A NEW NONCONFORMING MIXED FINITE ELEMENT SCHEME FOR THE STATIONARY NAVIER-STOKES EQUATIONS

doi:10.1016/S0252-9602(11)60238-5

数学物理学报(英文版) ›› 2011, Vol. 31 ›› Issue (2): 367-382.doi: 10.1016/S0252-9602(11)60238-5

A NEW NONCONFORMING MIXED FINITE ELEMENT SCHEME FOR THE STATIONARY NAVIER-STOKES EQUATIONS

石东洋¹|任金城²|龚伟³

1.Department of Mathematics, Zhengzhou University, Zhengzhou 450052, China；2.Department of Mathematics, Shangqiu Normal University, Shangqiu 476000, China；3.Institute of Computational Mathematics, Academy of Mathematics and Systems Science, |Chinese Academy of Sciences, Beijing 100190, China

收稿日期:2007-09-07 修回日期:2010-02-27 出版日期:2011-03-20 发布日期:2011-03-20
基金资助:
The research is supported by NSF of China (10671184)

A NEW NONCONFORMING MIXED FINITE ELEMENT SCHEME FOR THE STATIONARY NAVIER-STOKES EQUATIONS

SHI Dong-Yang¹, REN Jin-Cheng², GONG Wei³

1.Department of Mathematics, Zhengzhou University, Zhengzhou 450052, China；2.Department of Mathematics, Shangqiu Normal University, Shangqiu 476000, China；3.Institute of Computational Mathematics, Academy of Mathematics and Systems Science, |Chinese Academy of Sciences, Beijing 100190, China

Received:2007-09-07 Revised:2010-02-27 Online:2011-03-20 Published:2011-03-20
Supported by:
The research is supported by NSF of China (10671184)

摘要/Abstract

摘要：

In this article, a new stable nonconforming mixed finite element scheme is proposed for the stationary Navier-Stokes equations, in which a new low order Crouzeix-Raviart type nonconforming rectangular element is taken for approximating space for the velocity and the piecewise constant element for the pressure. The optimal order error estimates for the approximation of both the velocity and the pressure in L²-norm are established, as well as one in broken H¹-norm for the velocity. Numerical experiments are given which are consistent with our theoretical analysis.

关键词: Stationary Navier-Stokes equations, nonconforming mixed finite element scheme, optimal order error estimates

Abstract:

Key words: Stationary Navier-Stokes equations, nonconforming mixed finite element scheme, optimal order error estimates

中图分类号:

65N15

石东洋, 任金城, 龚伟. A NEW NONCONFORMING MIXED FINITE ELEMENT SCHEME FOR THE STATIONARY NAVIER-STOKES EQUATIONS[J]. 数学物理学报(英文版), 2011, 31(2): 367-382.

SHI Dong-Yang, REN Jin-Cheng, GONG Wei. A NEW NONCONFORMING MIXED FINITE ELEMENT SCHEME FOR THE STATIONARY NAVIER-STOKES EQUATIONS[J]. Acta mathematica scientia,Series B, 2011, 31(2): 367-382.

参考文献

[1] Nicolaides R A. Analysis and convergence of the MAC scheme II: Navier-Stokes equations. SIAM J Numer Anal, 1992, 65(213): 29--44

[2] Girault V, Raviart P A. Finite Element Method for Navier-Stokes Equations: Theory and Algorithms. New York: Springer-Verlag, 1986

[3] Temam R. Navier-Stokes Equation, Theory and Numerical Analysis. Amstedam, New York: North-Hoolland, 1984

[4] Thomasset F. Implementation of Finite Element Methods for Navier-Stokes Equations. Berlin: Springer, 1981

[5] Eymard R, Herbin R. A staggered finite volume scheme on general meshes for the Navier-Stokes Equations in two space dimensions. Int J Finite Volumes, 2005, 2(1) (electronic)

[6] Douglas J Jr, Santos J E, Sheen D, Ye X. Nonconforming Galerkin methods based on quadrilateral elements for second order
elliptic problems. RAIRO Math Model Anal Numer, 1999, 33(4): 747--770

[7] Crouzeix M, Raviart P A. Conforming and nonconforming finite element methods for solving the stationary Stokes equations. RAIRO Numer Anal, 1973, 7(R-3): 33--76

[8] Rannacher R, Turek S. Simple nonconforming quadrilateral Stokes element. Numer Meth PDE, 1992, 8(97): 97--111

[9] Cai Z Q, Douglas J Jr, Ye X. A stable nonconforming quadrilateral finite element method for the stationary Stokes and Navier-Stokes equations. Calcolo, 1999, 36(4): 215--232

[10] Li K T, Huang A X, Huang Q H. The Finite Element Methods and Applications(II). Xi'an: Xi'an Jiaotong University Press, 1987

[11] Han H D. Nonconforming elements in the mixed finite element method. J Comput Math, 1984, 2(3): 223--233

