Optimal control finite element approximation for penalty variational formulation of three-dimensional Navier-Stokes problem

An optimal control finite element approximation of the penalization type for the solution of incompressible viscous Navier-Stokes equations is presented. This paper has proved the convergence of its minimizing sequence and the existence of the solution for the optimal control problem corresponding to the penalty variational problem of the Navier-Stokes equations. Based on this method and the conjugate gradient algorithm, the program for solving the three-dimensional Navier-Stokes problem has been developed, and some numerical results have also been provided.