Multigrid preconditioners for the Newton-Krylov method in the optimal control of the stationary Navier-Stokes equations
For researchers solving optimal control problems with nonlinear Navier-Stokes constraints, this provides efficient preconditioners that extend prior linear work, though the improvement is incremental.
This work constructs and analyzes optimal-order multigrid preconditioners for the Newton-Krylov method in optimal control of stationary Navier-Stokes equations, achieving significant reductions in iteration count and wall-clock time for low to moderate Reynolds numbers.
The focus of this work is on the construction and analysis of optimal-order multigrid preconditioners to be used in the Newton-Krylov method for a distributed optimal control problem constrained by the stationary Navier-Stokes equations. As in our earlier work [7] on the optimal control of the stationary Stokes equations, the strategy is to eliminate the state and adjoint variables from the optimality system and solve the reduced nonlinear system in the control variables. While the construction of the preconditioners extends naturally the work in [7], the analysis shown in this paper presents a set of significant challenges that are rooted in the nonlinearity of the constraints. We also include numerical results that showcase the behavior of the proposed preconditioners and show that for low to moderate Reynolds numbers they can lead to significant drops in number of iterations and wall-clock savings.