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

This paper gives numerical results for a 3D extension of the 2D DDFV scheme. Our scheme is of the same inspiration as the one called CeVe-DDFV ([9]), with a more straightforward dual mesh construction. We sketch the construction in which, starting from a given 3D mesh (which can be non conformal and have arbitrary polygonal faces), one defines a dual mesh and a diamond mesh, reconstructs a discrete gradient, and proves the discrete duality property. Details are given in [1]. 1 DDFV methods in 2D and in 3D. A 3D CeVe-DDFV scheme. DDFV (“Discrete Duality Finite Volume”) scheme was introduced in 2D by Hermeline in [15] and by Domelevo and Omnès in [13]. To handle anisotropic problems or nonlinear problems, or in order to work on general distorted meshes, full gradient reconstruction from point values is a popular strategy. It is well known that reconstruction of a discrete gradient is facilitated by adding unknowns that are new with respect to those of standard cell-centered finite volume schemes. The 2D DDFV method consists in adding new unknowns at the vertices of the initial mesh (this initial mesh is often called the primal one), and in use of new control volumes (called dual cells, or co-volumes) around these points. A family of diamond cells is naturally associated to this construction, each diamond being built on two neighbor cell centers xK,xL and the two vertices of the edge K|L that separates them. On a diamond, one can construct a discrete gradient direction per direction (cell-cell and vertex-vertex), following the idea of [8]. It turns out that this discrete gradient is related by a discrete analogue of integraton-by-parts formula, called “discrete duBoris Andreianov CNRS UMR 6623, Besanc, on, France, e-mail: [email protected] Florence Hubert LATP, Université de Provence, Marseille, France, e-mail: [email protected] Stella Krell INRIA, Lille, France, e-mail: [email protected]