Optimal Control of the Stokes Equations: A Priori Error Analysis for Finite Element Discretization with Postprocessing

Abstract

An optimal control problem for 2d and 3d Stokes equations is investigated with pointwise control constraints. This paper is concerned with the discretization of the control by piecewise constant functions. The state and the adjoint state are discretized by finite element schemes. In the paper a postprocessing strategy is suggested, which allows for significant improvement of the accuracy.

MSC codes

  1. 49K20
  2. 49M25
  3. 65N30

Keywords

  1. PDE‐constrained optimization
  2. finite elements
  3. error estimates
  4. Stokes equations
  5. numerical approximation
  6. control constraints

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References

1.
N. Arada, E. Casas, and F. Tröltzsch, Error estimates for a semilinear elliptic optimal control problem, Comput. Optim. Appr., 23 (2002), pp. 201–229.
2.
R. Becker and M. Braack, A finite element pressure gradient stabilization for the Stokes equations based on local projections, Calcolo, 38 (2001), pp. 137–199.
3.
M. Bergounioux, K. Ito, and K. Kunisch, Primal‐dual strategy for constrained optimal control problems, SIAM J. Control Optim., 37 (1999), pp. 1176–1194.
4.
P. Bochev and M. Gunzburger, Least‐squares finite‐element methods for optimization and control problems for the Stokes equations, Comput. Math. Appl., 48 (2004), pp. 1035–1057.
5.
D. Braess, Finite Elemente, Springer, Berlin, 1992.
6.
E. Burman and P. Hansbo, Edge stabilization for the generalized Stokes problem: a continuous interior penalty method, Comput. Methods Appl. Mech. Engrg., 195 (2006), pp. 2393–2410.
7.
E. Casas, Using piecewise linear functions in the numerical approximation of semilinear elliptic control problems, submitted.
8.
E. Casas, M. Mateos, and F. Tröltzsch, Error estimates for the numerical approximation of boundary semilinear elliptic control problems, Comput. Optim. Appl., 31 (2005), pp. 193–219.
9.
E. Casas and F. Tröltzsch, Error estimates for linear‐quadratic elliptic control problems, in Analysis and Optimization of Differential Systems, V. B. et al., ed., Boston, Kluwer Academic Publishers, Boston, 2003, pp. 89–100.
10.
P. Ciarlet, Basic error estimates for elliptic problems, in Finite Element Methods, vol. II of Handbook of Numerical Analysis, North‐Holland, Amsterdam, 1991, pp. 17–351.
11.
P. Clément, Approximation by finite element functions using local regularization, Revue Franc. Automat. Inform. Rech. Operat., 9 (1975), pp. 77–84.
12.
S. S. Collis and M. Heinkenschloss, Analysis of the streamline upwind/Petrov Galerkin method applied to the solution of optimal control problems, CAAM TR02‐01, Rice University, Houston, TX, (2002).
13.
M. Dauge, Stationary Stokes and Navier–Stokes systems on two‐ or three‐dimensional domains with corners I, SIAM J. Math. Anal., 20 (1989), pp. 74–97.
14.
K. Deckelnick and M. Hinze, Semidiscretization and error estimates for distributed control of the instationary Navier–Stokes equations, Numer. Math., 97 (2004), pp. 297–320.
15.
R. Falk, Approximation of a class of optimal control problems with order of convergence estimates, J. Math. Anal. Appl., 44 (1973), pp. 28–47.
16.
T. Geveci, On the approximation of the solution of an optimal control problem governed by an elliptic equation, RAIRO Anal. Numér., 13 (1979), pp. 313–328.
17.
V. Girault and P.‐A. Raviart, Finite element methods for Navier–Stokes equations. Theory and algorithms, Springer Ser. Comput. Math. 5, Berlin, 1986.
18.
M. Gunzburger, L. Hou, and T. Svobodny, Analysis and finite element approximation of optimal control problems for the stationary Navier–Stokes equations with Dirichlet controls, Math. Model. Numer. Anal., 25 (1991), pp. 711–748.
19.
M. Gunzburger, L. Hou, and T. Svobodny, Analysis and finite element approximation of optimal control problems for the stationary Navier–Stokes equations with distributed and Neumann controls, Math. Comp., 57 (1991), pp. 123–151.
20.
M. Hinze, A variational discretization concept in control constrained optimization: The linear‐quadratic case, Comput. Optim. Appl., 30 (2005), pp. 45–61.
21.
P. Hood and C. Taylor, A numerical solution of the Navier–Stokes equations using the finite element technique, Internat. J. Comput. and Fluids, 1 (1973), pp. 73–100.
22.
K. Kunisch and A. Rösch, Primal‐dual active set strategy for a general class of constrained optimal control problems, SIAM J. Optim., 13 (2002), pp. 321–334.
23.
J. L. Lions, Contrôle Optimal de Systems Gouvernés par des Équations aux Dérivées Partielles, Dunvol Gauthier–Villars, Paris, 1968.
24.
K. Malanowski, Convergence of approximations vs. regularity of solutions for convex, control‐constrained optimal control problems, Appl. Math. Opt., 8 (1982), pp. 69–95.
25.
C. Meyer and A. Rösch, Superconvergence properties of optimal control problems, SIAM J. Control Optim., 43 (2004), pp. 970–985.
26.
C. Meyer and A. Rösch, $L^{\infty}$‐estimates for approximated optimal control problems, SIAM J. Control Optim., 44 (2005), pp. 1636–1649.
27.
A. Rösch, Error estimates for parabolic optimal control problems with control constraints, Z. Anal. Anwendungen, 23 (2004), pp. 353–376.
28.
A. Rösch, Error estimates for linear‐quadratic control problems with control constraints, Optim. Methods Softw., 21 (2006), pp. 121–134.

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Published In

cover image SIAM Journal on Numerical Analysis
SIAM Journal on Numerical Analysis
Pages: 1903 - 1920
ISSN (online): 1095-7170

History

Submitted: 2 August 2005
Accepted: 6 April 2006
Published online: 29 September 2006

MSC codes

  1. 49K20
  2. 49M25
  3. 65N30

Keywords

  1. PDE‐constrained optimization
  2. finite elements
  3. error estimates
  4. Stokes equations
  5. numerical approximation
  6. control constraints

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