The Finite Element Operator Splitting Methods For Unsteady Incompressible Navier-Stokes Equations

Navier-Stokes equations are a typical system to describe viscous incompress-ible fluid.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 two fully discrete finite ele-ment schemes for the unsteady incompressible Navier-Stokes equations where high Reynolds numbers are allowed.We present in this paper finite element operator splitting methods based on a subgrid model for the numerical solution of the incom-pressible Navier-Stokes equations.The schemes use conforming finite element pairs for spatial discretization and difference formula for temporal discretization which is of first-order.The main idea of these two methods is to use operator splitting method to separate the nonlinear and incompressible terms.At each time step of finite element operator splitting method,a linear Burger’s system is first solved,and then a Stokes problem is solved.We proved the stability of the numerical schemes,and error bound of the approximate velocity which is first-order in time.Numerical experiments verify the theoretical predictions and demonstrate the effectiveness of the proposed method.The main works of this paper are as follows:(1)We mainly introduce the development background of the finite element method for solving the unsteady incompressible Navier-Stokes equations,and give some basis theoretical knowledge and symbolic annotation.(2)Two numerical schemes for the finite element operator splitting method based on subgrid model are given and the stability of the schemes are proved.The prior error estimates for the fully discrete solution is further derived.Finally,we give the numerical experiments.