ANDREIANOV, Boris; BENDAHMANE, Mostafa; KARLSEN, Kenneth Hvistendahl

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ANDREIANOV, Boris
Laboratoire de Mathématiques de Besançon (UMR 6623) [LMB]

BENDAHMANE, Mostafa
Centro de Investigación en Ingeniería Matemática [Concepción] [CI²MA]
Institut de Mathématiques de Bordeaux [IMB]

KARLSEN, Kenneth Hvistendahl
Center of Mathematics for Applications [Oslo] [CMA]

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Ce document a été publié dans

Finite Volumes for Complex Applications V, Finite Volumes for Complex Applications V, Finite volumes for complex applications V, 2008-06, Aussois. 2008p. pp. 161-168

ISTE, London; John Wiley & Sons

Résumé en anglais

We point out a simple 2D formula to reconstruct the discrete gradient on a polygon from the values given at the vertices. Together with a finite volume kind divergence reconstruction, this discrete gradient can be used for discretization of various PDEs, such as fully nonlinear (or linear anisotropic) diffusion problems, starting from rather general meshes. Its key advantage is the discrete integration-by-parts formula, known as the discrete duality property. Our approach allows us to preserve the crucial properties of the continuous diffusion operators (such as the monotonicity, the coercivity, the variational structure) at the discrete level. Further, we apply the same formula in the context of 3D “double” schemes, in the spirit of [Hermeline 98, 07] and [Domelevo, Omnes 05]; we give the associated discrete duality formula. In the case of meshes with the orthogonality condition, we also give a discrete entropy dissipation formula. As an example, we obtain convergence of “double” finite volume discretizations to the entropy solution of a model doubly nonlinear hyperbolic-parabolic equation.< Réduire

Mots clés en anglais

Finite volume approximation

Discrete gradient

Discrete duality

DDFV

Consistency

Dimension three

Anisotropic elliptic problems

General mesh

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A gradient reconstruction formula for finite volume schemes and discrete duality

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Mots clés en anglais

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