consistent finite elements for otimal control problems in computational fluid dynamics

对于流体动力学领域的最优控制问题，由于流体动力学方程的离散化是非常微妙的，因此与纯椭圆型或抛物型问题相比，产生了额外的问题。造成这种情况的原因是多方面的：(a)对流项的处理通常需要稳定化或上旋技术，(b)连续供能条件不能延续到许多离散方案，这可能导致解的存在性和唯一性的丧失，(c)额外的能量或熵条件应该保持。当伴随问题被表述和求解时，这些点对优化问题有影响。例如，离散化和优化在一般情况下是不能互换的。本项目建议书的重点是分析离散化方法和解决这一领域的优化问题。特别是，我们检查，从连续到离散水平的交叉，使有益的性质得以保持。流体动力学方程的变化从不可压缩情况开始，发展到可压缩多相流体。从长远来看，我们有包括多物理场在内的优化问题，例如铝生产的优化。

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.