Robust local problem error estimation for a singularly perturbed problem on anisotropic finite element meshes

Gerd Kunert

ESAIM: Mathematical Modelling and Numerical Analysis (2010)

  • Volume: 35, Issue: 6, page 1079-1109
  • ISSN: 0764-583X

Abstract

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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.

How to cite

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Kunert, Gerd. "Robust local problem error estimation for a singularly perturbed problem on anisotropic finite element meshes." ESAIM: Mathematical Modelling and Numerical Analysis 35.6 (2010): 1079-1109. <http://eudml.org/doc/197504>.

@article{Kunert2010,
abstract = { 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. },
author = {Kunert, Gerd},
journal = {ESAIM: Mathematical Modelling and Numerical Analysis},
keywords = {Error estimator; anisotropic solution; stretched elements; reaction diffusion equation; singularly perturbed problem.; error estimator; singular perturbation; error bounds; numerical example},
language = {eng},
month = {3},
number = {6},
pages = {1079-1109},
publisher = {EDP Sciences},
title = {Robust local problem error estimation for a singularly perturbed problem on anisotropic finite element meshes},
url = {http://eudml.org/doc/197504},
volume = {35},
year = {2010},
}

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
DA - 2010/3//
PB - EDP Sciences
VL - 35
IS - 6
SP - 1079
EP - 1109
AB - 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.
LA - eng
KW - Error estimator; anisotropic solution; stretched elements; reaction diffusion equation; singularly perturbed problem.; error estimator; singular perturbation; error bounds; numerical example
UR - http://eudml.org/doc/197504
ER -

References

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