An augmented Lagrangian preconditioner for the control of the Navier--Stokes equations

TL;DR

This work tackles the distributed control of steady incompressible Navier–Stokes flow by solving the KKT system with an inexact Newton method. The main contribution is an augmented Lagrangian preconditioner, combined with a practical Schur-complement approximation and multigrid inner solves, that remains robust to changes in viscosity $\nu$, mesh size $h$, and regularization $\beta$ in 2D. The authors compare against a block pressure–convection–diffusion preconditioner and show that, under inexact Newton updates, the augmented Lagrangian approach yields stable, scalable convergence and can handle highly convection-dominated regimes. Numerical experiments on lid-driven cavity and backward-facing step problems demonstrate fast convergence with near-linear complexity, validating the method’s practicality for PDE-constrained Navier–Stokes control and motivating future extensions to time-dependent problems and larger-scale applications.

Abstract

We address the solution of the distributed control problem for the steady, incompressible Navier--Stokes equations. We propose an inexact Newton linearization of the optimality conditions. Upon discretization by a finite element scheme, we obtain a sequence of large symmetric linear systems of saddle-point type. We use an augmented Lagrangian-based block triangular preconditioner in combination with the flexible GMRES method at each Newton step. The preconditioner is applied inexactly via a suitable multigrid solver. Numerical experiments indicate that the resulting method appears to be fairly robust with respect to viscosity, mesh size, and the choice of regularization parameter when applied to 2D problems.

Figures (1)

• Figure 4.1: Solver diagram for the augmented Lagrangian preconditioner applied to stationary Navier--Stokes control problems.