Efficient Solution of Large-Scale Saddle Point Systems Arising in Riccati-Based Boundary Feedback Stabilization of Incompressible Stokes Flow

Peter Benner, Jens Saak, Martin Stoll, Heiko K. Weichelt

Abstract: We investigate numerical methods for solving large-scale saddle point systems which arise during the feedback control of flow problems. We focus on the instationary Stokes equations that describe instationary, incompressible flows for moderate viscosities. After a mixed finite element discretization we get a differential-algebraic system of differential index two [J. Weickert, Navier-Stokes Equations as a Differential-Algebraic System, Preprint SFB393/96-08, Department of Mathematics, Chemnitz University of Technology, Chemnitz, Germany, 1996]. To reduce this index, we follow the analytic ideas of [J.-P. Raymond, SIAM J. Control Optim., 45 (2006), pp. 790--828] coupled with the projection idea of [M. Heinkenschloss, D. C. Sorensen, and K. Sun, SIAM J. Sci. Comput., 30 (2008), pp. 1038--1063]. Avoiding this explicit projection leads to solving a series of large-scale saddle point systems. In this paper we construct iterative methods to solve such saddle point systems by deriving efficient preconditioners based on the approaches of Wathen and colleagues, e.g., [M. Stoll and A. Wathen, J. Comput. Phys., 232 (2013), pp. 498--515]. In addition, the main results can be extended to the nonsymmetric case of linearized Navier--Stokes equations. We conclude with numerical examples showcasing the performance of our preconditioned iterative saddle point solver.