论文标题
用于分布式椭圆形的最佳控制问题的自适应有限元方法与可变的能量正则化
An adaptive finite element method for distributed elliptic optimal control problems with variable energy regularization
论文作者
论文摘要
We analyze the finite element discretization of distributed elliptic optimal control problems with variable energy regularization, where the usual $L^2(Ω)$ norm regularization term with a constant regularization parameter $\varrho$ is replaced by a suitable representation of the energy norm in $H^{-1}(Ω)$ involving a variable, mesh-dependent regularization parameter $\varrho(x)$.事实证明，计算的有限元状态$ \ widetilde {u} _ {\ varrho h} $与所需的状态$ \ overline {u} $（target）在$ l^2（ω）$ norm中是最佳的，只要$ \ varrho（x）$ contance是$ \ varrho（x）$ contuly本地mesh尺寸。当使用自适应网格以近似不连续的目标函数时，这一点尤其重要。自适应方案可以由可计算和可本质的错误norm $ \ |驱动。 \ widetilde {u} _ {\ varrho h} - \ overline {u} \ | _ {l^2（ω）} $之间的有限元素状态$ \ wideTilde {u} _ {数值结果不仅说明了我们的理论发现，而且还表明，离散降低的最佳系统的迭代求解器非常有效且稳健。
We analyze the finite element discretization of distributed elliptic optimal control problems with variable energy regularization, where the usual $L^2(Ω)$ norm regularization term with a constant regularization parameter $\varrho$ is replaced by a suitable representation of the energy norm in $H^{-1}(Ω)$ 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 $L^2(Ω)$ 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 $\| \widetilde{u}_{\varrho h} - \overline{u}\|_{L^2(Ω)}$ 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.
