## 留言板

 引用本文: 黄鹏展, 何银年, 冯新龙. 解Stokes特征值问题的一种两水平稳定化有限元方法[J]. 应用数学和力学, 2012, 33(5): 588-597.
HUANG Peng-zhan, HE Yin-nian, FENG Xin-long. A Two-Level Stabilized Finite Element Method for the Stokes Eigenvalue Problem[J]. Applied Mathematics and Mechanics, 2012, 33(5): 588-597. doi: 10.3879/j.issn.1000-0887.2012.05.007
 Citation: HUANG Peng-zhan, HE Yin-nian, FENG Xin-long. A Two-Level Stabilized Finite Element Method for the Stokes Eigenvalue Problem[J]. Applied Mathematics and Mechanics, 2012, 33(5): 588-597.

## 解Stokes特征值问题的一种两水平稳定化有限元方法

##### doi: 10.3879/j.issn.1000-0887.2012.05.007

###### 通讯作者: 黄鹏展(1983—),男,浙江人,博士生(E-mail:hpzh007@yahoo.cn);何银年(1953—),男,陕西人,教授,博士(联系人. Tel: +86-29-82665242; E-mail: heyn@mail.xjtu.edu.cn);冯新龙(1976—),男,江苏人,教授,博士(E-mail: fxlmath@yahoo.cn).
• 中图分类号: O242.21; O241.82; O351.3

