This paper focuses on the quasi-optimality of an adaptive nonconforming FEM for a distributed optimal control problem governed by the Stokes equations. The nonconforming lowest order Crouzeix-Raviart element and piecewise constant spaces are used to discretize the velocity and pressure variables, respectively. The control variable is discretized using a variational approach. We present error equivalence results at both continuous and discrete levels, leading to a priori and a posteriori error estimates. The quasi-optimal convergence rates of the adaptive algorithm are established based on a general axiomatic framework that includes stability, reduction, discrete reliability, and quasi-orthogonality.

翻译：本文聚焦于由 Stokes 方程支配的分布最优控制问题的自适应非协调有限元法的拟最优性。采用非协调最低阶 Crouzeix-Raviart 元及分片常数空间分别对速度与压力变量进行离散。控制变量采用变分方法进行离散。我们在连续与离散两个层面给出误差等价性结果，进而推导出先验与后验误差估计。基于包含稳定性、缩减性、离散可靠性及拟正交性的一般公理化框架，建立了自适应算法的拟最优收敛率。