Error estimates for fully discrete BDF finite element approximations of the Allen–Cahn equation

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, draa065, https://doi.org/10.1093/imanum/draa065
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
Appears in Collections:	Journal/Magazine Article

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