consistent finite elements for otimal control problems in computational fluid dynamics

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For optimal control problems in the field of fluid dynamics additional questions arise compared to pure elliptic or parabolic problems, because the discretization of fluid dynamical equations is very delicate. The reasons for this are multifaceted: (a) the treatment of convective terms usually require stabilization or upwinding techniques, (b) the continuous inf-sup condition do not carry over to many discrete schemes, which may lead to the lost of existence and uniqueness of solutions, (c) additional energy or entropy conditions should be maintained. These points have consequences in optimization problems when the adjoint problems are formulated and solved. For instance, discretization and optimization may not commute in general.The focal point of this project proposal is the analysis of discretization methods and solving aspects in this field of optimization problems. In particular, we examine, the crossover from the continuous to the discrete level, so that beneficial properties became maintained. The variety of fluid dynamical equations starts with the incompressible case and reaches compressible multiphase fluids. On the long-term, we have optimization problems including multi-physics in mind, as e.g. the optimization of aluminum production.