Robust a posteriori error estimates for finite element discretizations of the heat equation with discontinuous coefficients

Stefano Berrone

ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique (2006)

  • Volume: 40, Issue: 6, page 991-1021
  • ISSN: 0764-583X

Abstract

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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 θ -scheme with 1 / 2 ≤ θ ≤ 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.

How to cite

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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 40.6 (2006): 991-1021. <http://eudml.org/doc/244686>.

@article{Berrone2006,
abstract = {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.},
author = {Berrone, Stefano},
journal = {ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique},
keywords = {a posteriori error estimates; parabolic problems; discontinuous coefficients; finite elements; heat equation},
language = {eng},
number = {6},
pages = {991-1021},
publisher = {EDP-Sciences},
title = {Robust a posteriori error estimates for finite element discretizations of the heat equation with discontinuous coefficients},
url = {http://eudml.org/doc/244686},
volume = {40},
year = {2006},
}

TY - JOUR
AU - Berrone, Stefano
TI - Robust a posteriori error estimates for finite element discretizations of the heat equation with discontinuous coefficients
JO - ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique
PY - 2006
PB - EDP-Sciences
VL - 40
IS - 6
SP - 991
EP - 1021
AB - 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.
LA - eng
KW - a posteriori error estimates; parabolic problems; discontinuous coefficients; finite elements; heat equation
UR - http://eudml.org/doc/244686
ER -

References

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