[12] Adams R A. Sobolev Spaces. New York: Academic Press, 1975

[13] Li K T, Zhou L. Finite element nonlinear Galerkin methods for penalty Navier-Stokes equations. Math Numer Sinic, 1995, 17(4): 360--380

[14] Shi D Y, Ren J C. Nonconforming mixed finite element method for the stationary Conduction-Convection problem. Inter J Numer Anal Modeling, 2009, 6(2): 293--310

[15] Ciarlet P G. The Finite Element Method for Elliptic Problems. Amstedam, New York: North-Hoolland, 1978

[16] Apel T, Nicaise S, Schp\"{o}berl L. Crouzeix-Raviart type finite elements on anisotropic meshes. Numer Math, 2001, 89(2): 193--223

[17] Ming P B. Nonconforming elements vs locking problem
[Ph D Thesis]. Beijing: Institute of Computational Mathematics, CAS, 1999

[18] Hu J. Quadrilateral locking free elements in elasticity
[Ph D Thesis]. Beijing: Institute of Computational Mathematics, CAS, 2004

[19] Shi D Y, Mao S P, Chen S C. An anisotropic nonconforming finite element with some superconvergence results. J Comput Math, 2005, 23(3): 261--274

[20] Lin Q, Tobiska L, Zhou A H. Superconvergence and extrapolation of nonconforming low order elements applied to the Poisson equation. IMA J Numer Anal, 2005, 25(1): 160--181

[21] Lee C O, Lee J, Sheen D. A locking-free nonconforming finite element method for planar linear elasticity. Advances Comput Math, 2003, 19(1--3): 277--291

[22] Wang L H, Qi H. On Locking-free finite element schemes for the pure displacement boundary value problem in the planar linear elasticity. Math Numer Sinica, 2002, 24(2): 243--256

[23] Shi D Y, Mao S P, Chen S C. A Locking-free anisotropic nonconforming finite element for planar linear elasticity problem. Acta Mathematica Scientia, 2007, 27B(1): 193--202

[24] He Y N, Wang A W. A simplified two-level method for the steady Navier-Stokes equations. Comput Methods Appl Mech Engrg, 2008, 197(17/18): 1568--1576

A NEW NONCONFORMING MIXED FINITE ELEMENT SCHEME FOR THE STATIONARY NAVIER-STOKES EQUATIONS

A NEW NONCONFORMING MIXED FINITE ELEMENT SCHEME FOR THE STATIONARY NAVIER-STOKES EQUATIONS

PDF (PC)

摘要/Abstract

引用本文

使用本文

参考文献

相关文章 9

Metrics

本文评价

推荐阅读 0

[1]	祝鹏, 王筱沈. A LEAST SQUARE BASED WEAK GALERKIN FINITE ELEMENT METHOD FOR SECOND ORDER ELLIPTIC EQUATIONS IN NON-DIVERGENCE FORM[J]. 数学物理学报(英文版), 2020, 40(5): 1553-1562.
[2]	杜光芝. EXPANDABLE PARALLEL FINITE ELEMENT METHODS FOR LINEAR ELLIPTIC PROBLEMS[J]. 数学物理学报(英文版), 2020, 40(2): 572-588.
[3]	王培珍, 陈绍春. A POSTERIORI ERROR ESTIMATION OF THE NEW MIXED ELEMENT SCHEMES FOR SECOND ORDER ELLIPTIC PROBLEM ON ANISOTROPIC MESHES[J]. 数学物理学报(英文版), 2014, 34(5): 1510-1518.
[4]	赵纪坤, 陈绍春. EXPLICIT ERROR ESTIMATE FOR THE NONCONFORMING WILSON´S ELEMENT[J]. 数学物理学报(英文版), 2013, 33(3): 839-846.
[5]	陈绍春, 郑艳君, 毛士鹏. NEW SECOND ORDER NONCONFORMING TRIANGULAR ELEMENT FOR PLANAR ELASTICITY PROBLEMS[J]. 数学物理学报(英文版), 2011, 31(3): 815-825.
[6]	李明霞, 陈竑焘, 毛士鹏. ACCURACY ENHANCEMENT FOR THE SIGNORINI PROBLEM WITH FINITE ELEMENT METHOD[J]. 数学物理学报(英文版), 2011, 31(3): 897-908.
[7]	祝鹏, 谢资清, 周叔子. A COUPLED CONTINUOUS-DISCONTINUOUS FEM APPROACH FOR CONVECTION DIFFUSION EQUATIONS[J]. 数学物理学报(英文版), 2011, 31(2): 601-612.
[8]	石东洋; 毛士鹏; 陈绍春. A LOCKING-FREE ANISOTROPIC NONCONFORMING FINITE ELEMENT FOR PLANAR LINEAR ELASTICITY PROBLEM[J]. 数学物理学报(英文版), 2007, 27(1): 193-202.
[9]	芮洪兴. A FINITE VOLUME ELEMENT METHOD FOR THERMAL CONVECTION PROBLEMS[J]. 数学物理学报(英文版), 2004, 24(1): 129-138.