Multigrid preconditioners for the Newton-Krylov method in the optimal control of the stationary Navier-Stokes equations

TL;DR

This paper develops and analyzes multigrid preconditioners for the Newton-Krylov method applied to optimal control problems constrained by stationary Navier-Stokes equations, demonstrating efficiency improvements at moderate Reynolds numbers.

Contribution

It extends multigrid preconditioner construction to nonlinear Navier-Stokes constraints and provides analysis and numerical validation for their effectiveness.

Findings

Preconditioners significantly reduce iteration counts at moderate Reynolds numbers.

Numerical results show wall-clock time savings with proposed preconditioners.

Analysis addresses challenges from nonlinearity in the constraints.

Abstract

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.