Semi-discrete a priori error analysis for the optimal control of the unsteady Navier-Stokes equations with variational multiscale stabilization

In this work, the optimal control problems of the unsteady Navier-Stokes equations with variational multiscale stabilization (VMS) are considered. At first, the first order continuous optimality conditions are obtained. Since the adjoint equation of the Navier-Stokes problem is a convection diffusion type system, then the same stabilization is applied to it. Semi discrete a priori error estimates are obtained for the state, adjoint state and control variables. Crank-Nicholson time discretization is used to get the fully discrete scheme. Numerical examples verify the theoretical findings and show the efficiency of the stabilization for higher Reynolds number.