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On a posteriori error control for the Allen–Cahn problem
Author(s) -
Georgoulis Emmanuil H.,
Makridakis Charalambos
Publication year - 2013
Publication title -
mathematical methods in the applied sciences
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.719
H-Index - 65
eISSN - 1099-1476
pISSN - 0170-4214
DOI - 10.1002/mma.2894
Subject(s) - mathematics , allen–cahn equation , a priori and a posteriori , partial differential equation , optimal control , finite element method , polynomial , simplicity , mathematical analysis , mathematical optimization , philosophy , physics , epistemology , thermodynamics
We derive a posteriori bounds in non‐Hilbertian norms for finite element approximations to the Allen–Cahn phase field model. The derivation of the bounds relies crucially on an approximate spectral estimate for the linearised steady part of the partial differential equation, yielding only polynomial dependence on the phase separation length. A second key ingredient in the proof of the a posteriori bounds is the use of the elliptic reconstruction framework, enabling the control of the error in the L ∞ ( L r )‐norms, r ∈ [2, ∞ ]. For simplicity of the presentation, only the spatially discrete method is considered here. Copyright © 2013 John Wiley & Sons, Ltd.