Robust a posteriori error estimates for finite element discretizations of the heat equation with discontinuous coefficients
ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique, Volume 40 (2006) no. 6, p. 991-1021

In this work we derive a posteriori error estimates based on equations residuals for the heat equation with discontinuous diffusivity coefficients. The estimates are based on a fully discrete scheme based on conforming finite elements in each time slab and on the A-stable $\theta$-scheme with $1/2\le \theta \le 1$. Following remarks of [Picasso, Comput. Methods Appl. Mech. Engrg. 167 (1998) 223-237; Verfürth, Calcolo 40 (2003) 195-212] it is easy to identify a time-discretization error-estimator and a space-discretization error-estimator. In this work we introduce a similar splitting for the data-approximation error in time and in space. Assuming the quasi-monotonicity condition [Dryja et al., Numer. Math. 72 (1996) 313-348; Petzoldt, Adv. Comput. Math. 16 (2002) 47-75] we have upper and lower bounds whose ratio is independent of any meshsize, timestep, problem parameter and its jumps.

DOI : https://doi.org/10.1051/m2an:2006034
Classification:  65M60,  65M15,  65M50
Keywords: a posteriori error estimates, parabolic problems, discontinuous coefficients
@article{M2AN_2006__40_6_991_0,
author = {Berrone, Stefano},
title = {Robust a posteriori error estimates for finite element discretizations of the heat equation with discontinuous coefficients},
journal = {ESAIM: Mathematical Modelling and Numerical Analysis - Mod\'elisation Math\'ematique et Analyse Num\'erique},
publisher = {EDP-Sciences},
volume = {40},
number = {6},
year = {2006},
pages = {991-1021},
doi = {10.1051/m2an:2006034},
zbl = {1121.65098},
mrnumber = {2297102},
language = {en},
url = {http://www.numdam.org/item/M2AN_2006__40_6_991_0}
}

Berrone, Stefano. Robust a posteriori error estimates for finite element discretizations of the heat equation with discontinuous coefficients. ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique, Volume 40 (2006) no. 6, pp. 991-1021. doi : 10.1051/m2an:2006034. http://www.numdam.org/item/M2AN_2006__40_6_991_0/

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