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Monotone corrections for generic cell-centered finite volume approximations of anisotropic diffusion equations

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Abstract

We present a nonlinear technique to correct a general finite volume scheme for anisotropic diffusion problems, which provides a discrete maximum principle. We point out general properties satisfied by many finite volume schemes and prove the proposed corrections also preserve these properties. We then study two specific corrections proving, under numerical assumptions, that the corresponding approximate solutions converge to the continuous one as the size of the mesh tends to zero. Finally we present numerical results showing that these corrections suppress local minima produced by the original finite volume scheme.

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Notes

  1. Using additionnal unknowns \(u_{\sigma }\) playing the role of approximation of \(\bar{u}\) on the boundary edges, assuming (13) is nothing but assuming that the scheme is exact when applied to constant families: \(\mathcal A ^\mathcal{D }(u)=0\) if \(u=(({u_{K}})_{{K \in \mathcal M }},(u_{\sigma })_{{\sigma }\in \mathcal{E }_{\mathrm{ext}}})=\mathrm{constant}\).

  2. This expectation is rigorously demonstrated in Remark 6.

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Acknowledgments

The authors would like to thank Jérôme Droniou for precious advices and discussions.

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Authors and Affiliations

  1. LJLL-UPMC Paris 06, 4 place Jussieu, 75005, Paris, France

    Clément Cancès

  2. Institut de Mathématiques et de Modélisation de Montpellier, Université Montpellier 2, CC051, Place Eugène Bataillon, 34095, Montpellier, France

    Mathieu Cathala

  3. CEA-Saclay, DEN, DM2S, STMF, LMEC, 91191, Gif-sur-Yvette, France

    Christophe Le Potier

Authors
  1. Clément Cancès
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  2. Mathieu Cathala
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  3. Christophe Le Potier
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Correspondence to Mathieu Cathala.

Additional information

This work was supported by the ANR project VFSitCom and by the GNR MoMaS.

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Cancès, C., Cathala, M. & Le Potier, C. Monotone corrections for generic cell-centered finite volume approximations of anisotropic diffusion equations. Numer. Math. 125, 387–417 (2013). https://doi.org/10.1007/s00211-013-0545-5

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  • DOI: https://doi.org/10.1007/s00211-013-0545-5

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