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 - u^*_h* , p - p^*_h*)|| ≤ C(h^2k+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 u_h, u^h, u* be the SG, NG and their approximate solutions respectively, then |u - u_h|₁≤ Ch, |u - u^h|₁ ≤ h², |u -u^*|₁ ≤ Ch³, and h^*≈ h² for NG, h^*_≈ h^3/2 for theirs.

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