Numerical analysis of a nonlinear time relaxation model of fluids

This report investigates a time relaxation model for the regularization of the strong solution of the Navier-Stokes equations. Following the earlier works of Adams, Stolz and Kleiser [37,38], Ervin, Layton and Neda [11], Layton and Neda [26], the nonlinear term investigated herein aims to better control the generalized turbulent fluctuations, thus providing better stabilization effect on the underlying model. We study the numerical properties of the resulting model and prove that the addition of the nonlinear term does not affect the overall order of convergence to the strong solution of the Navier-Stokes equation. Our theoretical results based on the finite element method are confirmed by the numerical tests performed in the last section of the paper.