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

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 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.