Parallel Grad-div Stabilized Finite Element Algorithms For Unsteady Incompressible Flows

N-S equation is an equation describing fluid flow,its application in a wide range of fields,playing an important role in weather prediction,blood flow,aircraft,aerodynamics and other fields,so its research is very important for people to understand and grasp the law of fluid movement.However,because the N-S equation is a second order nonlinear system,the properties of the solutions of are often understood by numerical experiments.Since the birth of the finite element method,with the improvement of the theory,handling complex boundary value problems has become one of the indispensable tools in computational fluid dynamics.However,when the finite element method is used for numerical calculation of unsteady incompressible flow,it usually faces two di culties.One is that the pressure will a?ect the approximate solution of velocity,which leads to the poor e?ect of velocity simulation.The second is large-scale calculation,especially unsteady problems.Therefore,it is necessary to use stabilized method and parallel algorithm to realize the large-scale computation of unsteady incompressible flow.On the basis of previous work,the main research work of this paper is as follows:Based on the Stokes projection,in Chapter 3,a grad-div stabilized method is studied,which is used to numerically simulate the unsteady and incompressible Stokes equations,aiming to improve the pressure robustness.This method uses the grad-div stabilization term to improve the pressure robustness,can be easily implemented with existing serial solvers.Only a small modification of the existing program is needed.The final numerical experiments show that compared with the method without grad-div stabilization,the standard grad-div stabilized method proposed in this chapter can improve the accuracy of approximate solutions of velocity.Based on the complete overlapping domain decomposition technique,a parallel grad-div stabilized method is proposed and studied for the unsteady incompressible Stokes equations in Chapter 4.The stable method aims to improve the pressure robustness and reduce the CPU required for numerical simulation,giving the stability of the grad-div stabilized finite element solution and the local prior estimates.Based on the local prior estimation and the error estimation in Chapter 3,the local parallel grad-div stabilized algorithm is designed in this chapter,and the error estimates of the approximate solutions of velocity and pressure are derived.Finally,numerical experiments show that: on the one hand,compared with the parallel non-stabilized algorithm,the parallel stabilized algorithm proposed in this chapter yields more accurate solutions.On the other hand,parallel stabilized algorithms greatly reduce the computation time,and obtain numerical solutions with accuracy comparable to standard stabilized algorithms.In Chapter 5,a parallel grad-div stabilized finite element method for numerically solving the unsteady incompressible N-S equation is proposed based on the two-grid method,aiming at improving the robustness of pressure and comparing the influence of three di?erent linear strategies(Newton,Oseen,Stokes).In this chapter,the local parallel grad-div stabilized algorithm is designed.The algorithm firstly solves a stabilized nonlinear problem on a global coarse grid,and then solves the grad-div stabilized residual problem in parallel on fine grid to correct the coarse grid solutions.Numerical experiments show that three parallel stabilized algorithms with di?erent linear strategies can reach the optimal convergence order.Compared with the parallel non-stabilized algorithm,the parallel stabilized algorithm proposed in this chapter can improve the accuracy of the approximate solutions of velocity.