Discontinuous Galerkin gradient discretisations for the approximation of second-order differential operators in divergen
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Discontinuous Galerkin gradient discretisations for the approximation of second-order differential operators in divergence form Robert Eymard1
· Cindy Guichard2,3
Received: 11 August 2017 / Revised: 18 November 2017 / Accepted: 16 December 2017 © SBMAC - Sociedade Brasileira de Matemática Aplicada e Computacional 2017
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
Communicated by Raphaèle Herbin.
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Robert Eymard [email protected] Cindy Guichard [email protected]
1
Laboratoire d’Analyse et de Mathématiques Appliquées, UPEC, UPEM, UMR8050 CNRS, Université Paris-Est Marne-la-Vallée, 77454 Marne-la-Vallée, France
2
Laboratoire Jacques-Louis Lions, UMR CNRS 7598, Sorbonne Universités, UPMC Univ. Paris 6, 75005 Paris, France
3
ANGE Project-Team (Inria, Cerema, UPMC, CNRS), 2 rue Simone Iff, CS 42112, 75589 Paris, France
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R. Eymard, C. Guichard
1 Introduction Discontinuous Galerkin (DG) methods have received a lot of attention over the last decade at least, and are still a subject of interest. They present the advantage of being suited to elliptic and parabolic problems, while opening the possibility to closely approximate weakly regular functions on general meshes. Although the convergence of DG methods has been proved on a variety of problems (see Di Pietro and Ern 2012 and references therein), note that the stabilisation of DG schemes for elliptic or parabolic problems has to be specified with respect to the problem, and that there are numerous possible choices (Arnold et al. 2001/2002). On the other hand, convergence and error estimate results for a wide variety of numerical methods applied to some elliptic, parabolic, coupled, linear and nonlinear problems are proved on the generic “gradient scheme” issued from the gradient discretisation method framework (see Droniou et al. 2016 and references therein). This framework is shown to include conforming Galerkin methods with or without mass lumping, nonconforming P1 finite elements, mixed finite elements and a variety of schemes issued from extensions of the finite volume method. Convergence and error estimate results are then proved in Droniou et al. (2016)
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