Streamline Diffusion Nonconforming Finite Element Method for the Time-Dependent Linearized Navier-Stokes Equations

CHEN Yu-mei; XIE Xiao-ping

doi:10.3879/j.issn.1000-0887.2010.07.007

Volume 31 Issue 7

Turn off MathJax

Article Contents

Abstract

References

Article Navigation > Applied Mathematics and Mechanics > 2010 > 31(7): 822-834

CHEN Yu-mei, XIE Xiao-ping. Streamline Diffusion Nonconforming Finite Element Method for the Time-Dependent Linearized Navier-Stokes Equations[J]. Applied Mathematics and Mechanics, 2010, 31(7): 822-834. doi: 10.3879/j.issn.1000-0887.2010.07.007

Citation:

PDF( 442 KB)

Streamline Diffusion Nonconforming Finite Element Method for the Time-Dependent Linearized Navier-Stokes Equations

doi: 10.3879/j.issn.1000-0887.2010.07.007

CHEN Yu-mei^1,2,
XIE Xiao-ping¹

School of Mathematics, Sichuan University, Chengdu 610064, P. R. China;
College of Mathematics and Information, China West Normal University, Nanchong, Sichuan 637002, P. R. China

Received Date: 1900-01-01
Rev Recd Date: 2010-05-28
Publish Date: 2010-07-15

Abstract

Abstract

A finite difference streamline diffusion non conforming finite element approxmiation was proposed for solving the time-dependent linearized Navier-Stokes equations. Stream line diffusion finite element method was used to discretize the space variables in order to cope with the usual instabilities caused by the convection term and finite difference discretization was used in the time domain. Noncon forming finite element approxmiations were used for the velocity and pressure fields: the velocity is approxmiated by discontinuous piecewise linear and the pressure by piecewise constant. Stability and optimal error estimates for the discrete solutions are obtained.
- stream line diffusion method,
- nonconforming,
- time-dependent linearized Navier-Stokes,
- error estimate

FullText(HTML)

References(16)

References

[1]	Hughes T J R, Brooks A N.A multi-dimensional up-wind scheme with no crosswind diffusion[C]Hughes T J R. Finite Element Methods for Convection Dominated Flows. ASME Monograph AMD-34, 1979:19-35.
[2]	Nvert U.A finite element for convection-diffusion problems[D]. PhD thesis. Goteborg: Chalmers University of Technology, 1982.
[3]	Johnson C.Finite element methods for convection-diffusion problems[C]Glowinski R, Lions J L. Computing Methods in Engineering and Applied Sciences Ⅴ.Amsterdam: North-Holland, 1981, 311-323.
[4]	Johnson C, Nvert U.An analysis of some finite element methods for advection-diffusion[C]Axelsson O, Frank L S, Van der Sluis A.Analytical and Numerical Approaches to Asymptotic Problems in Analysis.Amsterdam: North-Holland, 1981:99-118.
[5]	Johnson C, Nvert U, Pitkranta J.Finite element methods for linear hyperbolic problems[J].Computer Methods in Applied Mechanics and Engineering, 1984, 45(1/3):285-312. doi: 10.1016/0045-7825(84)90158-0
[6]	Johnson C, Saranen J.Streamline diffusion methods for the incompressible Euler and Navier-Stokes equations[J].Mathematics of Computation,1986, 47(175):1-18. doi: 10.1090/S0025-5718-1986-0842120-4
[7]	Tobiska L, Verfürth R.Analysis of a streamline diffusion finite element method for the Stokes and Navier-Stokes equations[J].SIAM Journal on Numerical Analysis,1996, 33(1):107-127. doi: 10.1137/0733007
[8]	Zhou G H.How accurate is the streamline diffusion finite element method[J].Journal of Computational Mathematics, 1997, 66(217):31-44. doi: 10.1090/S0025-5718-97-00788-6
[9]	Sun C, Shen H.The finite difference streamline diffusion method for time-dependent convection-diffusion equations[J].Numerical Mathematics（A Journal of Chinese Universities English Series）,1998, 7(1):72-85.
[10]	张强.不可压N-S方程的差分流线扩散法[J].计算数学, 2003, 25(3):311-320.
[11]	张强, 孙澈.非线性对流扩散方程的差分流线扩散有限元方法[J].计算数学, 1998, 20(2):211-224.
[12]	Sun T J, Ma K Y.The finite difference streamline diffusion method for the incompressible Navier-Stokes equations[J].Applied Mathematics and Computation, 2004, 149:493-505. doi: 10.1016/S0096-3003(03)00156-5
[13]	John V, Maubach J, Tobiska L.Nonconforming streamline-diffusion-finite-element-methods for convection-diffusion problems[J].Numerische Mathematik, 1997, 78(2):165-188. doi: 10.1007/s002110050309
[14]	John V, Matthies G, Schiewech F, Tobiska L.A streamline-diffusion method for nonconforming finite element approximations applied to convection-diffusion problems[J].Computer Methods in Applied Mechanics and Engineering, 1998, 166(1/2):85-97. doi: 10.1016/S0045-7825(98)80014-5
[15]	Matthies G, Tobiska L. The streamline-diffusion method for conforming and nonconforming finite elements of lowest order applied to convection-diffusion peoblems[J].Computing, 2001, 66(4): 343-364. doi: 10.1007/s006070170019
[16]	Knobloch P, Tobiska L.A streamline diffusion method for nonconforming finite element approximations applied to the linearized incompressible Navier-Stokes equations[C] Iliev O P. Proceedings of the 4th International Conference on Numerical Methods in Application. Singapore, Sofia: World Scientific, 1999:530-538.

