ANALYSIS AND COMPUTATIONS OF LEAST-SQUARES METHOD FOR OPTIMAL CONTROL PROBLEMS FOR THE STOKES EQUATIONS

- Journal title : Journal of the Korean Mathematical Society
- Volume 46, Issue 5, 2009, pp.1007-1025
- Publisher : The Korean Mathematical Society
- DOI : 10.4134/JKMS.2009.46.5.1007

Title & Authors

ANALYSIS AND COMPUTATIONS OF LEAST-SQUARES METHOD FOR OPTIMAL CONTROL PROBLEMS FOR THE STOKES EQUATIONS

Choi, Young-Mi; Kim, Sang-Dong; Lee, Hyung-Chun;

Choi, Young-Mi; Kim, Sang-Dong; Lee, Hyung-Chun;

Abstract

First-order least-squares method of a distributed optimal control problem for the incompressible Stokes equations is considered. An optimality system for the optimal solution are reformulated to the equivalent first-order system by introducing the vorticity and then the least-squares functional corresponding to the system is defined in terms of the sum of the squared and norms of the residual equations of the system. Finite element approximations are studied and optimal error estimates are obtained. Resulting linear system of the optimality system is symmetric and positive definite. The V-cycle multigrid method is applied to the system to test computational efficiency.

Keywords

optimal control;least-squares finite element method;multigrid method;stokes equations;

Language

English

Cited by

1.

ANALYSIS OF VELOCITY-FLUX FIRST-ORDER SYSTEM LEAST-SQUARES PRINCIPLES FOR THE OPTIMAL CONTROL PROBLEMS FOR THE NAVIER-STOKES EQUATIONS,;;

References

1.

H. K. Baek, S. D. Kim, and H.-C. Lee, A Multigrid method for an optimal control problem of a diffusion-convection equation, submitted

2.

M. Bergounioux, Augmented Lagrangian method for distributed optimal control problems with state constraints, J. Optim. Theory Appl. 78 (1993), no. 3, 493.521

3.

P. B. Bochev and M. D. Gunzburger, Analysis of least squares finite element methods for the Stokes equations, Math. Comp. 63 (1994), no. 208, 479.506

4.

P. B. Bochev and M. D. Gunzburger, Least-squares methods for the velocity-pressure-stress formulation of the Stokes equations, Comput. Methods Appl. Mech. Engrg. 126 (1995), no. 3-4, 267.287

5.

P. B. Bochev and M. D. Gunzburger, Finite element methods of least-squares type, SIAM Rev. 40 (1998), no. 4, 789.837

6.

P. B. Bochev and M. D. Gunzburger, Least-squares finite-element methods for optimization and control problems for the Stokes equations, Comput. Math. Appl. 48 (2004), no. 7-8, 1035.1057

7.

P. B. Bochev and M. D. Gunzburger, Least-squares finite element methods for optimality systems arising in optimization and control problems, SIAM J. Numer. Anal. 43 (2006), no. 6, 2517.2543

8.

P. B. Bochev, T. A. Manteuffel, and S. F. McCormick, Analysis of velocity-flux firstorder system least-squares principles for the Navier-Stokes equations. I., SIAM J. Numer. Anal. 35 (1998), no. 3, 990.1009

9.

P. B. Bochev, T. A. Manteuffel, and S. F. McCormick, Analysis of velocity-flux least-squares principles for the Navier-Stokes equations. II., SIAM J. Numer. Anal. 36 (1999), no. 4, 1125.1144.

10.

J. H. Bramble, R. D. Lazarov, and J. E. Pasciak, A least-squares approach based on a discrete minus one inner product for first order systems, Math. Comp. 66 (1997), no. 219, 935.955

11.

J. H. Bramble and J. E. Pasciak, Least-squares methods for Stokes equations based on a discrete minus one inner product, J. Comput. Appl. Math. 74 (1996), no. 1-2, 155.173

12.

Z. Cai, T. Manteuffel, and S. McCormick, First-order system least squares for velocityvorticity- pressure form of the Stokes equations, with application to linear elasticity, Electron. Trans. Numer. Anal. 3 (1995), Dec., 150.159

13.

Z. Cai, First-order system least squares for the Stokes equations, with application to linear elasticity, SIAM J. Numer. Anal. 34 (1997), no. 5, 1727.1741.

14.

C. Chang and M. Gunzburger, A finite element method for first order elliptic systems in three dimensions, Appl. Math. Comput. 23 (1987), no. 2, 171.184

15.

Y. Choi, H.-C. Lee, and B.-C. Shin, A least-squares/penalty method for distributed optimal control problems for Stokes equations, Comput. Math. Appl. 53 (2007), no. 11, 1672.1685

16.

P. Ciarlet, The Finite Element Method for Elliptic Problems, Studies in Mathematics and its Applications, Vol. 4. North-Holland Publishing Co., Amsterdam-New York- Oxford, 1978

17.

V. Girault and P.-A. Raviart, Finite Element Methods for Navier-Stokes Equations, Theory and algorithms. Springer Series in Computational Mathematics, 5. Springer-Verlag, Berlin, 1986

18.

R. Glowinski and J. He, On shape optimization and related issues, Computational methods for optimal design and control (Arlington, VA, 1997), 151.179, Progr. Systems Control Theory, 24, Birkhauser Boston, Boston, MA, 1998

19.

M. D. Gunzburger, Perspectives in Flow Control and Optimization, Advances in Design and Control, 5. Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 2003

20.

M. D. Gunzburger, L. S. Hou, and T. Svobodney, Analysis and finite element approximation of optimal control problems for the stationary Navier-Stokes equations with distributed and Neumann controls, Math. Comp. 57 (1991), no. 195, 123.151

21.

M. D. Gunzburger and H.-C. Lee, Analysis and approximation of optimal control problems for first-order elliptic systems in three dimensions, Appl. Math. Comput. 100 (1999), no. 1, 49.70

22.

M. D. Gunzburger and H.-C. Lee, A penalty/least-squares method for optimal control problems for first-order elliptic systems, Appl. Math. Comput. 107 (2000), no. 1, 57.75

23.

J.-W. He, R. Glowinski, R. Metcalfe, A. Nordlander, and J. Periaux, Active control and drag optimization for flow past a circular cylinder, J. Comput. Phys. 163 (2000), no. 1, 83.117

24.

J.-W. He, M. Chevalier, R. Glowinski, R. Metcalfe, A. Nordlander, and J. Periaux, Drag reduction by active control for flow past cylinders, Computational mathematics driven by industrial problems (Martina Franca, 1999), 287.363, Lecture Notes in Math., 1739, Springer, Berlin, 2000

25.

B.-N. Jiang and L. A. Povinell, Optimal least-squares finite element method for elliptic problems, Comput. Methods Appl. Mech. Engrg. 102 (1993), no. 2, 199.212

26.

H.-C. Lee and Y. Choi, A least-squares method for optimal control problems for a secondorder elliptic system in two dimensions, J. Math. Anal. Appl. 242 (2000), no. 1, 105.128