Convergence of Adaptive FEM for Distributed Optimal Control Problems Governed by Stokes Equation

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.