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

中文翻译：

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变能量正则化分布椭圆最优控制问题的自适应有限元方法

我们分析了具有可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，其中具有恒定正则化参数的通常范数正则化项被涉及可变的、依赖于网格的正则化参数的能量范数的合适表示所取代。事实证明，计算的有限元状态与期望状态（目标）之间的误差在范数中是最佳的，前提是其行为类似于局部网格尺寸的平方。当使用自适应网格来逼近不连续目标函数时，这一点尤其重要。自适应方案可以由有限元状态和目标之间的可计算和可定位的误差范数驱动。数值结果不仅说明了我们的理论发现，还表明离散化降低最优性系统的迭代求解器非常高效且鲁棒。