A NEW APPROXIMATE INERTIAL MANIFOLD AND ASSOCIATED ALGORITHM

• Published in: Acta Mathematica Scientia Vol. 26; no. 1; pp. 1 - 16
• Main Author: 李开泰 徐忠锋 杨晓忠
• Format: Journal Article
• Language: English
• Published: Department of Mathematics,Xi'an Jiaotong University,Xi'an 710049,China%Department of Mathematics,North China Electric Power University,Zhengzhou 450045,China 2006
• Subjects: Navier-Stokes方程; 惯性流形; 有限元分析; 联合算法; new approximation inertial manifold; Two level finite element; Navier-Stokes equations
• ISSN: 0252-9602; 1572-9087; 1003-3998
• DOI: 10.1016/S0252-9602(06)60021-0

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（h^2k＋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-uh^＊·,p-ph^＊·）｜｜｜≤C（h^2k＋1-ε＋h^＊（m＋1））,where （h, h＊） and （k, m） are co~trse and fine meshes and degree of finite element subspa~es, respectively. These results are much better them Standard G~tlerkin（SG） and nonlinear Galcrkin （NG） methods. For example, for 2D NS eqs and linear element, let uh,u^h, u^＊ be the SG, NG and their approximate solutions respectively, then ｜｜u-uh｜｜1≤Ch,｜｜u-u^h｜｜i≤Ch^2,｜｜u-u^＊｜｜1≤Ch^3,and h^＊ ≈ h^2 for NG, h^＊ ≈ h^3/2 for theirs.