A 3D Discrete Duality Finite Volume Method for Nonlinear Elliptic Equations

Abstract

Abstract. Discrete Duality Finite Volume (DDFV) schemes have recently been developed in 2D to approximate nonlinear diffusion problems on general meshes. In this paper, a 3D extension of these schemes is proposed. The construction of this extension is detailed and its main properties are proved: a priori bounds, well-posedness and error estimates. The practical implementation of this scheme is easy. Numerical experiments are presented to illustrate its good behavior.

Small-stencil 3D schemes for diffusive flows in porous media

“…The DDFV scheme, see [9,15,26] for the two dimensional case and [7,8,11,13,14,27,28] for the three dimensional case, may also be seen as a gradient scheme. Consider the case where the domain Ω is the union of octahedra which are the so-called diamond cells (such a cell is depicted in Fig.…”

A CeVeFE DDFV scheme for discontinuous anisotropic permeability tensors

“…The scheme in (4) originates a symmetric and positive-definite linear system of equations (see [6] for a thourough discussion of the other properties). Assembling the matrix of the system amounts to gathering the local contributions of the discrete gradient associated to each diamond cell.…”

“…This method is a variant of the DDFV formulation proposed by Y. Coudière and F. Hubert in [6] to extend to three-dimensional (3D) problems the original two-dimensional finite volume schemes by F. Hermeline [11] and K. Domelevo and P. Omnès [9]. In the DDFV approach the diffusive flux is approximated using a piecewise constant approximation of the solution gradient over a set of edge-based cells called diamond cells.…”

“…Here, we consider the alternative construction proposed in [6], which uses two families of additional unknowns. In the first family, the unknowns are located at the vertices of the mesh and are the solution of a finite volume method whose control volumes are built around the vertices.…”

An Error Estimate for the Approximation of Linear Parabolic Equations by the Gradient Discretization Method

“…The Gradient Discretization method (GDM) [5,3] provides a common mathematical framework for a number of numerical schemes dedicated to the approximation of elliptic or parabolic problems, linear or nonlinear, coupled or not; these include conforming and non conforming finite element, mixed finite element, hybrid mixed mimetic schemes [4] and some Multi-Point Flux Approximation [1] and Discrete Duality finite volume schemes [2] : we refer to [3, Part III] for more on this (note that in the present proceedings, it is shown that in some way the Discontinuous Galerkin schemes may also enter this framework [6]). Let us recall this framework in the case of the following linear elliptic problem:…”