Navier-Stokes方程的亚格子模型后处理混合有限元方法
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摘要
提出并考察了3种基于亚格子模型的后处理混合有限元方法,其主要思想是:第一步在粗网格上求解带有亚格子模型稳定项的Navier-Stokes方程,得到最后时刻T的有限元解u_H ;第二步在最后时刻T,对第一步所得解u_H进行后处理,主要通过在细网格上(或用高阶元)分别求解带有亚格子模型稳定项的Stokes问题、Newton问题或者Ossen问题.验结果表明:在选取适当的稳定化参数和网格尺寸的条件下,3种稳定化的后处理有限元方法提高了稳定化的混合有限元解的精确度,并且收敛阶较标准的有限元方法明显提高了一阶.从计算时间看,除ν =1以外,在其它情况下稳定化的Newton型后处理花费的时间相对较多,而稳定化的Ossen型后处理花费的时间相对较少.从精确度来看,Newton型后处理和Ossen型后处理方法所得速度的H~1-范误差和压力的L~2-范误差比Stokes型后处理方法更有效.
In this paper,we mainly study three postprocessed mixed finite element methods 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 stabilized nonlinear Navier-Stokes problem on a coarse grid to obtain an approximate solution u_Hat time T.The second step is to postprocess u_H on a finer grid(or by highorder finite elements),by solving a stabilized Stokes problem,a stabilized Newton-Type problem,or a stabilized Ossen problem.The numerical results show that under the conditions of selecting appropriate stabilizing parameters and grid sizes,the postprocessed finite element method can improve the precision of the mixed finite-element solution,and the order of convergence is obviously improved by one unit compared with the standard subgrid stabilized method.From the point of the computational time,in addition to ν=1,the stabilized Newton-type postprocessed method takes a relatively 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 precision of the computed solutions,the Newton and Oseen-type postprocessed methods are better than the Stokes-type postprocessed method.
In this paper,we mainly study three postprocessed mixed finite element methods 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 stabilized nonlinear Navier-Stokes problem on a coarse grid to obtain an approximate solution u_Hat time T.The second step is to postprocess u_H on a finer grid(or by highorder finite elements),by solving a stabilized Stokes problem,a stabilized Newton-Type problem,or a stabilized Ossen problem.The numerical results show that under the conditions of selecting appropriate stabilizing parameters and grid sizes,the postprocessed finite element method can improve the precision of the mixed finite-element solution,and the order of convergence is obviously improved by one unit compared with the standard subgrid stabilized method.From the point of the computational time,in addition to ν=1,the stabilized Newton-type postprocessed method takes a relatively 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 precision of the computed solutions,the Newton and Oseen-type postprocessed methods are better than the Stokes-type postprocessed method.
引文
[1]TEMAN R.Navier-Stokes Equations:Theory and Numerical Analysis[M].Amsterdam:North-Holland Publishing,1977.
[2]GIRAULT V,RAVIART P A.Finite Element Methods for Navier-Stokes Equations:Theory and Algorithms[M].Berlin:Springer-Verlag,1986.
[3]GLOWINSKI R.Finite Element Methods for Incompressible Viscous Flow[M]//Handbook of Numerical Analysis.Amsterdam:Elsevier,2003.
[4]ARCHILLA B G,NOVO J,TITI E S.Postprocessing the Galerkin Method:A Novel Approach to Approximate Inertial Manifolds[J].SIAM Journal on Numerical Analysis,1998,35(3):941-972.
[5]GARC B.Postprocessing the Galerkin Method:The Finite-Element Case[J].SIAM Journal on Numerical Analysis,2000,37(2):470-499.
[6]AYUSO B,ARCHILLA B G,NOVO J.The Postprocessed Mixed Finite-Element Methodfor the Navier-Stokes Equations[J].SIAM Journal on Numerical Analysis,2005,43(3):1091-1111.
[7]FRUTOS J D,ARCHILLA B G,NOVO J.Static Two-Grid Mixed Finite-Element Approximations to the Navier-Stokes Equations[J].Journal of Scientific Computing,2012,52(3):619-637.
[8]DURANGO F,NOVO J.Two-Grid Mixed Finite-Element Approximations to the Navier-StokesEquations Based on a Newton-Type Step[J].Journal of Scientific Computing,2018,74(1):456-473.
[9]SHANG Y Q.A Two-Level Subgrid Stabilized Oseen IterativeMethod for the Steady Navier-Stokes Equations[J].Journal of Computational Physics,2013,233(1):210-226.
[10]HECHT F.New Development in Freefem++[J].Journal of Numerical Mathematics,2012,20(3-4):159-344.
[11]SHANG Y Q.Error Analysis of a Fully Discrete Finite Element Variational Multiscale Method for Time-Dependent Incompressible Navier-Stokes Equations[J].Numerical Methods for Partial Differential Equations,2013,29(6):2025-2046.
[2]GIRAULT V,RAVIART P A.Finite Element Methods for Navier-Stokes Equations:Theory and Algorithms[M].Berlin:Springer-Verlag,1986.
[3]GLOWINSKI R.Finite Element Methods for Incompressible Viscous Flow[M]//Handbook of Numerical Analysis.Amsterdam:Elsevier,2003.
[4]ARCHILLA B G,NOVO J,TITI E S.Postprocessing the Galerkin Method:A Novel Approach to Approximate Inertial Manifolds[J].SIAM Journal on Numerical Analysis,1998,35(3):941-972.
[5]GARC B.Postprocessing the Galerkin Method:The Finite-Element Case[J].SIAM Journal on Numerical Analysis,2000,37(2):470-499.
[6]AYUSO B,ARCHILLA B G,NOVO J.The Postprocessed Mixed Finite-Element Methodfor the Navier-Stokes Equations[J].SIAM Journal on Numerical Analysis,2005,43(3):1091-1111.
[7]FRUTOS J D,ARCHILLA B G,NOVO J.Static Two-Grid Mixed Finite-Element Approximations to the Navier-Stokes Equations[J].Journal of Scientific Computing,2012,52(3):619-637.
[8]DURANGO F,NOVO J.Two-Grid Mixed Finite-Element Approximations to the Navier-StokesEquations Based on a Newton-Type Step[J].Journal of Scientific Computing,2018,74(1):456-473.
[9]SHANG Y Q.A Two-Level Subgrid Stabilized Oseen IterativeMethod for the Steady Navier-Stokes Equations[J].Journal of Computational Physics,2013,233(1):210-226.
[10]HECHT F.New Development in Freefem++[J].Journal of Numerical Mathematics,2012,20(3-4):159-344.
[11]SHANG Y Q.Error Analysis of a Fully Discrete Finite Element Variational Multiscale Method for Time-Dependent Incompressible Navier-Stokes Equations[J].Numerical Methods for Partial Differential Equations,2013,29(6):2025-2046.