A stabilised characteristic finite element method for transient Navier–Stokes equations

In this work, we consider a stabilised characteristic finite element method for the time-dependent Navier–Stokes equations based on the lowest equal-order finite element pairs. The diffusion term in these equations is discretised by using finite element method, the temporal differentiation and advection terms are treated by characteristic schemes. Unconditionally stable results and error estimates of optimal order for the velocity and pressure are established. Finally, some numerical results are provided to verify the performance of this method.

[1]  J. Douglas,et al.  Stabilized mixed methods for the Stokes problem , 1988 .

[2]  Yinnian He,et al.  The Euler implicit/explicit scheme for the 2D time-dependent Navier-Stokes equations with smooth or non-smooth initial data , 2008, Math. Comput..

[3]  U. Ghia,et al.  High-Re solutions for incompressible flow using the Navier-Stokes equations and a multigrid method , 1982 .

[4]  Yinnian He,et al.  A stabilized finite element method based on local polynomial pressure projection for the stationary Navier--Stokes equations , 2008 .

[5]  L. Franca,et al.  Stabilization arising from PGEM: A review and further developments , 2009 .

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

[7]  E. Erturk,et al.  Numerical solutions of 2‐D steady incompressible driven cavity flow at high Reynolds numbers , 2004, ArXiv.

[8]  Hirofumi Notsu,et al.  A Single-Step Characteristic-Curve Finite Element Scheme of Second Order in Time for the Incompressible Navier-Stokes Equations , 2009, J. Sci. Comput..

[9]  Yvon Maday,et al.  A high-order characteristics/finite element method for the incompressible Navier-Stokes equations , 1997 .

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

[11]  Béatrice Rivière,et al.  Subgrid Stabilized Defect Correction Methods for the Navier-Stokes Equations , 2006, SIAM J. Numer. Anal..

[12]  Endre Süli,et al.  Convergence and nonlinear stability of the Lagrange-Galerkin method for the Navier-Stokes equations , 1988 .

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

[14]  David J. Silvester,et al.  ANALYSIS OF LOCALLY STABILIZED MIXED FINITE-ELEMENT METHODS FOR THE STOKES PROBLEM , 1992 .

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

[16]  O. Pironneau.  On the transport-diffusion algorithm and its applications to the Navier-Stokes equations , 1982 .

[17]  B. Maury.  Characteristics ALE Method for the Unsteady 3D Navier-Stokes Equations with a Free Surface , 1996 .

[18]  C. Wang.  Characteristic finite analytic method (CFAM) for incompressible Navier-Stokes equations , 2000 .

[19]  Yinnian He,et al.  Stabilized finite element method for the non-stationaryNavier-Stokes problem , 2005 .

[20]  Jim Douglas,et al.  An absolutely stabilized finite element method for the stokes problem , 1989 .

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

[22]  Jinchao Xu,et al.  Some Estimates for a Weighted L 2 Projection , 1991 .

[23]  Brigitte Métivet.  Résolution spectrale des équations de Navier-Stokes par une méthode de sous-domaines courbes , 1987 .

[24]  Jean E. Roberts,et al.  Mixed and hybrid finite element methods , 1987 .

[25]  Rodolfo Bermejo,et al.  Finite element modified method of characteristics for the Navier–Stokes equations , 2000 .

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

[27]  T. F. Russell,et al.  NUMERICAL METHODS FOR CONVECTION-DOMINATED DIFFUSION PROBLEMS BASED ON COMBINING THE METHOD OF CHARACTERISTICS WITH FINITE ELEMENT OR FINITE DIFFERENCE PROCEDURES* , 1982 .

[28]  R. Rannacher,et al.  Finite element approximation of the nonstationary Navier-Stokes problem. I : Regularity of solutions and second-order error estimates for spatial discretization , 1982 .

[29]  Mary F. Wheeler,et al.  Some improved error estimates for the modified method of characteristics , 1989 .

[30]  Yinnian He,et al.  Two-Level Method Based on Finite Element and Crank-Nicolson Extrapolation for the Time-Dependent Navier-Stokes Equations , 2003, SIAM J. Numer. Anal..