Numerical criterion for the stabilization of steady states of the Navier-Stokes equations
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abstract
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 steady state solution to the closed-loop standard Galerkin (or nonlinear Galerkin) NSE discretization, it is shown that, if the number of modes involved in the computation is large enough, this controller stabilizes a nearby steady 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 between the stabilized numerical steady state and the actually stabilized steady state of the closed-loop Navier-Stokes equations. More accurate approximation procedures, based on the concept of postprocessing the Galerkin results, are also presented. All the criterion conditions are imposed on the computed numerical solution, and no a priori knowledge is required about the steady state solution of the full PDE. This criterion holds for a large class of unbounded linear feedback operators and can be slightly modified to include certain nonlinear parabolic systems such as reaction-diffusion systems.
