On discrete functional inequalities for some finite volume schemes
Provides theoretical foundations for finite volume schemes, benefiting numerical analysts working on nonlinear and non-isotropic PDEs.
The paper proves discrete Gagliardo-Nirenberg-Sobolev and Poincaré-Sobolev inequalities for finite volume meshes with arbitrary boundary values, using continuous BV embeddings. These inequalities are applied to discrete duality finite volume schemes for nonlinear elliptic and parabolic problems.
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 $BV(Ω)$ into $L^{N/(N-1)}(Ω)$ for a Lipschitz domain $ Ω\subset \mathbb{R}^{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.