The gradient discretisation method

Jérôme Droniou (1) , Robert Eymard (2) , Thierry Gallouët (3) , Cindy Guichard (4, 5) , Raphaele Herbin (3)

1 School of Mathematical Sciences [Clayton]
2 LAMA - Laboratoire d'Analyse et de Mathématiques Appliquées
3 I2M - Institut de Mathématiques de Marseille
4 LJLL (UMR_7598) - Laboratoire Jacques-Louis Lions
5 ANGE - Numerical Analysis, Geophysics and Ecology

This monograph is dedicated to the presentation of the gradient discretisation method (GDM) and to some of its applications. It is intended for masters students, researchers and experts in the field of the numerical analysis of partial differential equations. The GDM is a framework which contains classical and recent discretisation schemes for diffusion problems of different kinds: linear or non-linear, steady-state or time-dependent. The schemes may be conforming or non-conforming, low or high order, and may be built on very general meshes. In this monograph, the core properties that are required to prove the convergence of a GDM are stressed, and the analysis of the method is performed on a series of elliptic and parabolic problems. As a result, for these models, any scheme entering the GDM framework is known to converge. A key feature of this monograph is the presentation of techniques and results which enable a complete convergence analysis of the GDM on fully non-linear, and sometimes degenerate, models. The scope of some of these techniques and results goes beyond the GDM, and makes them potentially applicable to numerical schemes not (yet) known to fit into this framework. Appropriate tools are also provided to easily check whether a given scheme satisfies the core properties of a GDM. Using these tools, it is shown that a number of methods are GDMs; some of these methods are classical, such as the conforming finite elements, the non-conforming finite elements, and the mixed finite elements. Others are more recent, such as the discontinuous Galerkin methods, the hybrid mimetic mixed or nodal mimetic finite differences methods, some discrete duality finite volume schemes, and some multi-point flux approximation schemes.

Domaines: Analyse numérique [math.NA]

Jérôme Droniou, Robert Eymard, Thierry Gallouët, Cindy Guichard, Raphaele Herbin. The gradient discretisation method. Springer International Publishing AG, 82, 2018, Mathématiques et Applications, M. Hoffmann et V. Perrier, 978-3-319-79042-8. ⟨10.1007/978-3-319-79042-8⟩. ⟨hal-01382358v8⟩