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This paper focuses on the quasi-optimality of an adaptive nonconforming FEMfor a distributed optimal control problem governed by the Stokes equations. Thenonconforming lowest order Crouzeix-Raviart element and piecewise constantspaces 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-optimalconvergence rates of the adaptive algorithm are established based on a generalaxiomatic framework that includes stability, reduction, discrete reliability,and quasi-orthogonality.

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