A projection-based variational multiscale method for the optimal control problems governed by the stationary Navier-Stokes equations

In this work, we apply a variational multiscale stabilization (VMS) to the optimal control problems of Navier-Stokes equations. We first obtain the optimality conditions by using Lagrange approach. After stating the continuous optimality system, we formulate the discrete problem by a projection-based stabilized finite element scheme. We give the stability estimates for both state and adjoint state variables. Then, we present the a priori error analysis. The main issue of this paper is to show the efficiency of the variational multiscale stabilization for this optimal control problem. We verify our results by numerical examples.