On discrete functional inequalities for some finite volume schemes

We prove several discrete Gagliardo–Nirenberg–Sobolev and Poincaré–Sobolev inequalities for some approximations with arbitrary boundary values on finite volume meshes. The key point of our approach is to use the continuous embedding of the space BV(Ω) into LN/(N−1)(Ω) for a Lipschitz domain Ω ⊂ ℝN, with N≥2. Finally, we give several applications to discrete duality finite volume schemes which are used for the approximation of nonlinear and nonisotropic elliptic and parabolic problems.