# On a functional inequality arising in the analysis of finite-volume methods

Abstract : In the present paper we prove a functional inequality of the Poincaré--Wirtinger type on some particular domains with a precise estimate of the constant as a function of the geometry of the domain. This type of inequality arises, for instance, in the analysis of finite volume (FV) numerical methods, which was our main motivation. As an illustration, our result let us prove uniform a priori bounds for the FV approximate solutions of the heat equation with Ventcell boundary conditions in the natural energy space constituted by functions in $H^1(\Omega)$ whose trace belongs to $H^1(\partial \Omega)$. The main difficulty here comes from the fact that the approximation is performed on non-polygonal control volumes since the domain is itself non polygonal.
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https://hal.archives-ouvertes.fr/hal-01134988
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Submitted on : Friday, June 26, 2015 - 3:12:12 PM
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Pierre Bousquet, Franck Boyer, Flore Nabet. On a functional inequality arising in the analysis of finite-volume methods. Calcolo, Springer Verlag, 2016, 53 (3), pp.363-397. ⟨10.1007/s10092-015-0153-0⟩. ⟨hal-01134988v3⟩

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