Analysis and finite element error estimates for the velocity tracking problem for Stokes flows via a penalized formulation

Konstantinos Chrysafinos

ESAIM: Control, Optimisation and Calculus of Variations (2004)

  • Volume: 10, Issue: 4, page 574-592
  • ISSN: 1292-8119

Abstract

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A distributed optimal control problem for evolutionary Stokes flows is studied via a pseudocompressibility formulation. Several results concerning the analysis of the velocity tracking problem are presented. Semidiscrete finite element error estimates for the corresponding optimality system are derived based on estimates for the penalized Stokes problem and the BRR (Brezzi-Rappaz-Raviart) theory. Finally, the convergence of the solutions of the penalized optimality systems as ε → 0 is examined.

How to cite

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Chrysafinos, Konstantinos. "Analysis and finite element error estimates for the velocity tracking problem for Stokes flows via a penalized formulation." ESAIM: Control, Optimisation and Calculus of Variations 10.4 (2004): 574-592. <http://eudml.org/doc/245553>.

@article{Chrysafinos2004,
abstract = {A distributed optimal control problem for evolutionary Stokes flows is studied via a pseudocompressibility formulation. Several results concerning the analysis of the velocity tracking problem are presented. Semidiscrete finite element error estimates for the corresponding optimality system are derived based on estimates for the penalized Stokes problem and the BRR (Brezzi-Rappaz-Raviart) theory. Finally, the convergence of the solutions of the penalized optimality systems as $\varepsilon \rightarrow 0$ is examined.},
author = {Chrysafinos, Konstantinos},
journal = {ESAIM: Control, Optimisation and Calculus of Variations},
keywords = {optimal control; velocity tracking; finite elements; semidiscrete error estimates; Stokes equations; penalized formulation},
language = {eng},
number = {4},
pages = {574-592},
publisher = {EDP-Sciences},
title = {Analysis and finite element error estimates for the velocity tracking problem for Stokes flows via a penalized formulation},
url = {http://eudml.org/doc/245553},
volume = {10},
year = {2004},
}

TY - JOUR
AU - Chrysafinos, Konstantinos
TI - Analysis and finite element error estimates for the velocity tracking problem for Stokes flows via a penalized formulation
JO - ESAIM: Control, Optimisation and Calculus of Variations
PY - 2004
PB - EDP-Sciences
VL - 10
IS - 4
SP - 574
EP - 592
AB - A distributed optimal control problem for evolutionary Stokes flows is studied via a pseudocompressibility formulation. Several results concerning the analysis of the velocity tracking problem are presented. Semidiscrete finite element error estimates for the corresponding optimality system are derived based on estimates for the penalized Stokes problem and the BRR (Brezzi-Rappaz-Raviart) theory. Finally, the convergence of the solutions of the penalized optimality systems as $\varepsilon \rightarrow 0$ is examined.
LA - eng
KW - optimal control; velocity tracking; finite elements; semidiscrete error estimates; Stokes equations; penalized formulation
UR - http://eudml.org/doc/245553
ER -

References

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  3. [3] A. Fursikov,Optimal control of distributed systems. Theories and Applications. AMS Providence (2000). Zbl1027.93500
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  6. [6] M.D. Gunzburger and S. Manservisi, The velocity tracking problem for Navier-Stokes flows with bounded distributed control. SIAM J. Control Optim. 37 (2000) 1913-1945. Zbl0938.35118MR1720145
  7. [7] M.D. Gunzburger and S. Manservisi, Analysis and approximation of the velocity tracking problem for Navier-Stokes flows with distributed control. SIAM J. Numer. Anal. 37 (2000) 1481-1512. Zbl0963.35150MR1759904
  8. [8] L.S. Hou, Error estimates for semidiscrete finite element approximation of the Stokes equations under minimal regularity assumptions. J. Sci. Comput. 16 (2001) 287-317. Zbl0996.76048MR1873285
  9. [9] L.S. Hou and S.S. Ravindran, A penalized Neumann control approach for solving an optimal Dirichlet control problem for the Navier-Stokes equations. SIAM J. Control Optim. 36 (1998) 1795-1814. Zbl0917.49003MR1632548
  10. [10] Jie Shen, On error estimates of the penalty method for unsteady Navier-Stokes equations. SIAM J. Numer. Anal. 32 (1995) 386-403. Zbl0822.35008MR1324294
  11. [11] R. Temam, Navier-Stokes equations. North-Holland, Amsterdam (1979). Zbl0426.35003MR603444
  12. [12] R. Temam, Une méthode d’approximation de la solution des équations de Navier-Stokes. Bull. Soc. Math. France 98 (1968) 115-152. Zbl0181.18903
  13. [13] B.A. Ton, Optimal shape control problems for the Navier-Stokes equations. SIAM J. Control Optim. 41 (2003) 1733-1747. Zbl1037.49035MR1972531

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