New Feedback Control and Adaptive Evolve-Filter-Relax Regularization for the Navier-Stokes Equations in the Convection-Dominated Regime

Abstract

We propose, analyze, and investigate numerically a novel feedback control strategy for high Reynolds number flows. For both the continuous and the discrete (finite element) settings, we prove that the new strategy yields accurate results for high Reynolds numbers that were not covered by current results. We also show that the new feedback control yields more accurate results than the current control approaches in marginally-resolved numerical simulations of a two-dimensional flow past a circular cylinder at Reynolds numbers $Re=1000$. We note, however, that for realistic control parameters, the stabilizing effect of the new feedback control strategy is not sufficient in the convection-dominated regime. Our second contribution is the development of an adaptive evolve-filter-relax (aEFR) regularization that stabilizes marginally-resolved simulations in the convection-dominated regime and increases the accuracy of the new feedback control in realistic parameter settings. For the finite element setting, we prove that the novel feedback control equipped with the new aEFR method yields accurate results for high Reynolds numbers. Furthermore, our numerical investigation shows that the new strategy yields accurate results for reduced order models that dramatically decrease the size of the feedback control problem.