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

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 timedependent 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.