07-8   Berichtsreihe des Mathematischen Seminars der Universität Kiel

Malte Braack:

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 a partial differential equation: 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 SUPG are used, because they are non-symmetric. Consequently, the computed control are significantly affected by the way of defining the discrete optimality condition. Therefore, there is a need for symmetric stabilization so that discretization and building the adjoint commute. In this paper we formulate the use of the symmetric local projection stabilization in the context of optimal control problems for the Oseen system in order to obtain a full consistent and stable adjoint problem. Furthermore, we give a quasi-optimal a priori estimate.

Mail an Jens Burmeister
[Thu Feb 19 18:56:37 2009]