First-Order System Least-Squares Methods for an Optimal Control Problem by the Stokes Flow

Abstract

The least-squares approximations of an optimal control problem governed by the Stokes equations are considered, which leads to an unconstrained coupled optimization problem by the Lagrange multiplier method. The least-squares functionals for the two- and three-dimensional first-order coupled optimality systems are employed by modifying those functionals in [Z. Cai, T. A. Manteuffel, and S. F. McCormick, SIAM J. Numer. Anal., 34 (1997), pp. 1727–1741]. The established ellipticity and continuity in a product $H^1$ norm yield the optimal discretization error estimates in the finite element spaces. For numerical tests, we apply V-cycle multigrid methods to the whole discrete algebraic system.