In this paper, three general principles for constructing approximate inertial manifolds are provided, under which the associate approximate inertial form of origin problem, which is a finite dimensional ordinary differential equation, is well-possed and its solution will approximate the genuine solution at some degree. At last, for some kinds of approximate inertial manifolds and a family of approximate inertial manifolds, we indicate that the principles given here are suitable.
A optimum ffote element nonlinear Galerkin algorithm is presented for thetwo-dimensional nonstationary Navier-Stokes equations. The standard finite elemellt Galerkin algorithIn consists in solving a nonlinear equation on the fine gridfinite elemellt space Xh’ The optimum finite element nonlinear Galerkin algorithm consists in solving a nonlinear subproblem on a coarse grid finite elementspace XH(H > h) and solving a linear subproblem on a fine grid incremental finite element space Wh = (I - RH)Xh- If H is chosen such that H = O(h1/2),then two algorithms are of the c0nvergence rate of same order. However, sinceH >> h, dimXH << dimXh, the optimum finite elemellt nonlinear Galerkin algorithm can save a large amoullt of comPutational time. Finally, we give thenumerical test which shows the correctness of theoretical analysis.
