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Singularly perturbed problems often yield solutions with strong directional features, e.g. with boundary layers. Such anisotropic solutions lend themselves to adapted, anisotropic discretizations. The quality of the corresponding numerical solution is a key issue in any computational simulation. To this end we present a new robust error estimator for a singularly perturbed reaction-diffusion problem. In contrast to conventional estimators, our proposal is suitable for anisotropic finite element meshes. The estimator is based on the solution of a local problem, and yields error bounds uniformly in the small perturbation parameter. The error estimation is efficient, i.e. a lower error bound holds. The error estimator is also reliable, i.e. an upper error bound holds, provided that the anisotropic mesh discretizes the problem sufficiently well. A numerical example supports the analysis of our anisotropic error estimator.
@article{M2AN_2001__35_6_1079_0,
author = {Kunert, Gerd},
title = {Robust local problem error estimation for a singularly perturbed problem on anisotropic finite element meshes},
journal = {ESAIM: Mathematical Modelling and Numerical Analysis },
pages = {1079--1109},
publisher = {EDP-Sciences},
volume = {35},
number = {6},
year = {2001},
mrnumber = {1873518},
zbl = {1041.65072},
language = {en},
url = {http://geodesic.mathdoc.fr/item/M2AN_2001__35_6_1079_0/}
}
TY - JOUR AU - Kunert, Gerd TI - Robust local problem error estimation for a singularly perturbed problem on anisotropic finite element meshes JO - ESAIM: Mathematical Modelling and Numerical Analysis PY - 2001 SP - 1079 EP - 1109 VL - 35 IS - 6 PB - EDP-Sciences UR - http://geodesic.mathdoc.fr/item/M2AN_2001__35_6_1079_0/ LA - en ID - M2AN_2001__35_6_1079_0 ER -
%0 Journal Article %A Kunert, Gerd %T Robust local problem error estimation for a singularly perturbed problem on anisotropic finite element meshes %J ESAIM: Mathematical Modelling and Numerical Analysis %D 2001 %P 1079-1109 %V 35 %N 6 %I EDP-Sciences %U http://geodesic.mathdoc.fr/item/M2AN_2001__35_6_1079_0/ %G en %F M2AN_2001__35_6_1079_0
Kunert, Gerd. Robust local problem error estimation for a singularly perturbed problem on anisotropic finite element meshes. ESAIM: Mathematical Modelling and Numerical Analysis , Tome 35 (2001) no. 6, pp. 1079-1109. http://geodesic.mathdoc.fr/item/M2AN_2001__35_6_1079_0/