A multigrid approach to optimal control computations for Navier-Stokes flows

Optimal control computations with boundary and distributed controls are presented by using a new multigrid approach for reliable and robust optimization. The implementation is based on a local Vanca solver for the Navier-Stokes and the adjoint system. The solution is achieved by solving and relaxing element by element the optimal control problem. In combination with this approach an imbedded domain method is presented to avoid the difficulties of the boundary stress computation for the adjoint equation which is not very accurate and robust in the traditional approach.The boundary control computations are shown to be more reliable if the classical boundary control approach is substituted by the distributed control over a suitable extended domain. Numerical tests for steady and unsteady solutions are presented. Also with this multigrid-Vanca approach the solution of the unsteady optimal problem is achieved by solving iteratively several optimal problems over all the time steps but over a limited number of degrees of freedom in space. This enhances enormously the capability of the adjoint method approach for complex eometries. Unsteady optimal control is presented in the form of a time piecewise optimal control problem which proves to be a very effective and robust method for flow tracking.