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

• Main Author: Grosman, Serguei
• Other Authors: TU Chemnitz, SFB 393
• Format: Others
• Language: English
• Published: Universitätsbibliothek Chemnitz 2006
• Subjects: a posteriori error estimation; singular perturbations; stretched elements; ddc:510; Anisotropie; Finite-Elemente-Methode; Reaktions-Diffusionsgleichung
• Online Access: http://nbn-resolving.de/urn:nbn:de:swb:ch1-200600475; http://nbn-resolving.de/urn:nbn:de:swb:ch1-200600475; http://www.qucosa.de/fileadmin/data/qucosa/documents/5160/data/index.html; http://www.qucosa.de/fileadmin/data/qucosa/documents/5160/data/.htaccess; http://www.qucosa.de/fileadmin/data/qucosa/documents/5160/data/SFBcov.png; http://www.qucosa.de/fileadmin/data/qucosa/documents/5160/data/pdf.gif; http://www.qucosa.de/fileadmin/data/qucosa/documents/5160/data/pre.css; http://www.qucosa.de/fileadmin/data/qucosa/documents/5160/data/ps.gif; http://www.qucosa.de/fileadmin/data/qucosa/documents/5160/data/sfb02-07.pdf; http://www.qucosa.de/fileadmin/data/qucosa/documents/5160/data/sfb02-07.ps.gz; http://www.qucosa.de/fileadmin/data/qucosa/documents/5160/20060047.txt

Singularly perturbed reaction-diffusion problems exhibit in general solutions with anisotropic features, e.g. strong boundary and/or interior layers. This anisotropy is reflected in the discretization by using meshes with anisotropic elements. The quality of the numerical solution rests on the robustness of the a posteriori error estimator with respect to both the perturbation parameters of the problem and the anisotropy of the mesh. An estimator that has shown to be one of the most reliable for reaction-diffusion problem is the <i>equilibrated residual method</i> and its modification done by Ainsworth and Babuška for singularly perturbed problem. However, even the modified method is not robust in the case of anisotropic meshes. The present work modifies the equilibrated residual method for anisotropic meshes. The resulting error estimator is equivalent to the equilibrated residual method in the case of isotropic meshes and is proved to be robust on anisotropic meshes as well. A numerical example confirms the theory.