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A posteriori error estimates for the allen-cahn problem

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journal contribution
posted on 26.11.2020, 16:23 by K Chrysafinos, EH Georgoulis, D Plaka
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.

History

Citation

SIAM Journal on Numerical Analysis, Vol. 58, No. 5, pp. 2662–2683

Author affiliation

School of Mathematics and Actuarial Science

Version

VoR (Version of Record)

Published in

SIAM Journal on Numerical Analysis

Volume

58

Issue

5

Pagination

2662 - 2683

Publisher

Society for Industrial & Applied Mathematics (SIAM)

issn

0036-1429

eissn

1095-7170

Acceptance date

06/07/2020

Copyright date

2020

Available date

26/11/2020

Language

en