Parallel Finite Element Algorithms Based On Two-grid Discretizations For The Unsteady Incompressible Flow Problems

In fluid mechanics,Navier-Stokes equations are an important model to describe viscous incompressible fluids.The study of its numerical methods is very important for our country's national defense construction and industrial design.However,Navier-Stokes equations are a system of partial differential equations with the nonlinear convection term.It is difficult to solve and computationally intensive,and Oseen equations are a linear form of Navier-Stokes equations,therefore,the study of the Oseen equations is an important basis for studying the Navier-Stokes equations.In this paper,combining the two-grid discretizations method and the parallel finite element algorithm,this paper proposes and studies a fully discrete parallel finite element algorithm for numerically solving the unsteady incompressible Oseen equation and the Navier-Stokes equation.First,we adopt the finite element space discrete and backward Euler scheme time discrete,a parallel finite element algorithm based on two-grid discretizations is used to solve the unsteady incompressible Oseen equation.The main idea of the algorithm is to solve the unsteady Oseen problem on the global coarse grid,obtain a finite element solution of the coarse grid,and then a residual problem is solved in parallel on the local fine grid to correct the coarse grid solution at each time step.therefore,the calculation time is reduced on the premise that the accuracy is relatively improved.Numerical experiments are given to verify the correctness of the theoretical analysis and the efficiency of the algorithm.Then,a parallel finite element algorithm based on two-grid discretizations is used to solve the unsteady incompressible Navier-Stokes equations,which is to first solve the nonlinear problem by Oseen iteration method on a coarse grid,and then to solve the Oseen,Newton or Stokes linearized residual problem in parallel on a fine grid to correct the coarse grid solution at each time step,respectively.Numerical experiments verify that the proposed method can achieve the same convergence order as the standard finite element method,while saving a large amount of running time,in the case that the size ratio of the coarse and fine meshes is appropriate.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 Oseen equations and Navier-Stokes equations,and give some basis theoretical knowledge and symbolic annotation.(2)The numerical scheme of the unsteady incompressible Oseen equation based on two-grid discrete parallel finite element algorithm is given and theoretically analyzed.The error estimate of the algorithm solution is derived.Finally,we give the numerical experiments.(3)The numerical scheme of the unsteady incompressible Navier-Stokes equation based on two-grid discrete parallel finite element algorithm is given,The numerical experiments show that the proposed method is effective.