Parameter-Robust Preconditioning for Oseen Iteration Applied to Stationary and Instationary Navier--Stokes Control

TL;DR

This paper introduces new parameter-robust preconditioned iterative methods for solving steady and time-dependent Navier--Stokes control problems, improving efficiency and stability across different regimes.

Contribution

The authors develop a novel preconditioning strategy for Navier--Stokes control problems that is robust with respect to problem parameters and applicable to both stationary and unsteady cases.

Findings

Numerical experiments confirm the effectiveness of the proposed preconditioners.

The methods are applicable to discretizations using backward Euler and Crank--Nicolson schemes.

The approach enhances convergence stability for Navier--Stokes control problems.

Abstract

We derive novel, fast, and parameter-robust preconditioned iterative methods for steady and time-dependent Navier--Stokes control problems. Our approach may be applied to time-dependent problems which are discretized using backward Euler or Crank--Nicolson, and is also a valuable candidate for Stokes control problems discretized using Crank--Nicolson. The key ingredients of the solver are a saddle-point type approximation for the linear systems, an inner iteration for the $(1,1)$-block accelerated by a preconditioner for convection--diffusion control, and an approximation to the Schur complement based on a potent commutator argument applied to an appropriate block matrix. A range of numerical experiments validate the effectiveness of our new approach.