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

We present a detailed survey of discrete functional analysis tools (consistency results, Poincaré and Sobolev embedding inequalities, discrete $W^{1,p}$ compactness, discrete compactness in space and in time) for the so-called Discrete Duality (DDFV) Finite Volume schemes in three space dimensions. We concentrate mainly on the 3D CeVe-DDFV scheme presented in [3]. Some of our results are new, such as a general time-compactness result based upon the idea of Kruzhkov [65]; others generalize the ideas known for the 2D DDFV schemes or for traditional two-point finite volume schemes. We illustrate the use of these tools by studying convergence of discretizations of nonlinear elliptic-parabolic problems of Leray-Lions kind, and provide numerical results for this example.

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.. Computational Methods in Applied Mathematics, 2013, 13 (4), pp. 369-410. ⟨10.1515/cmam-2013-0011⟩. ⟨hal-00567342v3⟩