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International Conference on Finite Volumes for Complex Applications

FVCA 2017: Finite Volumes for Complex Applications VIII - Methods and Theoretical Aspects pp 215–223Cite as

Céa-Type Quasi-Optimality and Convergence Rates for (Adaptive) Vertex-Centered FVM

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Part of the Springer Proceedings in Mathematics & Statistics book series (PROMS,volume 199)

Abstract

For a general second order linear elliptic PDE, we show a generalized Céa lemma for a vertex-centered finite volume method (FVM). The latter implies, in particular, a comparison result between the solutions of FVM and the finite element method (FEM). Furthermore, for a symmetric PDE, i.e., no convection is present, we prove linear convergence with generically optimal algebraic rates for an adaptive FVM algorithm.

Keywords

  • Finite volume method
  • Céa-type quasi-optimality
  • A posteriori error estimators
  • Adaptive algorithm
  • Local mesh-refinement
  • Optimal convergence rates

MSC (2010)

  • 65N08
  • 65N30
  • 65N50
  • 65N15
  • 65N12
  • 41A25

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  • DOI: 10.1007/978-3-319-57397-7_14
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References

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  3. Erath, C., Praetorius, D.: Adaptive vertex-centered finite volume methods with convergence rates. SIAM J. Numer. Anal. 54(4), 2228–2255 (2016). doi:10.1137/15M1036701

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  4. Erath, C., Praetorius, D.: Convergence rates of adaptive vertex-centered finite volume methods for general second order linear elliptic PDEs. In preparation (May) (2017)

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  5. Erath, C., Of, G., Sayas, F.-J.: A non-symmetric coupling of the finite volume method and the boundary element method. Numer. Math. 135(3), 895–922 (2017). doi:10.1007/s00211-016-0820-3

  6. Feischl, M., Führer, T., Praetorius, D.: Adaptive FEM with optimal convergence rates for a certain class of nonsymmetric and possibly nonlinear problems. SIAM J. Numer. Anal. 52(2), 601–625 (2014). doi:10.1137/120897225

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Author information

Authors and Affiliations

  1. TU Darmstadt, Department of Mathematics, Dolivostr. 15, 64293, Darmstadt, Germany

    Christoph Erath

  2. Institute for Analysis and Scientific Computing, TU Wien, Wiedner Hauptstr. 8–10, 1040, Vienna, Austria

    Dirk Praetorius

Authors
  1. Christoph Erath
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  2. Dirk Praetorius
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Corresponding author

Correspondence to Christoph Erath .

Editor information

Editors and Affiliations

  1. Equipe RAPSODI, Inria Lille - Nord Europe, Villeneuve-d’Ascq, France

    Clément Cancès

  2. Commissariat à l'énergie atomique et aux énergies alternatives, Centre de Saclay, Gif-sur-Yvette, France

    Pascal Omnes

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Erath, C., Praetorius, D. (2017). Céa-Type Quasi-Optimality and Convergence Rates for (Adaptive) Vertex-Centered FVM. In: Cancès, C., Omnes, P. (eds) Finite Volumes for Complex Applications VIII - Methods and Theoretical Aspects . FVCA 2017. Springer Proceedings in Mathematics & Statistics, vol 199. Springer, Cham. https://doi.org/10.1007/978-3-319-57397-7_14

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