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Error estimates with smooth and nonsmooth data for a finite element method for the Cahn-Hilliard equation


Authors: Charles M. Elliott and Stig Larsson
Journal: Math. Comp. 58 (1992), 603-630, S33
MSC: Primary 65M60; Secondary 65M15
MathSciNet review: 1122067
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Abstract: A finite element method for the Cahn-Hilliard equation (a semilinear parabolic equation of fourth order) is analyzed, both in a spatially semidiscrete case and in a completely discrete case based on the backward Euler method. Error bounds of optimal order over a finite time interval are obtained for solutions with smooth and nonsmooth initial data. A detailed study of the regularity of the exact solution is included. The analysis is based on local Lipschitz conditions for the nonlinearity with respect to Sobolev norms, and the existence of a Ljapunov functional for the exact and the discretized equations is essential. A result concerning the convergence of the attractor of the corresponding approximate nonlinear semigroup (upper semicontinuity with respect to the discretization parameters) is obtained as a simple application of the nonsmooth data error estimate.


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DOI: https://doi.org/10.1090/S0025-5718-1992-1122067-1
Keywords: Cahn-Hilliard equation, nonlinear, semigroup, smoothing property, finite element, backward Euler method, error estimate, nonsmooth data, upper semicontinuity, attractor
Article copyright: © Copyright 1992 American Mathematical Society