On 3D DDFV discretization of gradient and divergence operators. II. Discrete functional analysis tools and applications to degenerate parabolic problems.

Boris Andreianov (1) , Mostafa Bendahmane (2) , Florence Hubert (3)

1 LMB - Laboratoire de Mathématiques de Besançon (UMR 6623)
2 IMB - Institut de Mathématiques de Bordeaux
3 LATP - Laboratoire d'Analyse, Topologie, Probabilités

This paper is the sequel of the paper [2] of S. Krell and the authors, where a family of 3D finite volume schemes on ``double'' meshes was constructed and the crucial discrete duality property was established. Heading towards applications, we state some discrete functional analysis tools (consistency results, Poincare and Sobolev embedding inequalities, discrete $W^{1,p}$ compactness, discrete $L^1$ compactness in space and time) for the DDFV scheme of [2]. We apply them to infer convergence of discretizations of nonlinear elliptic-parabolic problems of Leray-Lions kind. Applications to degenerate parabolic-hyperbolic PDEs and to a degenerate parabolic system known in electro-cardiology are briefly discussed.

Mots clés: Finite volume approximation; Discrete duality; CeVe-DDFV; Convergence; Consistency; Discrete compactness; Discrete Sobolev embeddings; Degenerate parabolic problems

Domaines: Analyse numérique [math.NA]

Boris Andreianov, Mostafa Bendahmane, Florence Hubert. On 3D DDFV discretization of gradient and divergence operators. II. Discrete functional analysis tools and applications to degenerate parabolic problems.. 2011. ⟨hal-00567342v1⟩