In this paper, we investigate some fully coupled parallel two-grid Lagrange–Newton–Krylov–Schwarz algorithms for the suboptimal distributed control of unsteady incompressible flows governed by the Navier–Stokes equations. The algorithms include two major parts: a two-grid Newton method for the nonlinear part of the problem and a two-level Schwarz preconditioner for the linear part of the problem. Most of the existing approaches for distributed control problems are based on the so-called reduced space method, which is easier to implement, but may have convergence issues in some situations. In the full space approach, we couple the state variables, the control variables, and the adjoint variables in a single large system of nonlinear equations. The coupled system is considerably more ill-conditioned than its sub-systems; however, with the powerful two-grid approach, we are able to solve these difficult systems efficiently on large scale parallel computers. We show numerically that such an approach is scalable in the sense that the number of Newton iterations and the number of linear iterations are both nearly independent of the grid size, the number of processors, and the Reynolds numbers. We present numerical experiments for some suboptimal control problems obtained on supercomputers with more than two thousand processors. Copyright © 2012 John Wiley & Sons, Ltd.
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