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Title: Error estimates for fully discrete BDF finite element approximations of the Allen–Cahn equation
Authors: Akrivis, G
Li, B 
Issue Date: 2020
Source: IMA journal of numerical analysis, Jan. 2022, v. 42, no. 1, p. 363-391
Abstract: For a class of compatible profiles of initial data describing bulk phase regions separated by transition zones, we approximate the Cauchy problem of the Allen–Cahn (AC) phase field equation by an initial-boundary value problem in a bounded domain with the Dirichlet boundary condition. The initial-boundary value problem is discretized in time by the backward difference formulae (BDF) of order 1⩽q⩽5 and in space by the Galerkin finite element method of polynomial degree r−1⁠, with r⩾2⁠. We establish an error estimate of O(τqε−q−12+hrε−r−12+e−c/ε) with explicit dependence on the small parameter ε describing the thickness of the phase transition layer. The analysis utilizes the maximum-norm stability of BDF and finite element methods with respect to the boundary data, the discrete maximal Lp-regularity of BDF methods for parabolic equations and the Nevanlinna–Odeh multiplier technique combined with a time-dependent inner product motivated by a spectrum estimate of the linearized AC operator.
Publisher: Oxford University Press
Journal: IMA journal of numerical analysis 
ISSN: 0272-4979
EISSN: 1464-3642
DOI: 10.1093/imanum/draa065
Rights: © The Author(s) 2020. Published by Oxford University Press on behalf of the Institute of Mathematics and its Applications. All rights reserved.
This is a pre-copyedited, author-produced PDF of an article accepted for publication in IMA Journal of Numerical Analysis following peer review. The version of record Georgios Akrivis, Buyang Li, Error estimates for fully discrete BDF finite element approximations of the Allen–Cahn equation, IMA Journal of Numerical Analysis, Volume 42, Issue 1, January 2022, Pages 363–391 is available online at:
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