On discrete functional inequalities for some finite volume schemes Article - Juillet 2014

Marianne Bessemoulin-Chatard, Claire Chainais-Hillairet, Francis Filbet

Marianne Bessemoulin-Chatard, Claire Chainais-Hillairet, Francis Filbet, « On discrete functional inequalities for some finite volume schemes  », IMA Journal of Numerical Analysis, juillet 2014, pp. 10-32. ISSN 0272-4979

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 \mathbbR^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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