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 $\bar{u}$ (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 $\| \widetilde{u}_{\varrho h} - \bar{u}\|_{L^2(\Omega)}$ between the finite element state $\widetilde{u}_{\varrho h}$ and the target $\bar{u}$. 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：我们分析了与可变能源正规化有关的分布式椭圆最佳控制问题的有限元素分解, 通常的美元( 2/\\ OMEGA) 常规规范规范条款以恒定规范参数 $\\ varrho$ 替换为 $H\ -1} (\ OMEGA) 美元( 美元) 的能源规范的适当表述。 当使用适应性 meshes 来接近不连续的目标功能时, 这一点尤其重要。 适应性方案可以由可计算性和可本地化的错误规范 $\\\ 全局性{ u{ varrho} 美元 -\ bar{ { { ru} (目标) 在 $( 2/\\\ \ \ Omega) 标准条款中是最佳的, 只要$( varrho) 美元( x) 的正常行为方式与本地的mesh 大小。 当使用适应性 meshes 以近似不连续目标功能时, 这尤其重要。 适应性方案可以由可调和可本地化的错误规范 $\\\\ divodealtialtile\\ r} res res rodual rodudeal fal res resmus 唯一的系统之间 唯一的结果。