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Numerische Mathematik

, Volume 132, Issue 4, pp 721–766 | Cite as

Uniform-in-time convergence of numerical methods for non-linear degenerate parabolic equations

  • Jérôme Droniou
  • Robert Eymard
Article

Abstract

Gradient schemes is a framework that enables the unified convergence analysis of many numerical methods for elliptic and parabolic partial differential equations: conforming and non-conforming finite element, mixed finite element and finite volume methods. We show here that this framework can be applied to a family of degenerate non-linear parabolic equations (which contain in particular the Richards’, Stefan’s and Leray–Lions’ models), and we prove a uniform-in-time strong-in-space convergence result for the gradient scheme approximations of these equations. In order to establish this convergence, we develop several discrete compactness tools for numerical approximations of parabolic models, including a discontinuous Ascoli–Arzelà theorem and a uniform-in-time weak-in-space discrete Aubin–Simon theorem. The model’s degeneracies, which occur both in the time and space derivatives, also requires us to develop a discrete compensated compactness result.

Mathematics Subject Classification

65M12 35K65 46N40 

Notes

Acknowledgments

The authors would like to thank Clément Cancès for fruitful discussions on discrete compensated compactness theorems.

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Copyright information

© Springer-Verlag Berlin Heidelberg 2015

Authors and Affiliations

  1. 1.School of Mathematical SciencesMonash UniversityVictoriaAustralia
  2. 2.Laboratoire d’Analyse et de Mathématiques Appliquées, UMR 8050Université Paris-EstMarne-la-Vallée Cedex 2France

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