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Robust a-posteriori estimator for advection-diffusion-reaction problems

Author: Giancarlo Sangalli
Journal: Math. Comp. 77 (2008), 41-70
MSC (2000): Primary 65N30, 65G99
Published electronically: May 14, 2007
MathSciNet review: 2353943
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Abstract: We propose an almost-robust residual-based a-posteriori estimator for the advection-diffusion-reaction model problem.

The theory is developed in the one-dimensional setting. The numerical error is measured with respect to a norm which was introduced by the author in 2005 and somehow plays the role that the energy norm has with respect to symmetric and coercive differential operators. In particular, the mentioned norm possesses features that allow us to obtain a meaningful a-posteriori estimator, robust up to a $ \sqrt{\log(Pe)}$ factor, where $ Pe$ is the global Péclet number of the problem. Various numerical tests are performed in one dimension, to confirm the theoretical results and show that the proposed estimator performs better than the usual one known in literature.

We also consider a possible two-dimensional extension of our result and only present a few basic numerical tests, indicating that the estimator seems to preserve the good features of the one-dimensional setting.

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Additional Information

Giancarlo Sangalli
Affiliation: Dipartimento di Matematica, Università di Pavia, Via Ferrata 1, 27100 Pavia, Italy

Received by editor(s): December 6, 2004
Received by editor(s) in revised form: November 29, 2006
Published electronically: May 14, 2007
Additional Notes: The author was supported in part by the PRIN 2004 project of the Italian MIUR
Article copyright: © Copyright 2007 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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