Scalable parallel algorithms for boundary control of thermally convective flows

Optimal control of fluid flows is an important and computationally challenging problem. In this paper, we investigate the application of a class of parallel and fully coupled two-grid Lagrange-Newton-Krylov-Schwarz (LNKSz) algorithms for the boundary control of thermally convective flows. The investigation focuses on the use of a two-grid inexact Newton solver for the necessary optimality condition obtained from the optimization problem and the use of a Krylov subspace solver together with an efficient two-level overlapping Schwarz preconditioner for the Jacobian system. Our parallel numerical results show that the proposed method is scalable with respect to the number of processors, the grid size, and robust with respect to some physical parameters such as the Reynolds number and the Grashof number. We also show some large scale calculations involving several million unknowns obtained on a supercomputer with more than two thousand processors.