On 3D DDFV Discretization of Gradient and Divergence Operators: Discrete Functional Analysis Tools and Applications to Degenerate Parabolic Problems

Abstract

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 Finite Volume (DDFV) schemes in three space dimensions. We concentrate mainly on the 3D CeVe-DDFV scheme presented in [IMA J. Numer. Anal., 32 (2012), pp. 1574-1603. Some of our results are new, such as a general time-compactness result based upon the idea of Kruzhkov (1969); others generalize the ideas known for the 2D DDFV schemes or for traditional two-point-flux 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.2010 Mathematical subject classification: 65N12, 65M12.

A nonlinear time compactness result and applications to discretization of degenerate parabolic–elliptic PDEs

“…Further, discrete variants of different time compactness results have been already proved both for many concrete applications (mainly in the context of finite element or finite volume methods) and in abstract form: we refer in particular to Eymard et al. [27,29] for Alt-Luckhaus kind technique for concrete applications, to Gallouët and Latché [31] for a discrete Aubin-Lions-Simon lemma, to Andreianov et al [7,4] for a discrete Kruzhkov lemma, and to Dreher and Jüngel [21] (see also [19]) for a discrete Dubinskii argument with uniform time stepping.…”

Benchmark 3D: a version of the DDFV scheme with cell/vertex unknowns on general meshes

“…This duality property greatly simplifies the theoretical analysis of finite volume schemes based on the DDFV construction, see e.g. [5,2]. This 2D strategy reveals to be particularly efficient in terms of gradient approximation (see [7,14]) and has been extended to a wide class of PDE problems (see [1,5,6,18,19] and references therein).…”

“…From the discrete duality (Prop. 1) which is a cornerstone of DDFV schemes, and from consistency properties of the projection, gradient and divergence operators (see [2]; cf. [5] for analogous properties in 2D) one easily derives that the scheme is well posed for l ≤ 4.…”

“…4 Given a family (T h ) h of CeVe-DDFV meshes, the associated discrete solutions u T h enjoy a uniform discrete H 1 estimate, and they converge to the exact solution u as the size h of the mesh tends to zero. Convergence analysis requires mild proportionality assumptions on the meshes T h in use, see [2].…”

Quick finite volume solver for incompressible Navier-Stokes equation by parallel Gram-Schmidt process based GMRES and HSS

“…Also, multiple variants of finite volume method are developed, and these variants include hybrid finite element finite volume (Nick and Matthäi, 2011), discontinuous finite volume method (Liu et al, 2011), Discrete Duality Finite Volume (Coudière and Manzini, 2010;Coudière and Hubert, 2011;Delcourte et al, 2005Delcourte et al, , 2007Delcourte et al, , 2009Delcourte, 2007Delcourte, , 2009Delcourte and Jennequin, 2008;Karlsen et al, 2010;Andreianov et al, 2007Andreianov et al, , 2012Andreianov et al, , 2013; Baron et al, 2013), etc.…”