We analyze the finite element discretization of distributed elliptic optimal
control problems with variable energy regularization, where the usual
L2(Ω) norm regularization term with a constant regularization parameter
ϱ is replaced by a suitable representation of the energy norm in
H−1(Ω) involving a variable, mesh-dependent regularization parameter
ϱ(x). It turns out that the error between the computed finite element
state uϱh and the desired state uˉ (target) is
optimal in the L2(Ω) norm provided that ϱ(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 ∥uϱh−uˉ∥L2(Ω) between the finite element
state uϱh and the target uˉ. 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