# 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 (2010)

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

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topChrysafinos, 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 (2010): 574-592. <http://eudml.org/doc/90744>.

@article{Chrysafinos2010,

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 ε → 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.; optimal control; velocity tracking; penalized formulation},

language = {eng},

month = {3},

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/90744},

volume = {10},

year = {2010},

}

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

DA - 2010/3//

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 ε → 0 is examined.

LA - eng

KW - Optimal control; velocity
tracking; finite elements; semidiscrete error estimates; Stokes equations;
penalized formulation.; optimal control; velocity tracking; penalized formulation

UR - http://eudml.org/doc/90744

ER -

## References

top- R. Adams, Sobolev Spaces. Academic Press, New York (1975).
- K. Chrysafinos and L.S. Hou, Error estimates for semidiscrete finite element approximations of linear and semilinear parabolic equations under minimal regularity assumptions. SIAM J. Numer. Anal.40 (2002) 282-306.
- A. Fursikov,Optimal control of distributed systems. Theories and Applications. AMS Providence (2000).
- V. Girault and P.-A. Raviart, Finite Element Methods for Navier-Stokes. Springer-Verlag, New York (1986).
- M.D. Gunzburger, L.S. Hou and T. Svobodny, Analysis and finite element approximation of optimal control problems for the stationary Navier-Stokes equations with Dirichlet controls. ESAIM: M2AN25 (1991) 711-748.
- 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.
- 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.
- 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.
- 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.
- Jie Shen, On error estimates of the penalty method for unsteady Navier-Stokes equations. SIAM J. Numer. Anal.32 (1995) 386-403.
- R. Temam, Navier-Stokes equations. North-Holland, Amsterdam (1979).
- R. Temam, Une méthode d'approximation de la solution des équations de Navier-Stokes. Bull. Soc. Math. France98 (1968) 115-152.
- B.A. Ton, Optimal shape control problems for the Navier-Stokes equations. SIAM J. Control Optim.41 (2003) 1733-1747.

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