American Institute of Mathematical Sciences

Advanced
  • Previous Article
    On the structure of the permanence region for competing species models with general diffusivities and transport effects
  • DCDS Home
  • This Issue
  • Next Article
    The asymptotic behavior of solutions of a semilinear parabolic equation
October  1996, 2(4): 497-524. doi: 10.3934/dcds.1996.2.497

Fourier nonlinear Galerkin method for Navier-Stokes equations

Kaitai Li 1, and  Yanren Hou 1,

1. 

Research Center for Applied Mathematics, Xi'an Jiaotong University, Xi'an, 710049, China

Received  October 1995 Revised  April 1996 Published  July 1996

Nonlinear Galerkin Methods (NGMs) are numerical schemes for evolutionary partial differential equations based on the theory of Inertial Manifolds (IMs) [1] and Approximate Inertial Manifolds (AIMs) [2]. In this paper, we focus our attention on the 2-D Navier- Stokes equations with periodic boundary conditions, and use Fourier methods to study its nonlinear Galerkin approximation which we call Fourier Nonlinear Galerkin Methods (FNGMs) here. The first part is contributed to the semidiscrete case. In this part, we derive the well-posedness of the nonlinear Galerkin form and the distance between the nonlinear Galerkin approximation and the genuine solution in Sobolev spaces of any orders. The second part is concerned about the full discrete case, in which, for a given numerical scheme based on NGMs, we investigate the stability and error estimate respectively. We derive the stability conditions for the scheme in any fractional Sobolev spaces. Finally, we give its error estimate in $H^r$ for any $r\geq 0$.
Keywords: Galerkin method, Navier-Stokes equations..
Mathematics Subject Classification: 35Q3.
Citation: Kaitai Li, Yanren Hou. Fourier nonlinear Galerkin method for Navier-Stokes equations. Discrete & Continuous Dynamical Systems, 1996, 2 (4) : 497-524. doi: 10.3934/dcds.1996.2.497
[1]

Yinnian He, R. M.M. Mattheij. Reformed post-processing Galerkin method for the Navier-Stokes equations. Discrete & Continuous Dynamical Systems - B, 2007, 8 (2) : 369-387. doi: 10.3934/dcdsb.2007.8.369
[2]

Evrad M. D. Ngom, Abdou Sène, Daniel Y. Le Roux. Boundary stabilization of the Navier-Stokes equations with feedback controller via a Galerkin method. Evolution Equations & Control Theory, 2014, 3 (1) : 147-166. doi: 10.3934/eect.2014.3.147
[3]

Hi Jun Choe, Hyea Hyun Kim, Do Wan Kim, Yongsik Kim. Meshless method for the stationary incompressible Navier-Stokes equations. Discrete & Continuous Dynamical Systems - B, 2001, 1 (4) : 495-526. doi: 10.3934/dcdsb.2001.1.495
[4]

Hi Jun Choe, Do Wan Kim, Yongsik Kim. Meshfree method for the non-stationary incompressible Navier-Stokes equations. Discrete & Continuous Dynamical Systems - B, 2006, 6 (1) : 17-39. doi: 10.3934/dcdsb.2006.6.17
[5]

Takayuki Kubo, Ranmaru Matsui. On pressure stabilization method for nonstationary Navier-Stokes equations. Communications on Pure & Applied Analysis, 2018, 17 (6) : 2283-2307. doi: 10.3934/cpaa.2018109
[6]

Pavel I. Plotnikov, Jan Sokolowski. Compressible Navier-Stokes equations. Conference Publications, 2009, 2009 (Special) : 602-611. doi: 10.3934/proc.2009.2009.602
[7]

Jan W. Cholewa, Tomasz Dlotko. Fractional Navier-Stokes equations. Discrete & Continuous Dynamical Systems - B, 2018, 23 (8) : 2967-2988. doi: 10.3934/dcdsb.2017149
[8]

Yinnian He, Kaitai Li. Nonlinear Galerkin approximation of the two dimensional exterior Navier-Stokes problem. Discrete & Continuous Dynamical Systems, 1996, 2 (4) : 467-482. doi: 10.3934/dcds.1996.2.467
[9]

Hui Peng, Qilong Zhai. Weak Galerkin method for the Stokes equations with damping. Discrete & Continuous Dynamical Systems - B, 2021  doi: 10.3934/dcdsb.2021112
[10]

Ben-Yu Guo, Yu-Jian Jiao. Mixed generalized Laguerre-Fourier spectral method for exterior problem of Navier-Stokes equations. Discrete & Continuous Dynamical Systems - B, 2009, 11 (2) : 315-345. doi: 10.3934/dcdsb.2009.11.315
[11]

Yueqiang Shang, Qihui Zhang. A subgrid stabilizing postprocessed mixed finite element method for the time-dependent Navier-Stokes equations. Discrete & Continuous Dynamical Systems - B, 2021, 26 (6) : 3119-3142. doi: 10.3934/dcdsb.2020222
[12]

Hermenegildo Borges de Oliveira. Anisotropically diffused and damped Navier-Stokes equations. Conference Publications, 2015, 2015 (special) : 349-358. doi: 10.3934/proc.2015.0349
[13]

Hyukjin Kwean. Kwak transformation and Navier-Stokes equations. Communications on Pure & Applied Analysis, 2004, 3 (3) : 433-446. doi: 10.3934/cpaa.2004.3.433
[14]

Vittorino Pata. On the regularity of solutions to the Navier-Stokes equations. Communications on Pure & Applied Analysis, 2012, 11 (2) : 747-761. doi: 10.3934/cpaa.2012.11.747
[15]

C. Foias, M. S Jolly, I. Kukavica, E. S. Titi. The Lorenz equation as a metaphor for the Navier-Stokes equations. Discrete & Continuous Dynamical Systems, 2001, 7 (2) : 403-429. doi: 10.3934/dcds.2001.7.403
[16]

Igor Kukavica. On regularity for the Navier-Stokes equations in Morrey spaces. Discrete & Continuous Dynamical Systems, 2010, 26 (4) : 1319-1328. doi: 10.3934/dcds.2010.26.1319
[17]

Igor Kukavica. On partial regularity for the Navier-Stokes equations. Discrete & Continuous Dynamical Systems, 2008, 21 (3) : 717-728. doi: 10.3934/dcds.2008.21.717
[18]

Susan Friedlander, Nataša Pavlović. Remarks concerning modified Navier-Stokes equations. Discrete & Continuous Dynamical Systems, 2004, 10 (1&2) : 269-288. doi: 10.3934/dcds.2004.10.269
[19]

Jean-Pierre Raymond. Stokes and Navier-Stokes equations with a nonhomogeneous divergence condition. Discrete & Continuous Dynamical Systems - B, 2010, 14 (4) : 1537-1564. doi: 10.3934/dcdsb.2010.14.1537
[20]

Yoshikazu Giga. A remark on a Liouville problem with boundary for the Stokes and the Navier-Stokes equations. Discrete & Continuous Dynamical Systems - S, 2013, 6 (5) : 1277-1289. doi: 10.3934/dcdss.2013.6.1277

2020 Impact Factor: 1.392

Tools

Metrics

  • PDF downloads (47)
  • HTML views (0)
  • Cited by (2)

Other articles
by authors

[Back to Top]

Copyright © 2021 American Institute of Mathematical Sciences