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

Content Provider	Society for Industrial and Applied Mathematics (SIAM)
Author	Benner, Peter Stoll, Martin Saak, Jens Weichelt, Heiko K.
Copyright Year	2013
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.
Starting Page	S150
Ending Page	S170
Page Count	21
File Format	PDF
ISSN	10648275
DOI	10.1137/120881312
e-ISSN	10957197
Issue Number	5
Volume Number	35
Language	English
Publisher	Society for Industrial and Applied Mathematics
Publisher Date	2013-10-28
Access Restriction	Subscribed
Subject Keyword	Schur complement approximation Iterative methods for linear systems saddle point systems Newton-type methods Riccati-based feedback Stabilization of systems by feedback Flow control and optimization flow control Preconditioners for iterative methods Stokes equations
Content Type	Text
Resource Type	Article
Subject	Applied Mathematics Computational Mathematics

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