For robust discretizations of the Navier-Stokes equations with small viscosity, standard Galerkin schemes have to be augmented by stabilization terms due to the indefinite convective terms and due to a possible lost of a discrete inf-sup condition. For optimal control problems for fluids such stabilization have in general an undesired effect in the sense that optimization and discretization do not commute. This is the case for the combination of streamline upwind Petrov-Galerkin (SUPG) and pressure stabilized Petrov-Galerkin (PSPG). In this work we study the effect of different stabilized finite element methods to distributed control problems governed by singular perturbed Oseen equations. In particular, we address the question whether a possible commutation error in optimal control problems lead to a decline of convergence order. Therefore, we give a priori estimates for SUPG/PSPG. In a numerical study for a flow with boundary layers, we illustrate to which extend the commutation error affects the accuracy.
Mots-clés : Oseen, Navier-Stokes, optimal control, finite elements, stabilized methods
@article{COCV_2012__18_4_987_0,
author = {Braack, Malte and Tews, Benjamin},
title = {Linear-quadratic optimal control for the {Oseen} equations with stabilized finite elements},
journal = {ESAIM: Control, Optimisation and Calculus of Variations},
pages = {987--1004},
publisher = {EDP-Sciences},
volume = {18},
number = {4},
year = {2012},
doi = {10.1051/cocv/2011201},
mrnumber = {3019469},
zbl = {1270.35017},
language = {en},
url = {http://archive.numdam.org/articles/10.1051/cocv/2011201/}
}
TY - JOUR AU - Braack, Malte AU - Tews, Benjamin TI - Linear-quadratic optimal control for the Oseen equations with stabilized finite elements JO - ESAIM: Control, Optimisation and Calculus of Variations PY - 2012 SP - 987 EP - 1004 VL - 18 IS - 4 PB - EDP-Sciences UR - http://archive.numdam.org/articles/10.1051/cocv/2011201/ DO - 10.1051/cocv/2011201 LA - en ID - COCV_2012__18_4_987_0 ER -
%0 Journal Article %A Braack, Malte %A Tews, Benjamin %T Linear-quadratic optimal control for the Oseen equations with stabilized finite elements %J ESAIM: Control, Optimisation and Calculus of Variations %D 2012 %P 987-1004 %V 18 %N 4 %I EDP-Sciences %U http://archive.numdam.org/articles/10.1051/cocv/2011201/ %R 10.1051/cocv/2011201 %G en %F COCV_2012__18_4_987_0
Braack, Malte; Tews, Benjamin. Linear-quadratic optimal control for the Oseen equations with stabilized finite elements. ESAIM: Control, Optimisation and Calculus of Variations, Tome 18 (2012) no. 4, pp. 987-1004. doi : 10.1051/cocv/2011201. http://archive.numdam.org/articles/10.1051/cocv/2011201/
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