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

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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.

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Authors and Affiliations

  1. Laboratoire Jacques-Louis Lions, Université Pierre et Marie Curie (Paris 6), B.C. 187, 4 place Jussieu, 75252, Paris, France

    Martin Vohralík

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  1. Martin Vohralík
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Correspondence to Martin Vohralík.

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This work was supported by the GdR MoMaS project “Numerical Simulations and Mathematical Modeling of Underground Nuclear Waste Disposal”, PACEN/CNRS, ANDRA, BRGM, CEA, EdF, IRSN, France.

The main part of this work was carried out during the author’s post-doc stay at Laboratoire de Mathématiques, Analyse Numérique et EDP, Université de Paris-Sud and CNRS, Orsay, France.

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Vohralík, M. Residual flux-based a posteriori error estimates for finite volume and related locally conservative methods. Numer. Math. 111, 121–158 (2008). https://doi.org/10.1007/s00211-008-0168-4

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