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Numerische Mathematik

, Volume 111, Issue 1, pp 121–158 | Cite as

Residual flux-based a posteriori error estimates for finite volume and related locally conservative methods

  • Martin VohralíkEmail author
Article

Abstract

We derive in this paper a posteriori error estimates for discretizations of convection–diffusion–reaction equations in two or three space dimensions. Our estimates are valid for any cell-centered finite volume scheme, and, in a larger sense, for any locally conservative method such as the mimetic finite difference, covolume, and other. We consider meshes consisting of simplices or rectangular parallelepipeds and also provide extensions to nonconvex cells and nonmatching interfaces. We allow for the cases of inhomogeneous and anisotropic diffusion–dispersion tensors and of convection dominance. The estimates are established in the energy (semi)norm for a locally postprocessed approximate solution preserving the conservative fluxes and are of residual type. They are fully computable (all occurring constants are evaluated explicitly) and locally efficient (give a local lower bound on the error times an efficiency constant), so that they can serve both as indicators for adaptive refinement and for the actual control of the error. They are semi-robust in the sense that the local efficiency constant only depends on local variations in the coefficients and becomes optimal as the local Péclet number gets sufficiently small. Numerical experiments confirm their accuracy.

Mathematics Subject Classification (2000)

65N15 76M12 76S05 

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Copyright information

© Springer-Verlag 2008

Authors and Affiliations

  1. 1.Laboratoire Jacques-Louis LionsUniversité Pierre et Marie Curie (Paris 6)ParisFrance

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