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Regularity of solutions and the convergence of the Galerkin method in the Ginzburg-Landau equation. (English) Zbl 0792.35096
Summary: 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 \(H^ 1_{\text{per}}((0,1)^ D)\) 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.

MSC:
35K60 Nonlinear initial, boundary and initial-boundary value problems for linear parabolic equations
35K55 Nonlinear parabolic equations
35A35 Theoretical approximation in context of PDEs
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