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.

翻译：我们分析了带有变能量正则化的分布式椭圆最优控制问题的有限元离散化，其中通常采用常数正则化参数$\varrho$的$L^2(\Omega)$范数正则化项被替换为涉及变量且依赖网格的正则化参数$\varrho(x)$的$H^{-1}(\Omega)$能量范数的合适表示。结果表明，当$\varrho(x)$的行为类似于局部网格尺寸的平方时，计算得到的有限元状态$\widetilde{u}_{\varrho h}$与期望状态$\overline{u}$（目标）之间的误差在$L^2(\Omega)$范数下是最优的。这在采用自适应网格逼近不连续目标函数时尤为重要。自适应方案可由有限元状态$\widetilde{u}_{\varrho h}$与目标$\overline{u}$之间的可计算且可局部化的误差范数$\| \widetilde{u}_{\varrho h} - \overline{u}\|_{L^2(\Omega)}$驱动。数值结果不仅验证了我们的理论发现，还表明离散化简化最优性系统的迭代求解器非常高效且稳健。