no. 4
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, Tome 10 (2004) no. 4, pp. 574-592.

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.

DOI : 10.1051/cocv:2004021
Classification : 35B37, 65M60, 49J20
Mots clés : optimal control, velocity tracking, finite elements, semidiscrete error estimates, Stokes equations, penalized formulation 
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     author = {Chrysafinos, Konstantinos},
     title = {Analysis and finite element error estimates for the velocity tracking problem for {Stokes} flows via a penalized formulation},
     journal = {ESAIM: Control, Optimisation and Calculus of Variations},
     pages = {574--592},
     publisher = {EDP-Sciences},
     volume = {10},
     number = {4},
     year = {2004},
     doi = {10.1051/cocv:2004021},
     mrnumber = {2111081},
     zbl = {1072.49021},
     language = {en},
     url = {http://www.numdam.org/articles/10.1051/cocv:2004021/}
}
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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, Tome 10 (2004) no. 4, pp. 574-592. doi : 10.1051/cocv:2004021. http://www.numdam.org/articles/10.1051/cocv:2004021/

[1] R. Adams, Sobolev Spaces. Academic Press, New York (1975). | MR | Zbl

[2] 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. | MR | Zbl

[3] A. Fursikov,Optimal control of distributed systems. Theories and Applications. AMS Providence (2000). | Zbl

[4] V. Girault and P.-A. Raviart, Finite Element Methods for Navier-Stokes. Springer-Verlag, New York (1986). | MR

[5] 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: M2AN 25 (1991) 711-748. | Numdam | MR | Zbl

[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. | MR | Zbl

[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. | MR | Zbl

[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. | MR | Zbl

[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. | MR | Zbl

[10] Jie Shen, On error estimates of the penalty method for unsteady Navier-Stokes equations. SIAM J. Numer. Anal. 32 (1995) 386-403. | MR | Zbl

[11] R. Temam, Navier-Stokes equations. North-Holland, Amsterdam (1979). | MR | Zbl

[12] R. Temam, Une méthode d'approximation de la solution des équations de Navier-Stokes. Bull. Soc. Math. France 98 (1968) 115-152. | Numdam | Zbl

[13] B.A. Ton, Optimal shape control problems for the Navier-Stokes equations. SIAM J. Control Optim. 41 (2003) 1733-1747. | MR | Zbl

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