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://www.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://www.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://www.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://www.numdam.org/articles/10.1051/cocv/2011201/
[1] , and , The effect of stabilization in finite element methods for the optimal boundary control of the Oseen equations. Finite Elem. Anal. Des. 41 (2004) 229-251. | MR
[2] and , A two-level stabilization scheme for the Navier-Stokes equations, in Numerical Mathematics and Advanced Applications, ENUMATH 2003. edited by, M. Feistauer et al., Springer (2004) 123-130. | MR | Zbl
[3] and , Optimal control of the convection-diffusion equation using stabilized finite element methods. Numer. Math. 106 (2007) 349-367. | MR | Zbl
[4] , Optimal control in fluid mechanics by finite elements with symmetric stabilization. SIAM J. Control Optim. 48 (2009) 672-687. | MR | Zbl
[5] and , Local projection stabilization for the Oseen problem and its interpretation as a variational multiscale method. SIAM J. Numer. Anal. 43 (2006) 2544-2566. | MR | Zbl
[6] , , and , Stabilized finite element methods for the generalized Oseen problem. Comput. Methods Appl. Mech. Engrg. 196 (2007) 853-866. | MR | Zbl
[7] and , Streamline upwind Petrov-Galerkin formulation for convection dominated flows with particular emphasis on the incompressible Navier-Stokes equations. Comput. Methods Appl. Mech. Engrg. 32 (1982) 199-259. | MR | Zbl
[8] and , Analysis of the streamline upwind/Petrov Galerkin method applied to the solution of optimal control problems. Technical report 02-01, Rice University, Houston, TX (2002).
[9] and , Optimal control and numercal adaptivity for advection-diffusion equations. ESIAM : M2AN 39 (2005) 1019-1040. | Numdam | MR | Zbl
[10] and , Finite Elements for the Navier Stokes Equations. Springer, Berlin (1986).
[11] and , Local error estimates for SUPG solutions of advection-dominated elliptic linear-quadratic optimal control problems. SIAM J. Numer. Anal. 47 (2010) 4607-4638. | MR | Zbl
[12] , and , Variational discretization for optimal control governed by convection dominated diffusion equations. J. Comput. Math. 27 (2009) 237-253. | MR | Zbl
[13] and , Streamline diffusion methods for the incompressible Euler and Navier-Stokes equations. Math. Comput. 47 (1986) 1-18. | MR | Zbl
[14] and , Residual-based stabilized higher-order FEM for a generalized Oseen problem. Math. Models Methods Appl. Sci. 16 (2006) 949-966. | MR | Zbl
[15] and , Residual-based stabilized higher-order FEM for a generalized Oseen problem. Math. Models Methods Appl. Sci. 16 (2006) 949-966. | MR | Zbl
[16] and , Optimal control of singularly perturb advection-diffusion-reaction problems. Math. Models Appl. Sci. 20 (2010) 1-21. | MR | Zbl
[17] , and , A unified convergence analysis for local projection stabilisations applied ro the Oseen problem. ESAIM : M2AN 41 (2007) 713-742. | Numdam | MR | Zbl
[18] and , A priori and a posteriori error estimates of streamline diffusion finite element method for optimal control problems governed by convection dominated diffusion equation. NMTMA 1 (2008) 297-320. | MR | Zbl
[19] and , A priori and a posteriori error analysis of edge stabilization Galerkin method for the optimal control problem governed by convection dominated diffusion equation. J. Comput. Appl. Math. 223 (2009) 198-217. | MR | Zbl
Cité par Sources :
