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计算数学  2011, Vol. 33 Issue (2): 133-144    DOI:
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Darcy-Stokes问题的统一稳定化有限体积法分析
谢春梅1, 骆艳1, 冯民富2
1. 电子科技大学数学科学学院, 成都, 610054;
2. 四川大学数学学院, 成都, 610064
ANALYSIS OF A UNIFIED STABILIZED FINITE VOLUME METHOD FOR THE DARCY-STOKES PROBLEM
Xie Chunmei1, Luo Yan1, Feng Minfu2
1. School of mathematical sciences, University of Electronic Science and Technology, Chengdu 610054, China;
2. School of Mathematics, Sichuan University, Chengdu 610064, China
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摘要 

本文对Darcy-Stokes问题提出了一种统一的稳定化有限体积法.在离散问题中, 采用两种剖分, 一种为三角形剖分, 一种为其对偶四边形剖分. 速度及压力分别采用非协调线性元及分片常数元来做逼近. 经证明, 文中的统一格式, 具有稳定性及最优误差估计. 最后用数值算例验证了本文的理论结果.

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关键词有限体积法   Darcy-Stokes问题   稳定化     
Abstract

A unified stabilized finite volume element method is proposed for the Darcy-Stokes problem. For the discretization form, two grids are needed; one is triangulation and the other is quadrilateral meshes. The velocity is approximated using nonconforming piecewise linears and the pressure piecewise constants. The proposed unified method in this paper is shown to be stable and optimally convergent for both the velocity and the pressure. Moreover, numerical experiments are given to demonstrate the theoretical results.

Key wordsfinite volume element   the Darcy-Stokes problem   stabilization   
收稿日期: 2009-09-10;
基金资助:

四川省科技攻关课题(编号05GG006-006-2)资助项目.

引用本文:   
. Darcy-Stokes问题的统一稳定化有限体积法分析[J]. 计算数学, 2011, 33(2): 133-144.
. ANALYSIS OF A UNIFIED STABILIZED FINITE VOLUME METHOD FOR THE DARCY-STOKES PROBLEM[J]. Mathematica Numerica Sinica, 2011, 33(2): 133-144.
 
[1] Brenner S C, Ridgway S L. The mathematical theory of finite element methods[M]. Springer-Verlag, 1994. MR 95f: 65001.
[2] Burman E, Hansbo P. A unified stabilized method for Stokes’ and Darcy’s equations[J]. J. Comput. Appl. Math., 2007, 198(1): 35-51.
[3] Burman E, Hansbo P. Stabilized Crouzeix-Raviart element for the Darcy-Stokes problem[J]. Nu-mer. Meth. PDEs., 2005, 21(5): 986-997.
[4] Chou S H. Analysis and convergence of a covolume method for the generalized Stokes problem[J]. Math. comput., 1997, 66(217): 85-104.
[5] Chou S H, Kwak D Y. A covolume method based on rotated bilinears for the generalized Stokes problem[J]. SIAM. J. Numer. Anal., 1998, 35(2): 494-507.
[6] Chou S H, Kwak D Y. Analysis and convergence of a MAC scheme for the generalized Stokes problem[J]. Numer. Meth. PDEs., 1997, 13(2): 147-162.
[7] Chou S H, Vassilevski P S. A general mixed covolume framework for constructing conservative schemes for elliptic problems[J]. Math. Comput., 1999, 68(227): 991-1011.
[8] Crouzeix M, Raviart P A. Conforming and nonconforming finite element methods for solving the stationary Stokes equations[J]. Rairo Anal. Numer., 1973, 7: 33-76.
[9] Gao Fuzheng, Yuan Yirang. The upwind finite volume element method based on straight triangular prism partition for nonlinear convection-diffusion problem[J]. Appl. Math. Comput., 2006, 181(2): 1229-1242.
[10] Gastaldi L, Nochetto R. Optimal L1-error estimates for nonconforming and mixed finite element methods of lowest order[J]. Numer. Math., 1987, 50(5): 587-611.
[11] He Guoliang, He Yinnian. The finite volume method based on stabilized finite element for the stationary Navier-Stokes problem[J]. J. Comput. Appl. Math., 2007, 205(1): 651-665.
[12] Wu Haijun, Li Ronghua. Error estimates for finite volume element methods for general second-order elliptic problems[J]. Numer. Meth. PDEs., 2003, 19(6): 693-708.
[13] Rui Hongxing. Symmetric modified finite volume element methods for self-adjoint elliptic and parabolic problems[J]. J. Comput. Appl. Math., 2002, 146(2): 373-386.
[14] Lazarov R,Michev I and Vassilevski P. Finite volume methods for convection-diffusion problems[J]. SIAM. J. Numer. Anal., 1996, 33(1): 31-55.
[15] Mardal K A, Tai X.C, Winther R. A robust finite element method for Darcy-Stokes flow[J]. SIAM. J. Numer. Anal., 2002, 40(5): 1605-1631.
[16] Zhou Tianxiao, Feng Minfu, A least square petrov-Galerkin finite element method for the station-ary Navier-Stokes equations[J]. Math. Comp., 1993, 60(202): 531-543.
[17] Xie Xiaoping, Xu Jinchao, Xue Guangrui. Uniformly-stable finite element methods for Darcy-Stokes-Brinkman models[J]. J. Comput. Math., 2008, 26(3): 437-455.
[18] Ye Xiu. A discontinuous finite volume method for the Stokes problems[J]. SIAM. J. Numer. Anal., 2006, 44(1): 183-198.
[19] Luo Zhendong, Zhu Jiang. Convergence of simplified and stabilized mixed element formats based on bubble function for the Stokes problem[J]. Appl. Math. Mech., 2002, 23(10): 1207-1214.
[20] 胡俊, 满红英, 石钟慈. 带约束非协调旋转Q1元在Stokes的平面弹性问题的应用[J]. 计算数学, 2005, 27(3): 311-324. 浏览
[21] 骆艳, 冯民富. Stokes方程的间断有限元法[J]. 计算数学, 2006, 28(2): 163-174. 浏览
[22] 骆艳, 冯民富. Stokes方程的压力梯度局部投影间断有限元法[J]. 计算数学, 2008, 30(1): 25-36. 浏览
[23] 石东洋, 毛士鹏. 三维Stokes问题各项异性混合元分析[J]. 应用数学学报, 2006, 29(3): 502-517.
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[11] 段火元. 稳定化有限元方法中逆估计常数的确定[J]. 计算数学, 1998, 20(4): 403-408.

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