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Parallel solvers for flow control based on domain decomposition
Author(s) -
Ketelaer Eva,
Heuveline Vincent,
Heinkenschloss Mathias
Publication year - 2013
Publication title -
pamm
Language(s) - English
Resource type - Journals
ISSN - 1617-7061
DOI - 10.1002/pamm.201310193
Subject(s) - domain decomposition methods , schur complement , partial differential equation , preconditioner , optimal control , mathematics , quadratic equation , scalar (mathematics) , mathematical optimization , computer science , linear system , mathematical analysis , finite element method , eigenvalues and eigenvectors , physics , geometry , quantum mechanics , thermodynamics
We sketch a non‐overlapping domain decomposition method (DDM) for a linear quadratic optimal control problem governed by the Oseen equations. The DDM is applied to the system of necessary and sufficient optimality conditions. The approach extends balanced Neumann Neumann DDMs from single partial differential equations (PDEs) to the optimization control context, and it extends previous work on balanced Neumann Neumann DDMs for the optimal control of scalar elliptic PDEs to the optimal control of the Oseen equations. This extension requires a careful handling of the incompressibility constraint and resulting compatibility conditions, as well as a careful handling of the advection term. The DDM is used to parallelize the matrix‐vector operations for the optimality system, as well as to parallelize the preconditioner. We present two approaches. One tackles the global optimality system directly, the other forms the Schur complement corresponding to variables on the subdomain interfaces. We present numerical experiments which clearly show the potential of the approaches. (© 2013 Wiley‐VCH Verlag GmbH & Co. KGaA, Weinheim)