ON DISCRETE DUALITY FINITE VOLUME DISCRETIZATION OF GRADIENT AND DIVERGENCE OPERATORS IN 3D (original) (raw)

This work is intended to provide a convenient tool for the mathematical analysis of a particular kind of finite volume approximations which can be used, for instance, in the context of nonlinear and/or anisotropic diffusion operators. Following the approach developed by F. Hermeline and by K. Domelevo and P. Omnès, we consider a "double" covering T of a three-dimensional domain by a rather general primal mesh and by a well-chosen "dual" mesh. The associated discrete divergence operator div T is obtained by the standard finite volume approach. Then a consistent discrete gradient operator ∇ T is defined in such a way that −div T , ∇ T enjoy an analogue of the integration-by-parts formula known as the "discrete duality property". We discuss the implications of these properties and give a brief survey of other "discrete calculus" tools for "double" finite volume schemes.