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2018 | Book

Motivation and Basic Ideas Read chapter Read first chapter

The Gradient Discretisation Method

Authors: Prof. Jérôme Droniou, Prof. Dr. Robert Eymard, Prof. Thierry Gallouët, Dr. Cindy Guichard, Prof. Raphaèle Herbin

Publisher: Springer International Publishing

Book Series : Mathématiques et Applications

Part of: Springer Professional "Wirtschaft+Technik" , Springer Professional "Technik" , Springer Professional "Wirtschaft"

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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.

Table of Contents

Frontmatter

Elliptic Problems

Frontmatter
Chapter 1. Motivation and Basic Ideas
Abstract
After a brief review of a variety of discretisation methods for linear and non-linear elliptic problems, the basic ideas and motivations of GDM are presented.
Jérôme Droniou, Robert Eymard, Thierry Gallouët, Cindy Guichard, Raphaèle Herbin
Chapter 2. Dirichlet Boundary Conditions
Abstract
The GDM is presented, along with its convergence and error estimate properties, in the case of linear and quasi-linear elliptic problems with homogeneous and non-homogeneous Dirichlet boundary conditions.
Jérôme Droniou, Robert Eymard, Thierry Gallouët, Cindy Guichard, Raphaèle Herbin
Chapter 3. Neumann, Fourier and Mixed Boundary Conditions
Abstract
The GDM and its analysis are adapted here to cope with Neumann, Fourier and mixed boundary conditions. Properties of trace operators are detailed.
Jérôme Droniou, Robert Eymard, Thierry Gallouët, Cindy Guichard, Raphaèle Herbin

Parabolic Problems

Frontmatter
Chapter 4. Time-Dependent GDM
Abstract
The definition of gradient discretisations (GDs) for time-dependent problems is first given; it is followed by compactness results for the analysis of such problems.
Jérôme Droniou, Robert Eymard, Thierry Gallouët, Cindy Guichard, Raphaèle Herbin
Chapter 5. Non Degenerate Parabolic Problems
Abstract
Some non-degenerate parabolic problems are studied. An error estimate is obtained for a linear parabolic problem, followed by a convergence result for a quasi-linear problem. A class of non-linear parabolic problems under non-conservative form is then presented.
Jérôme Droniou, Robert Eymard, Thierry Gallouët, Cindy Guichard, Raphaèle Herbin
Chapter 6. Degenerate Parabolic Problems
Abstract
A generic non-linear parabolic model which includes both Richards’ model describing the flow of water in a heterogeneous anisotropic underground medium, and Stefan’s model which arises in the study of a simplified heat diffusion in a melting medium.
Jérôme Droniou, Robert Eymard, Thierry Gallouët, Cindy Guichard, Raphaèle Herbin

Examples of Gradient Discretisation Methods

Frontmatter
Chapter 7. Analysis Tools for Gradient Discretisations
Abstract
Analysis tools for GDM are presented. Polytopal toolboxes enable easy proofs of the coercivity, limit-conformity and compactness of gradient discretisations. The notion of local linearly exact gradient discretisations provides ways to analyse the consistency of GDs, as well as precise estimates on the consistency error.
Jérôme Droniou, Robert Eymard, Thierry Gallouët, Cindy Guichard, Raphaèle Herbin
Chapter 8. Conforming Approximations
Abstract
Conforming Galerkin methods are shown to fit into the GDM. Emphasis is put on conforming \(\mathbb {P}_{k}\) finite elements, with precise estimates on their consistency and analysis for a variety of boundary conditions. Mass-lumped \(\mathbb {P}_{1}\) finite elements are also shown to be GDMs.
Jérôme Droniou, Robert Eymard, Thierry Gallouët, Cindy Guichard, Raphaèle Herbin
Chapter 9. Non-conforming Finite Element Methods
Abstract
Non-conforming methods are presented in the context of the GDM. An abstract framework is developed that covers a wide range of non-conforming methods, and the special case of non-conforming \(\mathbb {P}_k\) finite elements is then considered. In the case \(k=1\), the presentation is given for all classical boundary conditions, and mass-lumped non-conforming \(Pfe_1\) finite elements are also shown to be GDs.
Jérôme Droniou, Robert Eymard, Thierry Gallouët, Cindy Guichard, Raphaèle Herbin
Chapter 10. Conforming GDs from Mixed Finite Element Methods
Abstract
GDMs are derived from mixed finite element schemes, using the initial and the hybrid formulations. High order estimates are proved in the case where the GDMs are issued from Raviart-Thomas mixed finite elements.
Jérôme Droniou, Robert Eymard, Thierry Gallouët, Cindy Guichard, Raphaèle Herbin
Chapter 11. Discontinuous Galerkin Methods
Abstract
Two GDMs are obtained from the Discontinuous Galerkin setting. The first one recovers the high order SIPG schemes in the case of linear problems, the second one, based on average jumps, leads to simpler computations.
Jérôme Droniou, Robert Eymard, Thierry Gallouët, Cindy Guichard, Raphaèle Herbin
Chapter 12. The Multi-point Flux Approximation MPFA-O Scheme
Abstract
GDMs are built using the multi-point flux approximation-O scheme on rectangular and simplicial meshes.
Jérôme Droniou, Robert Eymard, Thierry Gallouët, Cindy Guichard, Raphaèle Herbin
Chapter 13. Hybrid Mimetic Mixed Schemes
Abstract
GDMs are constructed from the hybrid mimetic mixed schemes, recovering in particular the mimetic finite difference schemes and the SUSHI scheme.
Jérôme Droniou, Robert Eymard, Thierry Gallouët, Cindy Guichard, Raphaèle Herbin
Chapter 14. Nodal Mimetic Finite Difference Methods
Abstract
GDMs are obtained from the nodal mimetic finite differences methods, and also cover some DDFV schemes.
Jérôme Droniou, Robert Eymard, Thierry Gallouët, Cindy Guichard, Raphaèle Herbin
Backmatter
Metadata
Title
The Gradient Discretisation Method
Authors
Prof. Jérôme Droniou
Prof. Dr. Robert Eymard
Prof. Thierry Gallouët
Dr. Cindy Guichard
Prof. Raphaèle Herbin
Copyright Year
2018
Electronic ISBN
978-3-319-79042-8
Print ISBN
978-3-319-79041-1
DOI
https://doi.org/10.1007/978-3-319-79042-8

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