Abstract
In this paper we investigate the long time stability of the implicit Euler scheme for the Cahn-Hilliard equation with polynomial nonlinearity. The uniform estimates in H−1 and Hαs (s=1,2,3) spaces independent of time discrete step-sizes are derived for the numerical solution produced by this classical scheme with variable time step-sizes. The uniform Hα3 bound is obtained on basis of the uniform H1 estimate for the discrete chemical potential which is derived with the aid of the uniform discrete Gronwall lemma. A comparison with the estimates for the continuous-in-time dynamical system reveals that the classical implicit Euler method can completely preserve the long time behaviour of the underlying system. Such a long time behaviour is also demonstrated by the numerical experiments with the help of Fourier pseudospectral space approximation.
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