A Posteriori Error Estimates for the Allen--Cahn Problem
Abstract
This work is concerned with the proof of a posteriori error estimates for fully discrete Galerkin approximations of the Allen--Cahn equation in two and three spatial dimensions. The numerical method comprises the backward Euler method combined with conforming finite elements in space. For this method, we prove conditional type a posteriori error estimates in the $L^{}_4(0,T;L^{}_4(\Omega))$-norm that depend polynomially upon the inverse of the interface length $\epsilon$. The derivation relies crucially on the availability of a spectral estimate for the linearized Allen--Cahn operator about the approximating solution in conjunction with a continuation argument and a variant of the elliptic reconstruction. The new analysis also appears to improve variants of known a posteriori error bounds in $L_2(H^1)$, $L_\infty^{}(L_2^{})$-norms in certain regimes.
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© 2020, Society for Industrial and Applied Mathematics.
History
Submitted: 26 July 2019
Accepted: 6 July 2020
Published online: 23 September 2020
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Hellenic Foundation for Research and Innovation https://doi.org/10.13039/501100013209 : 3270
Funding Information
Hellenic Foundation for Research and Innovation https://doi.org/10.13039/501100013209 : 998
Funding Information
Leverhulme Trust https://doi.org/10.13039/501100000275 : RPG-2015-306
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Stavros Niarchos Foundation https://doi.org/10.13039/501100004343 : ARCHERS
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