Finite element discretizations of optimal control flow problems with boundary layers

Abstract

We study the effect of different stabilized finite element methods to distributed control problems governed by singular perturbed Oseen equations. On the one hand, the residual based stabilized finite element method SUPG/PSPG leads to different optimality systems depending on the discretization approach: first discretize the state equation and then formulate the corresponding optimality system or derive first the optimality system on the continuous level and then discretize it. On the other hand, for symmetric stabilization as for instance the local projection stabilization (LPS) both approaches lead to the same symmetric optimality system. In particular, we address the question whether a possible commutation error in optimal control problems with boundary layers discretized by stabilized finite element methods may affect the accuracy significantly or not. © 2011 Springer-Verlag.