Commun. Comput. Phys., 7 (2010), pp. 738-758. |
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 doi:10.4208/cicp.2009.09.067 Abstract 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 Notice: Undefined variable: pac in /var/www/html/issue/abstract/readabs.php on line 164 Key words: Optimal control, optimization, least-squares finite element methods, Lagrange multiplier, Stokes equations. *Corresponding author. Email: sryu@knu.ac.kr (S. Ryu), skim@knu.ac.kr (S. D. Kim), hclee@ajou.ac.kr (H.-C. Lee) |