First-Order System Least-Squares Methods for a Flux Control Problem by the Stokes Flow
Soorok Ryu 1, Sang Dong Kim 2, Hyung-Chun Lee 3*1 Department of Industrial and Applied Mathematics, Kyungpook National University, Daegu 702-701, Korea.
2 Department of Mathematics, Kyungpook National University, Daegu 702-701, Korea.
3 Department of Mathematics, Ajou University, Suwon 443-749, Korea.
Received 23 April 2009; Accepted (in revised version) 4 September 2009
Available online 28 October 2009
This article deals with a first-order least-squares approach to the solution of an optimal control problem governed by Stokes equations. As with our earlier work on a velocity control by the Stokes flow in [S. Ryu, H.-C. Lee and S. D. Kim, SIAM J. Numer. Anal., 47 (2009), pp. 1524-1545], we recast the objective functional as a $H^1$ seminorm in the velocity control term. By introducing a velocity-flux variable and using the Lagrange multiplier rule, a first-order optimality system is obtained. We show that the least-squares principle based on $L^2$ norms applied to this system yields the optimal discretization error estimates for each variable in $H^1$ norm, including the velocity flux. For numerical tests, multigrid method is employed to the discrete algebraic system, so that the velocity and flux controls are obtained.
AMS subject classifications: 65M55, 65N30, 49J20, 49K20
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Key words: Optimal control, optimization, least-squares finite element methods, Lagrange multiplier, Stokes equations.
Email: email@example.com (S. Ryu), firstname.lastname@example.org (S. D. Kim), email@example.com (H.-C. Lee)