The two-grid finite element approximation to the stream function form of the stationary Navier-Stokes equations was analyzed. The algorithms involve solving one small, nonlinear coarse mesh system, one linear problem on the fine mesh system, and a linear correct problem on the coarse mesh. The algorithms with the correct problem and without the correct problem were discussed. The algorithms produce an approximate solution with the optimal, asymptotic accuracy for any fixed Reynolds number.

研究热传导方程在一类非局部时间的边界条件下非线性热源的反演问题，文中应用Sobolev紧性方法证明了热源在Hōlder空间中的（关于时间的）局部存在性。

讨论了一类耗散发展方程时间离散化之后惯性流形的存在性。证明了如果时间步长充分小，且主算子A满足一个谱间隔条件，则所讨论的发展方程的一个简单差分格式存在一个惯性流形Mh．与Demengel F。的研究结果相比较，不仅简化了存在性证明，而且所建立的理论适用于更一般的非线性项。更进一步。在所建立的框架下，可以构造一个惯性流形序列（MhM），在某种意义下，当M-∞时MhM→Mh当h趋于零时Mh的收敛性以及理论应用将在第Ⅱ部分讨论。

The paper first deals with the existence of the maximal attractor of classical solution for reaction diffusion equation with dispersion, and gives the sup-norln estimate for the attractor. This estimate is optimal for the attractor under Neumann boundary condition. Next, the same problem is discussed for reaction diffusion system with uniformly contracting rectangle, and it reveals that the maximal attractor of classical solution for such system in the whole space is only necessary to be established in some invariant region. Finally, a few examples of application are given.

A two-grid method for the steady penalized incomprcssiblc Navicr-Stokes equations is presented. Convergence results arc proved. If h=O(h3-s) and ∈=O(H5-2s) (s=0 (n=2); s=1/2 (n=3)) arc chosen, the convcrgcnce order of this two-grid method is the same as that of the usual finite element method. Numerical results show that this method is efficient and can save a lot of computation time.