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

TL;DR

This paper introduces an augmented Lagrangian preconditioner combined with inexact Newton and multigrid methods to efficiently solve large saddle-point systems arising from the control of steady Navier--Stokes equations, demonstrating robustness in 2D simulations.

Contribution

It presents a novel augmented Lagrangian block preconditioner integrated with inexact Newton and multigrid techniques for Navier--Stokes control problems.

Findings

Method is robust with respect to viscosity, mesh size, and regularization parameter.

Numerical experiments confirm efficiency and stability in 2D cases.

Preconditioner improves convergence of iterative solvers for large saddle-point systems.

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.