The purpose of this paper is to investigate the effects of the use of mass-lumping in the finite element discretization of the reduced first-order optimality system arising from a standard tracking-type, distributed elliptic optimal control problem with $L_2$ regularization. We show that mass-lumping will not affect the $L_2$ error between the desired state and the computed state, but will lead to a Schur-complement system that allows for a fast matrix-by-vector multiplication. We show that the use of the Schur-Complement Preconditioned Conjugate Gradient method in a nested iteration setting leads to an asymptotically optimal solver with respect to the complexity.

翻译：本文旨在研究质量集中方法在标准跟踪型分布式椭圆最优控制问题（含$L_2$正则化）约化一阶最优性系统有限元离散中的影响。研究表明，质量集中不会影响期望状态与计算状态之间的$L_2$误差，但能生成允许快速矩阵-向量乘法的舒尔补系统。我们证明，在嵌套迭代框架中使用舒尔补预处理共轭梯度法可得到复杂度上渐近最优的求解器。