The Subgrid Stabilized Finite Element Methods With Backtracking Technique For The Navier-Stokes Equations At High Reynolds Numbers

Navier-Stokes equations are the basic equations for fluid flow,which play an very important role in Computational Fluid Dynamics.They describe many important physical phenomena.The research is important for people to understand and master the fluid motion law.However,Navier-Stokes problem is the class of nonlinear problem in fluid mechanics.Acking of realizing the nature of nonlinear phenomenon,it is difficult to find the exact solutions of the Navier-Stokes equations,so people often learn the behavior of the solutions by numerical simulation.Therefore,since 1900 s,the theory and numerical methods of Navier-Stokes equations have been widely concerned by mathematicians,scientists and cngincers.It is well known that the classical Galerkin methods may fail to work in approximating the above Navier-Stokes equations at high Reynolds numbers.The reason is that as the Reynolds number grows,the flow becomes convection dominate for which layers appears where the velocity solution and its gradient exhibit rapid variation.The classical Galerkin methods lead to numerical oscillations in these layer regions which can spread quickly and pollute the entire solution domain.Focusing on the problem mentioned above,we propose three two-level finite element methods with backtracking technique for the stationary Navier-Stokes equations at high Reynolds numbers.Our methods combine the best algorithmic features of the backtracking technique and subgrid stabilization methods.The basic idea of our methods is to first solve a fully nonlinear Navier-Stokes equations with a subgrid stabilization term on a coarse grid,and then solve a subgrid stabilized linear fine grid problem based on one step of Newton(or Oseen or Stokes)iteration,and finally,solve an error check problem on the coarse mesh.Stability of the methods and error estimates of the discrete solution are analyzed.The algorithmic parameter scalings are also derived.The theoretical show that,with suitable scalings of algorithmic parameters,the method can yield an optimal convergence rate.Numerical results on an example with known analytical solution,the lid-driven cavity flow and the backward-facing step flow are given to verify the theoretical predictions and demonstrate the effectweness of the methods.