On discrete functional inequalities for some finite volume schemes

Marianne Bessemoulin-Chatard (1) , Claire Chainais-Hillairet (2) , Francis Filbet (3)

1 LMBP - Laboratoire de Mathématiques Blaise Pascal
2 LPP - Laboratoire Paul Painlevé - UMR 8524
3 ICJ - Institut Camille Jordan

We prove several discrete Gagliardo-Nirenberg-Sobolev and Sobolev-Poincar ́e inequali- ties for some approximations with arbitrary boundary values on finite volume admissible meshes. The keypoint 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 (DDFV) schemes which are used for the approximation of nonlinear and non isotropic elliptic and parabolic problems.

Mots clés: Discrete functional inequality; finite volume scheme; DDFV

Domaines: Analyse numérique [math.NA]

Marianne Bessemoulin-Chatard, Claire Chainais-Hillairet, Francis Filbet. On discrete functional inequalities for some finite volume schemes. 2012. ⟨hal-00672591v1⟩