The Subgrid Stabilizing Postprocessed Mixed Finite Element Method For The Navier-Stokes Equations

In this paper,we mainly study three postprocessed mixed finite element meth-ods for the incompressible Navier-Stokes equations,which are based on a subgrid model.These methods consist of two steps.The first step is to solve a subgrid sta-bilized nonlinear Navier-Stokes problem on a coarse grid,to obtain an approximate solution uH at time T.Then,the second step is to postprocess uH on a finer grid(or by high-order finite elements),by solving a stabilized Stokes problem,a stabilized Newton-Type problem,or a stabilized Ossen problem.Numerical analysis and error estimation of these three stabilization methods are carried out,and the effective-ness of the algorithms are verified by numerical experiments.The numerical results show that under the conditions of selecting appropriate stabilizing parameters and the grid sizes,the postprocessed finite element methods can improve the precision of the mixed finite element solution,and the order of convergence is obviously im-proved by one unit compared with the standard subgrid stabilized method.Under the premise of selecting the same experimental parameters,from the point of the computational time,the stabilized Newton-type postprocessed method takes a rel-atively more time than the others,while the stabilized Ossen-type postprocessed method takes the least time among the three methods.And from the point of pre-cision of the computed solutions,for three kinds of stabilization algorithm velocity of H1-norm error,the Newton and Oseen-type postprocessed methods are better than the Stokes-type postprocessed method.But for the stabilization pressure in L2-norm,the Newton and Stokes-type postprocessed methods are better than the Ossen-type postprocessed method.The main works of this paper are as follows:(1)First of all,we will introduce the development background of the postprocessing finite element method and a series of stabilization methods to solve the incom-pressible Navier-Stokes equations,and gives some basic theoretical knowledge and symbol labeling.(2)The numerical scheme of Stokes,Ossen and Newton postprocessing finite ele-ment methods based on a subgrid stabilization are given,and their numerical analysis and error estimation are carried out respectively.At the end of each chapter,some numerical experiments are carried out to verify the effectiveness of these three stabilization algorithms.(3)Finally,the three stabilizing postprocessed mixed finite element methods men-tioned above are compared and analyzed by numerical experiment results.