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Computational and Applied Mathematics

, Volume 37, Issue 4, pp 4023–4054 | Cite as

Discontinuous Galerkin gradient discretisations for the approximation of second-order differential operators in divergence form

  • Robert EymardEmail author
  • Cindy Guichard
Article

Abstract

We include in the gradient discretisation method (GDM) framework two numerical schemes based on discontinuous Galerkin approximations: the symmetric interior penalty Galerkin (SIPG) method, and the scheme obtained by averaging the jumps in the SIPG method. We prove that these schemes meet the main mathematical gradient discretisation properties on any kind of polytopal mesh, by adapting discrete functional analysis properties to our precise geometrical hypotheses. Therefore, these schemes inherit the general convergence properties of the GDM, which hold for instance in the cases of the p-Laplace problem and of the anisotropic and heterogeneous diffusion problem. This is illustrated by simple 1D and 2D numerical examples.

Keywords

Gradient discretisation method Discontinuous Galerkin method Symmetric interior penalty Galerkin scheme Discrete functional analysis Polytopal meshes 

Mathematics Subject Classification

65N30 

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Copyright information

© SBMAC - Sociedade Brasileira de Matemática Aplicada e Computacional 2017

Authors and Affiliations

  1. 1.Laboratoire d’Analyse et de Mathématiques Appliquées, UPEC, UPEM, UMR8050 CNRSUniversité Paris-Est Marne-la-ValléeMarne-la-ValléeFrance
  2. 2.Laboratoire Jacques-Louis Lions, UMR CNRS 7598Sorbonne Universités, UPMC Univ. Paris 6ParisFrance
  3. 3.ANGE Project-Team (Inria, Cerema, UPMC, CNRS)ParisFrance

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