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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
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