First-Order System Least-Squares Methods for a Flux Control Problem by the Stokes Flow

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 H1 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 L2 norms applied to this system yields the optimal discretization error estimates for each variable in H1 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.