Dimensions of attractors for discretizations for Navier-Stokes equations

In this paper, we discretize the 2-D incompressible Navier-Stokes equations with the periodic boundary condition by the finite difference method. We prove that with a shift for discretization, the global solutions exist. After proving some discrete Sobolev inequalities in the sense of finite differences, we prove the existence of the global attractors of the discretized system, and we estimate the upper bounds for the Hausdorff and the fractal dimensions of the attractors. These bounds are indepent of the mesh sizes and are considerably close to those of the continuous version.