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A Posteriori Error Estimates for Finite Volume Approximations

Published online by Cambridge University Press:  27 January 2009

S. Cochez-Dhondt ,
S. Nicaise  and
S. Repin
S. Cochez-Dhondt
Affiliation:
Université de Valenciennes et du Hainaut Cambrésis, LAMAV, FR CNRS 2956, ISTV, F59313 - Valenciennes Cedex 9, France
S. Nicaise*
Affiliation:
Université de Valenciennes et du Hainaut Cambrésis, LAMAV, FR CNRS 2956, ISTV, F59313 - Valenciennes Cedex 9, France
S. Repin
Affiliation:
Steklov Institute of Mathematics in St. Petersburg, Fontanka 27, 191023, St. Petersburg, Russia
*
[email protected]
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Abstract

We present new a posteriori error estimates for the finite volume approximationsof elliptic problems. They are obtained by applying functional a posteriorierror estimates to natural extensions of the approximate solution and its fluxcomputed by the finite volume method. The estimates give guaranteed upper boundsfor the errors in terms of the primal (energy) norm, dual norm (for fluxes), andalso in terms of the combined primal-dual norms. It is shown that the estimatesprovide sharp upper and lower bounds of the error and their practicalcomputation requires solving only finite-dimensional problems.

Keywords

finite volume methodselliptic problemsa posteriori error estimatesof the functional type
Type
Research Article
Information
Mathematical Modelling of Natural Phenomena , Volume 4 , Issue 1: Modelling and numerical methods in contact mechanics , 2009 , pp. 106 - 122
Copyright
© EDP Sciences, 2009

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References

A. Agouzal, F. Oudin. A posteriori error estimator for finite volume methods. C. R. Acad. Sci. Paris, Sér. 1, 343 (2006), 349–354.
S. Repin, S. Sauter, A. Smolianski. Two-Sided a posteriori error estimates for mixed formulations of elliptic problems. Preprint 21-2005, Institute of Mathematics, University of Zurich (to appear in SIAM J. Numer. Anal.).
R. Verfürth. A review of a posteriori error estimation and adaptive mesh–refinement techniques. Wiley, Teubner, New York, 1996.
Vohralík, M.. A posteriori error estimates for finite volume and mixed finite element discretizations of convection-diffusion-reaction equations. ESAIM: Proc., 18 (2007), 57–69. CrossRef