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The Gradient Discretisation Method

  • Jérôme Droniou
  • Robert Eymard
  • Thierry Gallouët
  • Cindy Guichard
  • Raphaèle Herbin

Part of the Mathématiques et Applications book series (MATHAPPLIC, volume 82)

Table of contents

  1. Front Matter
    Pages i-xxiv
  2. Elliptic Problems

    1. Front Matter
      Pages 1-1
    2. Jérôme Droniou, Robert Eymard, Thierry Gallouët, Cindy Guichard, Raphaèle Herbin
      Pages 3-16
    3. Jérôme Droniou, Robert Eymard, Thierry Gallouët, Cindy Guichard, Raphaèle Herbin
      Pages 17-65
    4. Jérôme Droniou, Robert Eymard, Thierry Gallouët, Cindy Guichard, Raphaèle Herbin
      Pages 67-97
  3. Parabolic Problems

    1. Front Matter
      Pages 99-99
    2. Jérôme Droniou, Robert Eymard, Thierry Gallouët, Cindy Guichard, Raphaèle Herbin
      Pages 101-121
    3. Jérôme Droniou, Robert Eymard, Thierry Gallouët, Cindy Guichard, Raphaèle Herbin
      Pages 123-166
    4. Jérôme Droniou, Robert Eymard, Thierry Gallouët, Cindy Guichard, Raphaèle Herbin
      Pages 167-202
  4. Examples of Gradient Discretisation Methods

    1. Front Matter
      Pages 203-204
    2. Jérôme Droniou, Robert Eymard, Thierry Gallouët, Cindy Guichard, Raphaèle Herbin
      Pages 205-262
    3. Jérôme Droniou, Robert Eymard, Thierry Gallouët, Cindy Guichard, Raphaèle Herbin
      Pages 263-284
    4. Jérôme Droniou, Robert Eymard, Thierry Gallouët, Cindy Guichard, Raphaèle Herbin
      Pages 285-305
    5. Jérôme Droniou, Robert Eymard, Thierry Gallouët, Cindy Guichard, Raphaèle Herbin
      Pages 307-324
    6. Jérôme Droniou, Robert Eymard, Thierry Gallouët, Cindy Guichard, Raphaèle Herbin
      Pages 325-341
    7. Jérôme Droniou, Robert Eymard, Thierry Gallouët, Cindy Guichard, Raphaèle Herbin
      Pages 343-351
    8. Jérôme Droniou, Robert Eymard, Thierry Gallouët, Cindy Guichard, Raphaèle Herbin
      Pages 353-375
    9. Jérôme Droniou, Robert Eymard, Thierry Gallouët, Cindy Guichard, Raphaèle Herbin
      Pages 377-393
  5. Back Matter
    Pages 395-497

About this book

Introduction

This monograph presents  the Gradient Discretisation Method (GDM), which is a unified convergence analysis framework for numerical methods for elliptic and parabolic partial differential equations. The results obtained by the GDM cover both stationary and transient models; error estimates are provided for linear (and some non-linear) equations, and convergence is established for a wide range of fully non-linear models (e.g. Leray–Lions equations and degenerate parabolic equations such as the Stefan or Richards models). The GDM applies to a diverse range of methods, both classical (conforming, non-conforming, mixed finite elements, discontinuous Galerkin) and modern (mimetic finite differences, hybrid and mixed finite volume, MPFA-O finite volume), some of which can be built on very general meshes.

Keywords

Gradient Discretisation Method Gradient schemes Elliptic partial differential equations Parabolic partial differential equations Unified convergence analysis Discrete Aubin-Simon compactness theorems Convergence by compactness Error estimates

Authors and affiliations

  • Jérôme Droniou
    • 1
  • Robert Eymard
    • 2
  • Thierry Gallouët
    • 3
  • Cindy Guichard
    • 4
  • Raphaèle Herbin
    • 5
  1. 1.School of Mathematical SciencesMonash UniversityClaytonAustralia
  2. 2.Laboratoire d’Analyse et de Mathématiques AppliquéesUniversité Paris-Est Marne-la-ValléeChamps-sur-MarneFrance
  3. 3.Institut de Mathématiques de MarseilleAix-Marseille Université, Ecole Centrale de Marseille, CNRSMarseilleFrance
  4. 4.Laboratoire Jacques-Louis LionsSorbonne UniversitéParisFrance
  5. 5.Institut de Mathématiques de MarseilleAix-Marseille Université, Ecole Centrale de Marseille, CNRSMarseilleFrance

Bibliographic information

  • DOI https://doi.org/10.1007/978-3-319-79042-8
  • Copyright Information Springer International Publishing AG, part of Springer Nature 2018
  • Publisher Name Springer, Cham
  • eBook Packages Mathematics and Statistics
  • Print ISBN 978-3-319-79041-1
  • Online ISBN 978-3-319-79042-8
  • Series Print ISSN 1154-483X
  • Series Online ISSN 2198-3275
  • Buy this book on publisher's site
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