## A Two-Level Stabilized Finite Element Method for the Stokes Eigenvalue Problem

• 摘要: 基于局部Gauss积分,研究了解Stokes特征值问题的一种两水平稳定化有限元方法．该方法涉及在网格步长为H的粗网格上解一个Stokes特征值问题,在网格步长为h=O(H2)的细网格上解一个Stokes问题．这样使其能够仍旧保持最优的逼近精度,求得的解和一般的稳定化有限元解具有相同的收敛阶,即直接在网格步长为h的细网格上解一个Stokes特征值问题．因此,该方法能够节省大量的计算时间．数值试验验证了理论结果．
•  [1] Babuska I, Osborn J E. Eigenvalue Problems[C]Ciarlet P G, Lions J L.Handbook of Numerical Analysis, vol Ⅱ, Finite Element Method (Part Ⅰ). Amsterdam: North-Holland, 1991. [2] Babuska I, Osborn J E. Finite Element-Galerkin approximation of the eigenvalues and eigenvectors of self adjoint problems[J]. Math Comp, 1989, 52(186): 275-297. [3] Lin Q, Xie H. Asymptotic error expansion and Richardson extrapolation of eigenvalue approximations for second order elliptic problems by the mixed finite element method[J]. Appl Numer Math, 2009, 59(8): 1884-1893. [4] Lin Q. Fourth order eigenvalue approximation by extrapolation on domains with reentrant corners[J]. Numer Math, 1991, 58(1): 631-640. [5] Jia S, Xie H, Yin X, Gao S. Approximation and eigenvalue extrapolation of Stokes eigenvalue problem by nonconforming finite element methods[J]. Appl Math, 2009, 54(1): 1-15. [6] Chen H, Jia S H, Xie H. Postprocessing and higher order convergence for the mixed finite element approximations of the eigenvalue problem[J]. Appl Numer Math, 2011, 61(4): 615-629. [7] Chen H, Jia S, Xie H. Postprocessing and higher order convergence for the mixed finite element approximations of the Stokes eigenvalue problems[J]. Appl Math, 2009, 54(3): 237-250. [8] Huang P Z, He Y N, Feng X L. Numerical investigations on several stabilized finite element methods for the Stokes eigenvalue problem[J]. Math Prob Engrg, 2011, 2011: 1-14. [9] Chen W, Lin Q. Approximation of an eigenvalue problem associated with the Stokes problem by the stream function-vorticity-pressure method[J]. Appl Math, 2006, 51(1): 73-88. [10] Mercier B, Osborn J, Rappaz J, Raviart P A. Eigenvalue approximation by mixed and hybrid methods[J]. Math Comput, 1981, 36(154): 427-453. [11] Xu J, Zhou A H. A two-grid discretization scheme for eigenvalue problems[J]. Math Comput, 2009, 70(233): 17-25. [12] Yin X, Xie H, Jia S, Gao S. Asymptotic expansions and extrapolations of eigenvalues for the Stokes problem by mixed finite element methods[J]. J Comput Appl Math, 2008, 215(1): 127-141. [13] Lovadina C, Lyly M, Stenberg R. A posteriori estimates for the Stokes eigenvalue problem[J]. Numer Meth Part Differ Equ, 2009, 25(1): 244-257. [14] Luo F, Lin Q, Xie H. Computing the lower and upper bounds of Laplace eigenvalue problem: by combining conforming and nonconforming finite element methods[J]. Sci China Math, 2012. doi: 10.1007/s11425-012-4382-2. [15] Bochev P, Dohrmann C R, Gunzburger M D. Stabilization of low-order mixed finite elements for the Stokes equations[J]. SIAM J Numer Anal, 2006, 44(1): 82-101. [16] Li J, He Y N. A stabilized finite element method based on two local Gauss integrations for the Stokes equations[J]. J Comput Appl Math, 2008, 214(1): 58-65. [17] Li J, Chen Z. A new local stabilized nonconforming finite element method for the Stokes equations[J]. Computing, 2008, 82(2): 157-170. [18] Li J, He Y N, Chen Z X. A new stabilized finite element method for the transient Navier-Stokes equations[J]. Comput Methods Appl Mech Engrg, 2007, 197(1): 22-35. [19] Li J. Investigations on two kinds of two-level stabilized finite element methods for the stationary Navier-Stokes equations[J]. Appl Math Comput, 2006, 182(2): 1470-1481. [20] Huang P Z, Zhang T, Si Z Y. A stabilized Oseen iterative finite element method for stationary conduction-convection equations[J]. Math Meth Appl Sci, 2012, 35(1): 103-118. [21] Xu J. A novel two-grid method for semilinear elliptic equations[J]. SIAM J Sci Comput, 1994, 15(1): 231-237. [22] Xu J. Two-grid discretization techniques for linear and nonlinear PDEs[J]. SIAM J Numer Anal, 1996, 33(5): 1759-1778. [23] Layton W, Tobiska L. A two-level method with backtracking for the Navier-Stokes equations[J]. SIAM J Numer Anal, 1998, 35(5): 2035-2054. [24] 马飞遥, 马逸尘, 沃维丰.基于二重网格的定常Navier-Stokes方程的局部和并行有限元算法[J]. 应用数学和力学, 2007, 28(1): 25-33.(MA Fei-yao, MA Yi-chen, WO Wei-feng. Local and parallel finite element algorithms based on two-grid discretization for steady Navier-Stokes equations[J]. Applied Mathematics and Mechanics(English Edition), 2007, 28(1): 27-35.) [25] 秦新强, 马逸尘, 章胤.二维非线性对流扩散方程特征有限元的两重网络算法[J]. 应用数学和力学, 2005, 26(11): 1365-1372.(QIN Xin-qiang, MA Yi-chen, ZHANG Yin. Two-grid method for characteristics finite-element solution of 2D nonlinear convection-dominated diffusion problem[J]. Applied Mathematics and Mechanics(English Edition), 2005, 26(11): 1506-1514.) [26] 王琤, 黄自萍, 李立康.二阶椭圆问题带单位分解技巧的两重网格方法[J]. 应用数学和力学, 2008, 29(4): 477-482.(WANG Cheng, HUANG Zi-ping, LI Li-kang. Two-grid partition of unity method for second order elliptic problems[J]. Applied Mathematics and Mechanics(English Edition), 2008, 29(4): 527-533.) [27] Zhang Y, He Y N. A two-level finite element method for the stationary Navier-Stokes equations based on a stabilized local projection[J]. Numer Meth Part Differ Equ, 2011, 27(2): 460-477. [28] Ervin V, Layton W, Maubach J. A posteriori error estimators for a two-level finite element method for the Navier-Stokes equations[J]. Numer Meth Part Differ Equ, 1996, 12(3): 333-346. [29] He Y N, Li K T. Two-level stabilized finite element methods for the steady Navier-Stokes problem[J]. Computing, 2005, 74(4): 337-351. [30] He Y N, Wang A W. A simplified two-level method for the steady Navier-Stokes equations[J]. Comput Methods Appl Mech Engrg, 2008, 197(17): 1568-1576. [31] 尚月强, 罗振东. Navier-Stokes方程的一种并行两水平有限元方法[J]. 应用数学和力学, 2010, 31(11): 1351-1359.(SHANG Yue-qiang, LUO Zhen-dong. A parallel two-level finite element method for the Navier-Stokes equations[J]. Applied Mathematics and Mechanics(English Edition), 2010, 31(11): 1429-1438.) [32] Becker R, Hansbo P. A simple pressure stabilization method for the Stokes equation[J]. Commun Numer Meth Engrg, 2008, 24(11): 1421-1430. [33] Hecht F, Pironneau O, Hyaric A L, Ohtsuka K. FREEFEM++ [34] [P/OL], version 2.3-3, Paris, 2008 [35] [2012-03-19], http:www.freefem.org.

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##### 出版历程
• 收稿日期:  2011-05-04
• 修回日期:  2012-02-10
• 刊出日期:  2012-05-15

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