An efficient two-step algorithm for the incompressible flow problem

A new two-step stabilized finite element method for the 2D/3D stationary Navier–Stokes equations based on local Gauss integration is introduced and analyzed in this paper. The method consists of solving one Navier–Stokes problem based on the P1−P1 finite element pair and then solving a general Stokes problem based on the P2−P2 finite element pair, i.e., computes a lower order predictor and a higher order corrector. Moreover, the stability and convergence of the present method are deduced, which show that the new method provides an approximate solution with the convergence rate of the same order as the P2−P2 stabilized finite element solution solving the Navier–Stokes equations on the same mesh width. However, our method can save a large amount of computational time. Finally, numerical tests confirm the theoretical results of the method.

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

[2]  R. Rannacher,et al.  Finite-element approximations of the nonstationary Navier-Stokes problem. Part IV: error estimates for second-order time discretization , 1990 .

[3]  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 .

[4]  P. Hansbo,et al.  A simple pressure stabilization method for the Stokes equation , 2007 .

[5]  William Layton,et al.  Two-level Picard and modified Picard methods for the Navier-Stokes equations , 1995 .

[6]  Yinnian He,et al.  A simplified two-level method for the steady Navier–Stokes equations , 2008 .

[7]  Li Jian,et al.  TWO-LEVEL METHODS BASED ON THREE CORRECTIONS FOR THE 2D/3D STEADY NAVIER-STOKES EQUATIONS , 2011 .

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

[9]  Yinnian He,et al.  Convergence of three iterative methods based on the finite element discretization for the stationary Navier–Stokes equations☆ , 2009 .

[10]  Jian Li,et al.  Investigations on two kinds of two-level stabilized finite element methods for the stationary Navier-Stokes equations , 2006, Appl. Math. Comput..

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

[12]  M. Marion,et al.  Nonlinear Galerkin methods and mixed finite elements: two-grid algorithms for the Navier-Stokes equations , 1994 .

[13]  Jinchao Xu.  Two-grid Discretization Techniques for Linear and Nonlinear PDEs , 1996 .

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

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

[16]  Pavel B. Bochev,et al.  An Absolutely Stable Pressure-Poisson Stabilized Finite Element Method for the Stokes Equations , 2004, SIAM J. Numer. Anal..

[17]  Yinnian He,et al.  Two-level Stabilized Finite Element Methods for the Steady Navier–Stokes Problem , 2005, Computing.

[18]  D. Gartling.  A test problem for outflow boundary conditions—flow over a backward-facing step , 1990 .

[19]  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..

[20]  William Layton,et al.  A posteriori error estimators for a two-level finite element method for the Navier-Stokes equations , 1996 .

[21]  Jinchao Xu,et al.  A Novel Two-Grid Method for Semilinear Elliptic Equations , 1994, SIAM J. Sci. Comput..

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

[23]  Yinnian He,et al.  Stabilized finite-element method for the stationary Navier-Stokes equations , 2005 .

[24]  Stig Larsson,et al.  The long-time behavior of finite-element approximations of solutions of semilinear parabolic problems , 1989 .

[25]  Vivette Girault,et al.  Finite Element Methods for Navier-Stokes Equations - Theory and Algorithms , 1986, Springer Series in Computational Mathematics.

[26]  Li Shan,et al.  A quadratic equal‐order stabilized method for Stokes problem based on two local Gauss integrations , 2010 .

[27]  Yinnian He,et al.  A two‐level finite element method for the stationary Navier‐Stokes equations based on a stabilized local projection , 2011 .

[28]  Lutz Tobiska,et al.  A Two-Level Method with Backtracking for the Navier--Stokes Equations , 1998 .

[29]  Zhangxin Chen,et al.  A new stabilized finite volume method for the stationary Stokes equations , 2009, Adv. Comput. Math..

[30]  William Layton,et al.  A two-level discretization method for the Navier-Stokes equations , 1993 .

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

[32]  Feng Shi,et al.  A finite element variational multiscale method for incompressible flows based on two local gauss integrations , 2009, J. Comput. Phys..

[33]  Arif Masud,et al.  A multiscale finite element method for the incompressible Navier-Stokes equations , 2006 .