A new local stabilized nonconforming finite element method for solving stationary Navier-Stokes equations

In this paper we study a new local stabilized nonconforming finite element method based on two local Gauss integrals for solving the stationary Navier-Stokes equations. This nonconforming method utilizes the lowest equal-order pair of mixed finite elements (i.e., NCP"1-P"1). Error estimates of optimal order are obtained, and numerical results agreeing with these estimates are demonstrated. Numerical comparisons with other mixed finite element methods for solving the Navier-Stokes equations are also presented to show the better performance of the present method.

[1]  Philippe G. Ciarlet,et al.  The finite element method for elliptic problems , 2002, Classics in applied mathematics.

[2]  Clark R. Dohrmann,et al.  Stabilization of Low-order Mixed Finite Elements for the Stokes Equations , 2004, SIAM J. Numer. Anal..

[3]  Yinnian He,et al.  A new stabilized finite element method for the transient Navier-Stokes equations , 2007 .

[4]  Sungyun Lee,et al.  Stable nonconforming quadrilateral finite elements for the Stokes problem , 2000, Appl. Math. Comput..

[5]  C. Dohrmann,et al.  A stabilized finite element method for the Stokes problem based on polynomial pressure projections , 2004 .

[6]  J. Douglas,et al.  A stable nonconforming quadrilateral finite element method for the stationary Stokes and Navier–Stokes equations , 1999 .

[7]  R. Temam,et al.  Navier-Stokes equations: theory and numerical analysis: R. Teman North-Holland, Amsterdam and New York. 1977. 454 pp. US $45.00 , 1978 .

[8]  O. Karakashian.  On a Galerkin--Lagrange Multiplier Method for the Stationary Navier--Stokes Equations , 1982 .

[9]  Yinnian He,et al.  A stabilized finite element method based on two local Gauss integrations for the Stokes equations , 2008 .

[10]  M. Fortin,et al.  A stable finite element for the stokes equations , 1984 .

[11]  Dongwoo Sheen,et al.  Nonconforming Galerkin methods based on quadrilateral elements for second order elliptic problems , 1999 .

[12]  Zhangxin Chen,et al.  Characteristic‐mixed covolume methods for advection‐dominated diffusion problems , 2006, Numer. Linear Algebra Appl..

[13]  Roland Becker,et al.  A finite element pressure gradient stabilization¶for the Stokes equations based on local projections , 2001 .

[14]  Zhangxin Chen.  Finite Element Methods And Their Applications , 2005 .

[15]  David J. Silvester,et al.  Fourier Analysis of Stabilized Q1 -Q1 Mixed Finite Element Approximation , 2001, SIAM J. Numer. Anal..

[16]  Richard E. Ewing,et al.  Multilevel preconditioners for mixed methods for second order elliptic problems , 1996 .

[17]  Zhangxin Chen,et al.  A new local stabilized nonconforming finite element method for the Stokes equations , 2008, Computing.

[18]  Michel Fortin,et al.  A minimal stabilisation procedure for mixed finite element methods , 2000, Numerische Mathematik.