$ behaves like
the local mesh size squared. This is especially important when adaptive meshes
are used in order to approximate discontinuous target functions. The adaptive
scheme can be driven by the computable and localizable error norm $\|
\widetilde{u}_{\varrho h} - \overline{u}\|_{L^2(\Omega)}$ between the finite
element state $\widetilde{u}_{\varrho h}$ and the target $\overline{u}$. The
numerical results not only illustrate our theoretical findings, but also show
that the iterative solvers for the discretized reduced optimality system are
very efficient and robust.
