非定常Navier-Stokes方程的质量集中各向异性非协调有限元逼近

Abstract

Abstract:

In this paper, a low order Crouzeix-Raviart type nonconforming triangular element is applied to the nonstationary Navier-Stokes equations. The approximation scheme of the lumped mass finite element methods for the problem is proposed. Without using Ritz-Volterra projection, the error estimates are derived both in the L²-norm and the energy norm for velocity and the L²-norm for pressure on anisotropic meshes through the technique of introducing two auxiliary finite element spaces to the boundary estimate.

Key words: Navier-Stokes equations, Lumped mass, Anisotropic meshes, Nonconforming finite element, Error estimate

CLC Number:

65N15

SHI Dong-Yang, WANG Hui-Min. The Lumped Mass Nonconforming Finite Element Approximation for the Nonstationary Navier-Stokes Equations on Anisotropic Meshes[J].Acta mathematica scientia,Series A, 2010, 30(4): 1018-1029.

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References

[1] Temam R. Navier-Stokes Equations. New York, Amsterdam: North-Holland, 1977

[2] Girault V, Raviart P A. Finite Element Approximation of the Navier-Stokes Equations. Berlin, New York: Springer-Verlag, 1981

[3] Heywood J G, Rannacher R. Finite element approximation of the nonstationary Navier-Stokes problem I: Regularity of solutions and second-order error estimates for spatial disretization. SIAM J Numer Anal, 1982, 19(2): 275--311

[4] Bernardi C, Raugel G. A nonconforming finite element method for the time-dependent Navier-Stokes equations. SIAM J Numer Anal, 1985, 22(3): 455--473

[5] Ciarlet P G. The Finite Element Method for Elliptic Problem. Amsterdam: North-Holland, 1978

[6] Zenisek A, Vanmaele M. The interpolation theory for narrow quadrilateral isoparametric finite elements. Numer Math, 1995, 72(1): 123--141

[7] Apel T, Lue G. Anisotropic mesh refinement in stabilized Galerkin methods. Numer Math, 1996, 74(3): 261--282

[8] Apel T, Dobrowlski M. Anisotropic interpolation with application to the finite element method. Computing, 1992, 47(3): 277--293

[9] Chen S C, Shi D Y, Zhao Y C. Anisotropic interpolation and quasi-Wilson element for narrow quadrilateral meshes. IMA J Numer Anal, 2004, 24: 77--95

[10] Shi D Y, Mao S P, Chen S C. A locking-free anisotropic nonconforming finite element for planar linear elasticity. Acta Math Sci, 2007, 27B(1): 193--202

[11] Chen S C, Zhao Y C, Shi D Y. Anisotropic interpolations with application to nonconforming elements. Appl Numer Math, 2004, 49(2): 135--152

[12] Shi D Y, Zhu H Q. The superconvergence analysis of an anisotropic element. J Sys Sci Complex, 2005, 18(4): 478--487

[13] Shi D Y, Mao S P, Chen S C. On the anisotropic accuracy analysis of ACM's nonconforming finite element. J Comput Math, 2005, 23(6): 635--646

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

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

[16] 李开泰,黄艾香,黄庆怀.有限元方法及其应用(II).西安:西安交通大学出版社,1988

[17] 黄明游. 发展方程数值计算方法.北京:科学出版社, 2004

[18] 戴培良. Navier-Stokes方程的集中质量非协调有限元法.工程数学学报, 2007, 24(2): 249--253

[19] Shi D Y, Zhang Y R. A nonconforming anisotropic finite element approximation with moving grids for Stokes problem. J Comput Math, 2006, 24(5): 561--578

[20] Apel T, Nicaise S, Sch\"{o}berl J. Crouzeix-Raviart type finite elements on anisotropic meshes. Numer Math, 2001, 89: 193--223

[21 }Apel T, Nicaise S, Schöberl J. A nonconforming finite elements method with anisotropic mesh groding for the Stokes problem in domains with edges. IMA J Numer Anal, 2001, 21: 843--856

[22] 石东洋, 张熠然. 非定常Stokes问题的矩形Crouzeix-Raviart型各向异性非协调元变网格方法.数学物理学报, 2006, 26A(5): 659--670

[23] Lin Q, Tobiska L, Zhou A H. Superconvergence and extrapolation of non-conforming low order finite elements applied to the Poisson equation. IMA J Numer Anal, 2005, 25: 160--181

[24] 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

[25] 石东洋, 汪松玉, 陈绍春. 两类各向异性非协调元某些超收敛性分析.计算数学, 2007, 29(3): 263--272

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

The Lumped Mass Nonconforming Finite Element Approximation for the Nonstationary Navier-Stokes Equations on Anisotropic Meshes

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