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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
*
snicaise@univ-valenciennes.fr
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Abstract

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

Keywords

finite volume methodselliptic problemsa posteriori error estimates of 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