REGULARITY OF SOLUTIONS AND THE CONVERGENCE OF THE GALERKIN METHOD IN THE GINZBURG-LANDAU EQUATION

In this paper an analytical explanation is given for two phenomena observed in numerical simulations of the Ginzburg-Landau equation on the domain [0, 1]D (D = 1, 2, 3) with periodic boundary conditions. First, it is shown that the solutions with H(per)1((0, 1)D) initial data become analytic (in the spatial variable). This behavior accounts for the numerically observed exponential decay of the Fourier-modes. Then, based on the regularity result, it is shown that the (linear) Galerkin method has an exponential rate of convergence. This gives an explanation of simulations which show that the Ginzburg-Landau equation can be approximated by very low dimensional Galerkin projections. Furthermore, we discuss the influence of the parameters in the Ginzburg-Landau equation on the decay rate of the Fourier-modes and on the rate of convergence of the Galerkin approximations.