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We analyze the finite element discretization of distributed elliptic optimalcontrol 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 elementstate $\widetilde{u}_{\varrho h}$ and the desired state $\overline{u}$ (target)is optimal in the $L^2(\Omega)$ norm provided that $\varrho(x)$ behaves likethe local mesh size squared. This is especially important when adaptive meshesare used in order to approximate discontinuous target functions. The adaptivescheme can be driven by the computable and localizable error norm $\|\widetilde{u}_{\varrho h} - \overline{u}\|_{L^2(\Omega)}$ between the finiteelement state $\widetilde{u}_{\varrho h}$ and the target $\overline{u}$. Thenumerical results not only illustrate our theoretical findings, but also showthat the iterative solvers for the discretized reduced optimality system arevery efficient and robust.

An adaptive finite element method for distributed elliptic optimal control problems with variable energy regularization