Linear-quadratic optimal control for the Oseen equations with stabilized finite elements

Malte Braack; Benjamin Tews

ESAIM: Control, Optimisation and Calculus of Variations (2012)

  • Volume: 18, Issue: 4, page 987-1004
  • ISSN: 1292-8119

Abstract

top
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.

How to cite

top

Braack, Malte, and Tews, Benjamin. "Linear-quadratic optimal control for the Oseen equations with stabilized finite elements." ESAIM: Control, Optimisation and Calculus of Variations 18.4 (2012): 987-1004. <http://eudml.org/doc/272857>.

@article{Braack2012,
abstract = {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.},
author = {Braack, Malte, Tews, Benjamin},
journal = {ESAIM: Control, Optimisation and Calculus of Variations},
keywords = {Oseen; Navier-Stokes; optimal control; finite elements; stabilized methods; quadratic cost functional; discretize-optimize; optimize-discretize; boundary layers},
language = {eng},
number = {4},
pages = {987-1004},
publisher = {EDP-Sciences},
title = {Linear-quadratic optimal control for the Oseen equations with stabilized finite elements},
url = {http://eudml.org/doc/272857},
volume = {18},
year = {2012},
}

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
PB - EDP-Sciences
VL - 18
IS - 4
SP - 987
EP - 1004
AB - 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.
LA - eng
KW - Oseen; Navier-Stokes; optimal control; finite elements; stabilized methods; quadratic cost functional; discretize-optimize; optimize-discretize; boundary layers
UR - http://eudml.org/doc/272857
ER -

References

top
  1. [1] F. Abraham, M. Behr and M. Heinkenschloss, 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. MR2097615
  2. [2] R. Becker and M. Braack, 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. Zbl1198.76062MR2121360
  3. [3] R. Becker and B. Vexler, Optimal control of the convection-diffusion equation using stabilized finite element methods. Numer. Math.106 (2007) 349–367. Zbl1133.65037MR2302057
  4. [4] M. Braack, Optimal control in fluid mechanics by finite elements with symmetric stabilization. SIAM J. Control Optim.48 (2009) 672–687. Zbl1186.35134MR2486088
  5. [5] M. Braack and E. Burman, Local projection stabilization for the Oseen problem and its interpretation as a variational multiscale method. SIAM J. Numer. Anal.43 (2006) 2544–2566. Zbl1109.35086MR2206447
  6. [6] M. Braack, E. Burman, V. John and G. Lube, Stabilized finite element methods for the generalized Oseen problem. Comput. Methods Appl. Mech. Engrg.196 (2007) 853–866. Zbl1120.76322MR2278180
  7. [7] A.N. Brooks and T.J.R. Hughes, 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. Zbl0497.76041MR679322
  8. [8] S.S. Collis and M. Heinkenschloss, 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. [9] L. Dedé and A. Quarteroni, Optimal control and numercal adaptivity for advection-diffusion equations. ESIAM : M2AN 39 (2005) 1019–1040. Zbl1075.49014MR2178571
  10. [10] V. Girault and P.-A. Raviart, Finite Elements for the Navier Stokes Equations. Springer, Berlin (1986). Zbl0413.65081
  11. [11] M. Heinkenschloss and D. Leykekhman, Local error estimates for SUPG solutions of advection-dominated elliptic linear-quadratic optimal control problems. SIAM J. Numer. Anal.47 (2010) 4607–4638. Zbl1218.49036MR2595051
  12. [12] M. Hinze, N. Yan and Z. Zhou, Variational discretization for optimal control governed by convection dominated diffusion equations. J. Comput. Math.27 (2009) 237–253. Zbl1212.65248MR2495058
  13. [13] C. Johnson and J. Saranen, Streamline diffusion methods for the incompressible Euler and Navier-Stokes equations. Math. Comput.47 (1986) 1–18. Zbl0609.76020MR842120
  14. [14] G. Lube and G. Rapin, Residual-based stabilized higher-order FEM for a generalized Oseen problem. Math. Models Methods Appl. Sci.16 (2006) 949–966. Zbl1095.76032MR2250026
  15. [15] G. Lube and G. Rapin, Residual-based stabilized higher-order FEM for a generalized Oseen problem. Math. Models Methods Appl. Sci.16 (2006) 949–966. Zbl1095.76032MR2250026
  16. [16] G. Lube and B. Tews, Optimal control of singularly perturb advection-diffusion-reaction problems. Math. Models Appl. Sci.20 (2010) 1–21. Zbl1193.65106MR2647025
  17. [17] G. Matthies, P. Skrzypacz and L. Tobiska, A unified convergence analysis for local projection stabilisations applied ro the Oseen problem. ESAIM : M2AN 41 (2007) 713–742. Zbl1188.76226MR2362912
  18. [18] N. Yan and Z. Zhou, A priori and a posteriori error estimates of streamline diffusion finite element method for optimal control problems governed by convection dominated diffusion equation. NMTMA1 (2008) 297–320. Zbl1174.65433MR2460012
  19. [19] N. Yan and Z. Zhou, 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. Zbl1156.65069MR2463111

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

                
Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.