On discrete functional inequalities for some finite volume schemes

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 $N\ge 2$. 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.