Numerical criterion for the stabilization of steady states of the Navier-Stokes equations

This paper introduces an explicit numerical criterion for the stabilization of steady state solutions of the Navier-Stokes equations (NSE) with linear feedback control. Given a linear feedback controller that stabilizes a ste ady state solution to the closed-loop standard Galerkin (or nonlinear Galer kin) NSE discretization, it is shown that, if the number of modes involved in the computation is large enough, this controller stabilizes a nearby ste ady state of the closed-loop NSE. We provide an explicit estimate, in terms of the physical parameters, for the number of modes required in order for this criterion to hold. Moreover, we provide an estimate for the distance b etween the stabilized numerical steady state and the actually stabilized st eady state of the closed-loop Navier-Stokes equations. More accurate approx imation procedures, based on the concept of postprocessing the Galerkin res ults, are also presented. All the criterion conditions are imposed on the c omputed numerical solution, and no a priori knowledge is required about the steady state solution of the full PDE. This criterion holds for a large cl ass of unbounded linear feedback operators and can be slightly modified to include certain nonlinear parabolic systems such as reaction-diffusion syst ems.