Overview
- Includes a complete convergence analysis of schemes for linear and non-linear PDEs, covering all standard boundary conditions for elliptic and parabolic models
- Presents a unified analysis of many classical, and less classical, numerical methods, including an analysis of degenerate models
- Provides very generic compactness results for stationary and time-dependent problems
Part of the book series: Mathématiques et Applications (MATHAPPLIC, volume 82)
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Table of contents (14 chapters)
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Front Matter
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Elliptic Problems
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Front Matter
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Parabolic Problems
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Front Matter
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Examples of Gradient Discretisation Methods
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Front Matter
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Back Matter
Keywords
About this book
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.
Authors and Affiliations
About the authors
Jérôme Droniou is Associate Professor at Monash University, Australia. His research focuses on elliptic and parabolic PDEs. He has published many papers on theoretical and numerical analysis of models with singularities or degeneracies, including convergence analysis of schemes without regularity assumptions on the data or solutions.
Robert Eymard is professor of mathematics at Université Paris-Est Marne-la-Vallée. His research concerns the design and analysis of numerical methods, mainly applied to fluid flows in porous media and incompressible Navier-Stokes equations.
Thierry Gallouet is professor at the University of Aix-Marseille. His research focuses on the analysis of partial differential equations and the approximation of their solutions by numerical schemes.
Cindy Guichard is assistant professor at Sorbonne Université. Her research is mainly focused on numerical methods for nonlinear fluid flows problems, including coupled elliptic or parabolic equations and hyperbolic equations.Raphaèle Herbin is professor at the University of Aix-Marseille. She is a specialist of numerical schemes for partial differential equations, with application to incompressible and compressible fluid flows.
Bibliographic Information
Book Title: The Gradient Discretisation Method
Authors: Jérôme Droniou, Robert Eymard, Thierry Gallouët, Cindy Guichard, Raphaèle Herbin
Series Title: Mathématiques et Applications
DOI: https://doi.org/10.1007/978-3-319-79042-8
Publisher: Springer Cham
eBook Packages: Mathematics and Statistics, Mathematics and Statistics (R0)
Copyright Information: Springer International Publishing AG, part of Springer Nature 2018
Softcover ISBN: 978-3-319-79041-1Published: 11 August 2018
eBook ISBN: 978-3-319-79042-8Published: 31 July 2018
Series ISSN: 1154-483X
Series E-ISSN: 2198-3275
Edition Number: 1
Number of Pages: XXIV, 497
Number of Illustrations: 19 b/w illustrations, 14 illustrations in colour
Topics: Computational Mathematics and Numerical Analysis, Partial Differential Equations