Error estimates for spectral approximation of flow optimal control problem with $ L^2 $-norm control constraint

In this paper, we are concerned with the Galerkin spectral approximation of an optimal control problem governed by the Stokes equation with \begin{document}$ L^2 $\end{document}-norm constraint on the control variable. By means of the derived optimality conditions for both the original control system and its spectral approximation one, we establish a priori error estimates and then obtain a posteriori error estimator. A numerical example is, subsequently, executed to illustrate the effectiveness of method and the high performance of estimators. Furthermore, we conjecture that the similar conclusions should hold for optimal control of the Navier-Stokes equation. It is then confirmed by another numerical example.