Summary.
We propose and analyze a semi-discrete and a fully discrete mixed finite element method for the Cahn-Hilliard equation u t + Δ(ɛΔu−ɛ−1f(u)) = 0, where ɛ > 0 is a small parameter. Error estimates which are quasi-optimal order in time and optimal order in space are shown for the proposed methods under minimum regularity assumptions on the initial data and the domain. In particular, it is shown that all error bounds depend on only in some lower polynomial order for small ɛ. The cruxes of our analysis are to establish stability estimates for the discrete solutions, to use a spectrum estimate result of Alikakos and Fusco [2], and Chen [15] to prove a discrete counterpart of it for a linearized Cahn-Hilliard operator to handle the nonlinear term on a stretched time grid. The ideas and techniques developed in this paper also enable us to prove convergence of the fully discrete finite element solution to the solution of the Hele-Shaw (Mullins-Sekerka) problem as ɛ → 0 in [29].
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Mathematics Subject Classification (1991): 65M60, 65M12, 65M15, 35B25, 35K57, 35Q99, 53A10
Acknowledgments. The first author would like to thank Nicholas Alikakos for explaining all the fascinating properties of the Allen-Cahn and Cahn-Hilliard equations to him. He would also like to thank Nicholas Alikakos and Xinfu Chen for answering his questions regarding the spectrum estimate in Proposition 1. The second author gratefully acknowledges financial support by the DFG.
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Feng, X., Prohl, A. Error analysis of a mixed finite element method for the Cahn-Hilliard equation. Numer. Math. 99, 47–84 (2004). https://doi.org/10.1007/s00211-004-0546-5
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DOI: https://doi.org/10.1007/s00211-004-0546-5