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.