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 ANGE - Numerical Analysis, Geophysics and Ecology
5 LJLL - Laboratoire Jacques-Louis Lions

This monograph is dedicated to the presentation of the Gradient Discretisation Method (GDM) and of 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 and may rely on very general polygonal or polyhedral 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, linear or non-linear, for which the GDM is particularly adapted. 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 then developed so as to easily check whether a given scheme satisfies the expected properties of a GDM. Thanks to these tools a number of methods can be shown to enter the GDM framework: some of these methods are classical, such as the conforming Finite Elements, the Raviart–Thomas Mixed Finite Elements, or the P 1 non-conforming Finite Elements. Others are more recent, such as the Hybrid Mixed Mimetic or Nodal Mimetic 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 : A framework for the discretisation and numerical analysis of linear and nonlinear elliptic and parabolic problems. 2016. ⟨hal-01382358v3⟩