A Posteriori Error Estimates for Finite Volume Approximations

S. Cochez-Dhondt; S. Nicaise; S. Repin

Mathematical Modelling of Natural Phenomena (2009)

  • Volume: 4, Issue: 1, page 106-122
  • ISSN: 0973-5348

Abstract

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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.

How to cite

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Cochez-Dhondt, S., Nicaise, S., and Repin, S.. "A Posteriori Error Estimates for Finite Volume Approximations." Mathematical Modelling of Natural Phenomena 4.1 (2009): 106-122. <http://eudml.org/doc/222441>.

@article{Cochez2009,
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.},
author = {Cochez-Dhondt, S., Nicaise, S., Repin, S.},
journal = {Mathematical Modelling of Natural Phenomena},
keywords = {finite volume methods; elliptic problems; a posteriori error estimates of the functional type; a posteriori error estimates; numerical examples},
language = {eng},
month = {1},
number = {1},
pages = {106-122},
publisher = {EDP Sciences},
title = {A Posteriori Error Estimates for Finite Volume Approximations},
url = {http://eudml.org/doc/222441},
volume = {4},
year = {2009},
}

TY - JOUR
AU - Cochez-Dhondt, S.
AU - Nicaise, S.
AU - Repin, S.
TI - A Posteriori Error Estimates for Finite Volume Approximations
JO - Mathematical Modelling of Natural Phenomena
DA - 2009/1//
PB - EDP Sciences
VL - 4
IS - 1
SP - 106
EP - 122
AB - 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.
LA - eng
KW - finite volume methods; elliptic problems; a posteriori error estimates of the functional type; a posteriori error estimates; numerical examples
UR - http://eudml.org/doc/222441
ER -

References

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  1. A. Agouzal, F. Oudin. A posteriori error estimator for finite volume methods. C. R. Acad. Sci. Paris, Sér. 1, 343 (2006), 349–354.  Zbl1023.65116
  2. 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.).  Zbl1185.35048
  3. R. Verfürth. A review of a posteriori error estimation and adaptive mesh–refinement techniques. Wiley, Teubner, New York, 1996.  Zbl0853.65108
  4. M. Vohralík. A posteriori error estimates for finite volume and mixed finite element discretizations of convection-diffusion-reaction equations. ESAIM: Proc., 18 (2007), 57–69.  Zbl05213256

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