Relative Articles

[1]	CUI Jianxuan, SHI Chengxin, LIU Mian, CHENG Hao. Source Identification for the Time-Fractional Diffusion Equation With Robin Boundary Conditions[J]. Applied Mathematics and Mechanics, 2022, 43(11): 1303-1312. doi: 10.21656/1000-0887.430004
[2]	LIU Mian, CHENG Hao, SHI Chengxin. Variational Regularization of the Inverse Problem of a Class of Nonlinear Time-Fractional Diffusion Equations[J]. Applied Mathematics and Mechanics, 2022, 43(3): 341-352. doi: 10.21656/1000-0887.420168
[3]	WU Di, LI Xiaolin. An Element-Free Galerkin Method for Time-Fractional Diffusion-Wave Equations[J]. Applied Mathematics and Mechanics, 2022, 43(2): 215-223. doi: 10.21656/1000-0887.420172
[4]	WANG Hong, LI Xiaolin. Analysis of 2D Transient Heat Conduction Problems With the Element-Free Galerkin Method[J]. Applied Mathematics and Mechanics, 2021, 42(5): 460-469. doi: 10.21656/1000-0887.410111
[5]	BAI Enpeng, XIONG Xiangtuan. A New Regularization Method for Solving Sideways Heat Equations[J]. Applied Mathematics and Mechanics, 2021, 42(5): 541-550. doi: 10.21656/1000-0887.410290
[6]	PANG Naihong, LI Hong. Error Estimates of Mixed Space-Time Finite Element Solutions to Sobolev Equations[J]. Applied Mathematics and Mechanics, 2020, 41(8): 834-843. doi: 10.21656/1000-0887.410053
[7]	TENG Fei, LUO Zhen-dong. A Reduced-Order Stabilized CNFVE Extrapolating Model for Non-Stationary Stokes Equations[J]. Applied Mathematics and Mechanics, 2014, 35(9): 986-1001. doi: 10.3879/j.issn.1000-0887.2014.09.005
[8]	LUO Zhen-dong, OU Qiu-lan, XIE Zheng-hui. A Reduced Finite Difference Scheme and Error Estimates Based on POD Method for the Non-Stationary Stokes Equation[J]. Applied Mathematics and Mechanics, 2011, 32(7): 795-806. doi: 10.3879/j.issn.1000-0887.2011.07.004
[9]	DENG Zhen-guo, MA He-ping. Optimal Error Estimates for Fourier Spectral Approxiation of the Generalized KdV Equation[J]. Applied Mathematics and Mechanics, 2009, 30(1): 30-39.
[10]	LUO Yan, FENG Min-fu. Discontinuous Element Pressure Gradient Stabilizations for the Compressible Navier-Stokes Equations Based on Local Projections[J]. Applied Mathematics and Mechanics, 2008, 29(2): 157-168.
[11]	SHI Dong-yang, WANG Hai-hong, GUO Cheng. Anisotropic Rectangular Nonconforming Finite Element Analysis for Sobolev Equations[J]. Applied Mathematics and Mechanics, 2008, 29(9): 1089-1100.
[12]	LUO Zhen-dong, MAO Yun-kui, ZHU Jiang. Petrov-Galerkin Least Squares Mixed Element Method for the Stationary Incompressible Magnetohydrodynamics[J]. Applied Mathematics and Mechanics, 2007, 28(3): 359-368.
[13]	LUO Zhen-dong, MAO Yun-kui, ZHU Jiang. Nonlinear Galerkin Mixed Element Methods for the Stationary Incompressible Magnetohydrodynamics[J]. Applied Mathematics and Mechanics, 2006, 27(12): 1486-1496.
[14]	LUO Zhen-dong, ZHU Jiang, ZENG Qing-cun, XIE Zheng-hui. Mixed Finite Element Methods for the Shallow Water Equations Including Current and Silt Sedimentation (Ⅱ)-The Discrete-Time Case Along Characteristics[J]. Applied Mathematics and Mechanics, 2004, 25(2): 166-180.
[15]	LUO Zhen-dong, ZHU Jiang, ZENG Qing-cun, XIE Zheng-hui. Mixed Finite Element Methods for the Shallow Water Equations Including Current and Silt Sedimentation (Ⅰ)-The Continuous-Time Case[J]. Applied Mathematics and Mechanics, 2004, 25(1): 74-84.
[16]	LUO Zhen-dong, ZHU Jiang, WANG Hui-jun. A Nonlinear Galerkin/Petrov-Least Squares Mixed Element Method for the Stationary Navier-Stokes Equations[J]. Applied Mathematics and Mechanics, 2002, 23(7): 697-706.
[17]	LUO Zhen-dong, ZHU Jiang. A Nonlinear Galerkin Mixed Element Method and a Posteriori Error Estimator for the Stationary Navier-Stokes Equations[J]. Applied Mathematics and Mechanics, 2002, 23(10): 1061-1072.
[18]	Duan Mei, Miyamoto Yutaka, Zhou Benkuan, Chen Dapeng. On the Error Estimate of h-Convergence in Quadrilateral Elements[J]. Applied Mathematics and Mechanics, 1995, 16(12): 1043-1050.
[19]	Yang Dan-ping. A Coupling Method of Difference with High Order Accuracy and Boundary Integral Equation for Evolutionary Equation and Its Error Estimates[J]. Applied Mathematics and Mechanics, 1991, 12(9): 831-844.
[20]	Hua Bo-hao, Wu Chang-chun, Liu Xiao-ling, Mao Zhao-lin. Plastic Limit Analysis of Incompatible Finite Element Method[J]. Applied Mathematics and Mechanics, 1990, 11(9): 797-803.