The Gradient Discretisation Method

Jerome / Eymard Droniou - Collection Yellow Sale 2023

497 pages, parution le 10/08/2018

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Résumé

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.

Part I Elliptic problems.- Part II Parabolic problems.- Part III Examples of gradient discretisation methods.- Part IV Appendix.

Jerome 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 Universite Paris-Est Marne-la-Vallee. 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 Universite. Her research is mainly focused on numerical methods for nonlinear fluid flows problems, including coupled elliptic or parabolic equations and hyperbolic equations.

Raphaele 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.