On discrete functional inequalities for some finite volume schemes

M. Bessemoulin-Chatard 1 C. Chainais-Hillairet 2, 3 F. Filbet 4, 5
2 MEPHYSTO - Quantitative methods for stochastic models in physics
LPP - Laboratoire Paul Painlevé - UMR 8524, ULB - Université Libre de Bruxelles [Bruxelles], Inria Lille - Nord Europe
4 MMCS - Modélisation mathématique, calcul scientifique
ICJ - Institut Camille Jordan [Villeurbanne]
5 KALiFFE - Kinetic models AppLIed for Future of Fusion Energy
Inria Grenoble - Rhône-Alpes, ICJ - Institut Camille Jordan [Villeurbanne], INSMI (CNRS)
Abstract : We prove several discrete Gagliardo-Nirenberg-Sobolev and Poincaré-Sobolev inequalities for some approximations with arbitrary boundary values on finite volume meshes. The keypoint of our approach is to use the continuous embedding of the space $BV(\Omega)$ into $L^{N/(N-1)}(\Omega)$ for a Lipschitz domain $ \Omega \subset \mathbb{R}^{N}$, with $N \geq 2$. Finally, we give several applications to discrete duality finite volume (DDFV) schemes which are used for the approximation of nonlinear and non isotropic elliptic and parabolic problems.
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Submitted on : Wednesday, January 15, 2014 - 11:43:24 AM
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M. Bessemoulin-Chatard, C. Chainais-Hillairet, F. Filbet. On discrete functional inequalities for some finite volume schemes. IMA Journal of Numerical Analysis, Oxford University Press (OUP), 2014, pp.10-32. ⟨10.1093/imanum/dru032⟩. ⟨hal-00672591v3⟩

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