数学物理学报(英文版)

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A NEW APPROXIMATE INERTIAL MANIFOLD AND ASSOCIATED ALGORITHM

李开泰; 徐忠锋; 杨晓忠   

  1. 西安交通大学数学系, 西安 710049
  • 收稿日期:2003-01-02 修回日期:2004-02-01 出版日期:2006-01-20 发布日期:2006-01-20
  • 通讯作者: 李开泰
  • 基金资助:

    Subsidized by the Special Funds for Major State Basic Research Projects (G1999 03 2801) and NFS of China (10001028 and 40375010).

A NEW APPROXIMATE INERTIAL MANIFOLD AND ASSOCIATED ALGORITHM

Li Kaitai; Xu Zhongfeng; Yang Xiaozhong   

  1. Department of Mathematics, Xi'an Jiaotong University, Xi'an 710049, China
  • Received:2003-01-02 Revised:2004-02-01 Online:2006-01-20 Published:2006-01-20
  • Contact: Li Kaitai

摘要:

In this article the authors propose a new approximate inertial manifold(AIM) to the Navier-Stokes equations. The solutions are in the neighborhoods of this AIM with thickness δ=o(h2k+1-ε). The article aims to investigate a two grids finite element approximation based on it and give error estimates of the approximate solution
||(u - u*h* , p - p*h*)|| ≤ C(h2k+1-ε + h*(m+1)),
where (h,h*) and (k,m) are coarse and fine meshes and degree of finite element subspaces, respectively. These results are much better than Standard Galerkin(SG) and nonlinear Galerkin (NG) methods. For example, for 2D NS eqs and linear element, let uh, uh, u* be the SG, NG and their approximate solutions respectively, then |u - uh|1≤ Ch, |u - uh|1 ≤ h2, |u -u*|1 ≤ Ch3, and h*≈ h2 for NG, h* h3/2 for theirs.

关键词: Two level finite element, Navier-Stokes equations, new approximation inertial manifold

Abstract:

In this article the authors propose a new approximate inertial manifold(AIM) to the Navier-Stokes equations. The solutions are in the neighborhoods of this AIM with thickness δ=o(h2k+1-ε). The article aims to investigate a two grids finite element approximation based on it and give error estimates of the approximate solution
||(u - u*h* , p - p*h*)|| ≤ C(h2k+1-ε + h*(m+1)),
where (h,h*) and (k,m) are coarse and fine meshes and degree of finite element subspaces, respectively. These results are much better than Standard Galerkin(SG) and nonlinear Galerkin (NG) methods. For example, for 2D NS eqs and linear element, let uh, uh, u* be the SG, NG and their approximate solutions respectively, then |u - uh|1≤ Ch, |u - uh|1 ≤ h2, |u -u*|1 ≤ Ch3, and h*≈ h2 for NG, h* h3/2 for theirs.

Key words: Two level finite element, Navier-Stokes equations, new approximation inertial manifold

中图分类号: 

  • 65N30