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Linear-quadratic optimal control for the Oseen equations withstabilized finite elements

Published online by Cambridge University Press:  16 January 2012

Malte Braack  and
Benjamin Tews
Malte Braack
Affiliation:
Mathematisches Seminar, Christian-Albrechts-Universität zu Kiel, Ludewig-Meyn-Str. 4, 24098 Kiel, Germany. braack@math.uni-kiel.de; tews@math.uni-kiel.de
Benjamin Tews
Affiliation:
Mathematisches Seminar, Christian-Albrechts-Universität zu Kiel, Ludewig-Meyn-Str. 4, 24098 Kiel, Germany. braack@math.uni-kiel.de; tews@math.uni-kiel.de
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Abstract

For robust discretizations of the Navier-Stokes equations with small viscosity, standardGalerkin schemes have to be augmented by stabilization terms due to the indefiniteconvective terms and due to a possible lost of a discrete inf-sup condition. For optimalcontrol problems for fluids such stabilization have in general an undesired effect in thesense that optimization and discretization do not commute. This is the case for thecombination of streamline upwind Petrov-Galerkin (SUPG) and pressure stabilizedPetrov-Galerkin (PSPG). In this work we study the effect of different stabilized finiteelement methods to distributed control problems governed by singular perturbed Oseenequations. In particular, we address the question whether a possible commutation error inoptimal control problems lead to a decline of convergence order. Therefore, we givea priori estimates for SUPG/PSPG. In a numerical study for a flow withboundary layers, we illustrate to which extend the commutation error affects theaccuracy.

Keywords

OseenNavier-Stokesoptimal controlfinite elementsstabilized methods
Type
Research Article
Information
ESAIM: Control, Optimisation and Calculus of Variations , Volume 18 , Issue 4 , October 2012 , pp. 987 - 1004
Copyright
© EDP Sciences, SMAI, 2012

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