The Variational Multiscale Methods For The Unsteady Incompressible Navier-Stokes Equations With High Reynolds Numbers

The turbulences described by the unsteady incompressible Navier-Stokes equa-tions where high Reynolds numbers are common in the field of nature and engi-neering technology.The study of its numerical methods is very important for our country's national defense construction and industrial design.Based on the finite element spatial discretization method,we present and study a fully discrete finite element variational multiscale scheme for the unsteady incompressible Navier-Stokes equations where high Reynolds numbers are allowed.Firstly,the unsteady incom-pressible Navier-Stokes equations are solved by the one-level finite element varia-tional multiscale method based on two local Gauss integrations.The scheme uses conforming finite element pairs for spatial discretization and a three point difference formula for temporal discretization which is of second-order.The method takes less CPU time with an improvement in accuracy of the computed solutions.Numerical results are conducted to verify the theoretical predictions and demonstrate the ef-fectiveness of the proposed numerical scheme.Then,a two-level fully discrete finite element variational multiscale method based on two local Gauss integrations for the unsteady Navier-Stokes equations is presented and studied.At each time step of this method,a stabilized nonlinear Navier-Stokes system is first solved on a coarse grid,and then a stabilized linear problem is solved on a fine grid to correct the coarse grid solution.We derive error estimates of the fully discrete solution which is second in time.Numerical experiments show that the method can save a lot of computation time compared with the finite element variational method which uses a one-level grid directly on the fine grid in the case of coarse grid matching.The main works of this paper are as follows:(1)We mainly introduce the development background of the finite element variational multiscale method for solving the unsteady incompressible Navier-Stokes equations,and give some basis theoretical knowledge and symbolic annotation.(2)The numerical scheme for the one-level finite element variational multiscale method based on two local Gauss integrations is given and the stability of the scheme is proved.The prior error estimates for the fully discrete solution is further derived.Finally,we give the numerical experiments.(3)The numerical scheme and the corresponding numerical analysis of the two-level finite variational multiscale method based on Gauss integeations are given.The numerical experiments show that the proposed method is effective.