A NEW APPROXIMATE INERTIAL MANIFOLD AND ASSOCIATED ALGORITHM
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...
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| Published in | Acta Mathematica Scientia Vol. 26; no. 1; pp. 1 - 16 |
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| Main Author | |
| 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
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| Subjects | |
| Online Access | Get full text |
| ISSN | 0252-9602 1572-9087 1003-3998 |
| DOI | 10.1016/S0252-9602(06)60021-0 |
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| 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（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. |
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| AbstractList | O1; 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 ≤ Ch2, ‖u- u*‖1 ≤ Ch3,and h* ≈ h2 for NG, h* ≈ h3/2 for theirs. 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. 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 delta = o(h2k+1-E). The article aims to investigate a two grids finite element approximation based on it and give error estimates of the approximate solution. |
| Author | 李开泰 徐忠锋 杨晓忠 |
| AuthorAffiliation | Department of Mathematics, Xi＇an Jiaotong University, Xi＇an 710049, China Department of Mathematics, North China Electric Power University, Zhengzhou 450055, China |
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| Cites_doi | 10.1137/0732054 10.1137/0726063 10.1002/(SICI)1098-2426(199605)12:3<333::AID-NUM4>3.0.CO;2-P 10.1137/0915016 |
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| Keywords | new approximation inertial manifold Two level finite element Navier-Stokes equations |
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| References | Lin (10.1016/S0252-9602(06)60021-0_bib1) 1997 Garcia-Archila (10.1016/S0252-9602(06)60021-0_bib3) 2000; 97 Girault (10.1016/S0252-9602(06)60021-0_bib5) 1986 Xu (10.1016/S0252-9602(06)60021-0_bib8) 1994; 15 Gunzburger (10.1016/S0252-9602(06)60021-0_bib6) 1989 Layton (10.1016/S0252-9602(06)60021-0_bib10) 1995; 80 Bezzi (10.1016/S0252-9602(06)60021-0_bib9) 1991 Ervin (10.1016/S0252-9602(06)60021-0_bib11) 1996; 12 Garcia-Achilla (10.1016/S0252-9602(06)60021-0_bib2) 1999; 68 Morion (10.1016/S0252-9602(06)60021-0_bib12) 1995; 32 Marion (10.1016/S0252-9602(06)60021-0_bib7) 1989; 26 Constantine (10.1016/S0252-9602(06)60021-0_bib4) 1988 |
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| Snippet | 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... 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... O1; 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... |
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| SubjectTerms | Navier-Stokes方程 惯性流形 有限元分析 联合算法 |
| Title | A NEW APPROXIMATE INERTIAL MANIFOLD AND ASSOCIATED ALGORITHM |
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