We consider energy norm a posteriori error analysis of conforming finite element approximations of singularly perturbed reaction–diffusion problems on simplicial meshes in arbitrary space dimension. Using an equilibrated flux reconstruction, the proposed estimator gives a guaranteed global upper bound on the error without unknown constants, and local efficiency robust with respect to the mesh size and singular perturbation parameters. Whereas previous works on equilibrated flux estimators only considered lowest-order finite element approximations and achieved robustness through the use of boundary-layer adapted submeshes or via combination with residual-based estimators, the present methodology applies in a simple way to arbitrary-order approximations and does not request any submesh or estimators combination. The equilibrated flux is obtained via local reaction–diffusion problems with suitable weights (cut-off factors), and the guaranteed upper bound features the same weights. We prove that the inclusion of these weights is not only sufficient but also necessary for robustness of any flux equilibration estimate that does not employ submeshes or estimators combination, which shows that some of the flux equilibrations proposed in the past cannot be robust. To achieve the fully computable upper bound, we derive explicit bounds for some inverse inequality constants on a simplex, which may be of independent interest.
Mots-clés : Singular perturbation, $$ error analysis, local efficiency, robustness, equilibrated flux
@article{M2AN_2020__54_6_1951_0,
author = {Smears, Iain and Vohral{\'\i}k, Martin},
title = {Simple and robust equilibrated flux \protect\emph{a posteriori} estimates for singularly perturbed reaction{\textendash}diffusion problems},
journal = {ESAIM: Mathematical Modelling and Numerical Analysis },
pages = {1951--1973},
publisher = {EDP-Sciences},
volume = {54},
number = {6},
year = {2020},
doi = {10.1051/m2an/2020034},
mrnumber = {4160328},
language = {en},
url = {http://www.numdam.org/articles/10.1051/m2an/2020034/}
}
TY - JOUR AU - Smears, Iain AU - Vohralík, Martin TI - Simple and robust equilibrated flux a posteriori estimates for singularly perturbed reaction–diffusion problems JO - ESAIM: Mathematical Modelling and Numerical Analysis PY - 2020 SP - 1951 EP - 1973 VL - 54 IS - 6 PB - EDP-Sciences UR - http://www.numdam.org/articles/10.1051/m2an/2020034/ DO - 10.1051/m2an/2020034 LA - en ID - M2AN_2020__54_6_1951_0 ER -
%0 Journal Article %A Smears, Iain %A Vohralík, Martin %T Simple and robust equilibrated flux a posteriori estimates for singularly perturbed reaction–diffusion problems %J ESAIM: Mathematical Modelling and Numerical Analysis %D 2020 %P 1951-1973 %V 54 %N 6 %I EDP-Sciences %U http://www.numdam.org/articles/10.1051/m2an/2020034/ %R 10.1051/m2an/2020034 %G en %F M2AN_2020__54_6_1951_0
Smears, Iain; Vohralík, Martin. Simple and robust equilibrated flux a posteriori estimates for singularly perturbed reaction–diffusion problems. ESAIM: Mathematical Modelling and Numerical Analysis , Tome 54 (2020) no. 6, pp. 1951-1973. doi : 10.1051/m2an/2020034. http://www.numdam.org/articles/10.1051/m2an/2020034/
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