Fourier spectral approximation to long-time behaviour of the derivative three-dimensional Ginzburg–Landau equation

In this paper, we consider a derivative Ginzburg–Landau equation with periodic initial-value condition in three-dimensional space. A fully discrete Galerkin–Fourier spectral approximation scheme is constructed, and then the dynamical behaviour of the discrete system is analysed. Firstly, the existence of global attractors ANτ of the discrete system are proved by a priori estimate of the discrete solution. Next, the convergence of approximate attractors is proved by error estimates of the discrete solution. Furthermore, the long-time convergence as N→∞N→∞ and τ→0τ→0 simultaneously as well as the numerical long-time stability of the discrete scheme are obtained.