On the long-time stability of finite element solutions of the navier-stokes equations in a rotating frame of reference

Abstract

This paper studies the long-time stability behavior of the Navier-Stokes equations (NSE) in a rotating frame of reference with atime accurate and adaptive finite element method. The proposed numerical scheme consists of two decoupled steps. In the first step, the Navier-Stokes equations are solved with the standard linearized backwardEuler finite element method (BE-FEM). In the second step, the approximate velocity solution obtained in the first step is post proceeded with a 2-step, linear time filter. It is proven that the approximate velocity solution is stable with respect to𝐿�2-norm at all times. The novelty of the stability analysis is that the stability bound obtained for the approximate velocity solution does not use any Gronwall-type estimate and is polynomially dependent on the Reynolds number, which is not common in long-time stability notion. The paper also provides two numerical experiments to test the algorithm. The first numerical experiment compares the 𝐿�2-norm of the velocity solution of the proposed algorithmusing pressure-robust and non pressure-robustFE over longer time intervals. The results reveal that the scheme gives much more accurate velocity solutions with pressure-robust methods, especially for the smaller 𝑣�. The second experiment, on the other hand, shows that the filter step increasesthe accuracy of the proposed numerical method over long-time intervals.