First-Order System Least-Squares Methods for a Flux Control Problem by the Stokes Flow (Communications in Computational Physics, 2010)
This article deals with a first-order least-squares approach to the solution ofan optimal control problem governed by Stokes equations. As with our earlier workon a velocity control by the Stokes flow in [S. Ryu, H.-C. Lee and S. D. Kim, SIAMJ. Numer. Anal., 47 (2009), pp. 1524-1545], we recast the objective functional as a H1 seminorm in the velocity control term. By introducing a velocity-flux variable and usingthe Lagrange multiplier rule, a first-order optimality system is obtained. We show thatthe least-squares principle based on L2 norms applied to this system yields the optimaldiscretization error estimates for each variable in H1 norm, including the velocity flux.For numerical tests, multigrid method is employed to the discrete algebraic system, sothat the velocity and flux controls are obtained.