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

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

Abstract

Abstract:

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.

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

CLC Number:

65N15

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.

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References

[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