Discrete gevrey regularity, attractors and upper-semicontinuity for a finite difference approximation to the Ginzburg-Landau equation

A semi-discrete spatial finite difference approximation to the complex Ginzburg-Landau equation with cubic non-linearity is considered. Using the fractional powers of a sectorial operator, discrete versions of the Sobolev spaces H-s, and Gevrey classes of regularity tau, G(tau), are introduced. Discrete versions of some standard Sobolev space norm inequalities are proved.