OPTIMAL CONTROL IN FLUID MECHANICS BY FINITE ELEMENTS WITH SYMMETRIC STABILIZATION

There are two main possibilities for the numerical computation of optimal control problems with constraints given by partial differential equations: One may consider first the discretized problem and then build the optimality condition. The other possibility is to formulate first the optimality condition on the continuous level and then discretize. Both approaches may lead to different discrete adjoint equations because discretization and building the adjoint do not commute in general. This type of inconsistency takes place when conventional stabilized finite elements for flow problems, as for instance, streamline diffusion (SUPG), are used, due to its nonsymmetry. Consequently, the computed control is significantly affected by the way of de. ning the discrete optimality condition. Hence, there is a need for symmetric stabilization so that discretization and building the adjoint commute. We formulate the use of this kind of stabilization and give a quasi-optimal a priori estimate in the context of optimal control problems for the Oseen system. In particular, we show that local projection stabilization and edge-oriented stabilization result to be quasi-optimal for optimal control problems.