Special Lecture - 10/01, 4:00PM, Keck Hall 105

Heiko K. Weichelt

Dept. of Mathematics, Chemnitz University of Technology, Germany
Research Group Computational Methods in Systems and Control Theory (CSC)
Max-Planck-Institute for Dynamics of Complex Technical Systems, Magdeburg, Germany

"Numerical Solution of the Feedback Control Problem for Navier-Stokes and Multi-Field Flow Problems: Ideas, Approaches, and Open Questions"

We discuss the feedback stabilization of flow problems described by the incompressible Navier-Stokes equations. In the last decade, a series of papers by Raymond and co-workers showed that for small perturbations, the deviation from a nominal flow, defined by a possibly unstable solution of the steady Navier-Stokes equations, can be steered to zero at an exponential convergence rate using an LQR problem for the velocity field projected onto a suitable space of divergence-free functions. We show how to solve this LQR problem numerically using the associated algebraic (operator) Riccati equation. The key idea is to avoid the explicit Helmholtz projection onto the divergence-free vector fields by employing a saddle point formulation discussed already by Heinkenschloss et al. (Heinkenschloss/Sorensen/Sun '08) in the context of balanced truncation model reduction. Also, a number of other issues such as initializing Newton's method for the algebraic Riccati equations, need to be solved to derive a working algorithm for the numerical solution of the flow stabilization problem. We will show how the computed feedback control using this approach effectively stabilizes unstable flows using as test examples von Karman vortex shedding and the coupled systems of a reactive substance transported by an incompressible fluid.

Furthermore, we investigate the arising large-scale saddle point systems which have to be solved in a threefold nested iteration. For obtaining a fast iterative solution of those non-symmetric systems we derive efficient preconditioners based on the approaches due to Wathen et al. (Elman/Silvester/Wathen '05, Stoll/Wathen '11). Finally we show recent numerical results regarding the arising nested iteration.

Joint work with Peter Benner and Jens Saak.