Finite volume schemes for diffusion equations: Introduction to and review of modern methods

Droniou, Jérôme

doi:10.1142/s0218202514400041

Cited by 260 publications

(195 citation statements)

References 114 publications

(92 reference statements)

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“…To satisfy this requirement as far as possible for the finite volume scheme we have generated the grid using the constrained Delaunay triangulation tool of the CGAL library. For more information on the finite volume method, the interested reader may consult [39], or the recent review [35]. We set homogeneous Dirichlet boundary conditions where ε can vary.…”

Section: Examplementioning

confidence: 99%

Mimetic finite difference approximation of flows in fractured porous media

et al. 2016

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Abstract.We present a possible framework for the numerical simulation of flow in fractured porous media that couples mimetic finite differences for the porous matrix with a finite volume scheme for the flow in the fractures. The resulting method is theoretically analyzed in the case of a single fracture. Moreover, several numerical experiments show the capability of the method to deal also with complicated networks of fractures. Thanks to the implementation of rather general coupling conditions, it encompasses both "conductive fractures", i.e., fractures with high permeability and "sealed fractures", i.e., fractures with low permeability which act as a flow barrier.Mathematics Subject Classification. 65N30, 35Q86, 76S05.

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Section: Examplementioning

confidence: 99%

Mimetic finite difference approximation of flows in fractured porous media

et al. 2016

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“…Although its ability to ensure stability, the classical upwind finite volume method does not permit to handle anisotropic diffusion even if the mesh verifies the orthogonality condition. Various "multi-point" schemes, where the approximation of the flux through an edge involves several scalar unknowns, have been proposed for anisotropic diffusion problems, see for example [22,18,14,2,15] for a detailed review of modern finite volume methods for diffusion equations. However, nonlinear corrections have been proposed in [11] in order to enforce the monotony, but no complete convergence proof have been provided for such methods yet.…”

Section: Introduction and Modelmentioning

confidence: 99%

Positive nonlinear CVFE scheme for degenerate anisotropic Keller-Segel system

Cancès¹,

Ibrahim²,

Saad³

2017

The SMAI Journal of Computational Mathematics

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Abstract. In this paper, a nonlinear control volume finite element (CVFE) scheme for a degenerate Keller-Segel model with anisotropic and heterogeneous diffusion tensors is proposed and analyzed. In this scheme, degrees of freedom are assigned to vertices of a primal triangular mesh, as in finite element methods. The diffusion term which involves an anisotropic and heterogeneous tensor is discretized on a dual mesh (Donald mesh) using the diffusion fluxes provided by the conforming finite element reconstruction on the primal mesh. The other terms are discretized using a nonclassical upwind finite volume scheme on the dual mesh. The scheme ensures the validity of the discrete maximum principle without any restriction on the transmissibility coefficients. The convergence of the scheme is proved under very general assumptions. Finally, some numerical experiments are carried out to prove the ability of the scheme to tackle degenerate anisotropic and heterogeneous diffusion problems over general meshes.Math. classification. 65N08, 65N30.

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“…Another emerging formalism that deserves to be mentioned is that of Compatible Discrete Operators (CDO) recently proposed by Bonelle and Ern [55]. We would also like to mention at this point the recent review by Droniou [56], which contains several complements with respect to the present work. In what follows we collectively refer to these methods as Variational Lowest-Order (VLO) methods.…”

Section: Da DI Pietro and M Vohralıḱ / A Review Of Recent Advancesmentioning

confidence: 91%

A Review of Recent Advances in Discretization Methods,a PosterioriError Analysis, and Adaptive Algorithms for Numerical Modeling in Geosciences

Pietro

Vohralı́k

2014

Oil Gas Sci. Technol. – Rev. IFP Energies nouvelles

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Re´sume´-Une revue des avance´es re´centes autour des me´thodes de discre´tisation, de l'analyse a posteriori, et des algorithmes adaptatifs pour la mode´lisation nume´rique en ge´osciences -Cet article de revue traite de deux the´matiques de recherche en ge´osciences qui ont connu d'importants de´veloppements au cours des dernie`res anne´es. Dans la premie`re partie, on conside`re un ingre´dient cle´pour la re´solution nume´rique du proble`me d'e´coulement de Darcy, a`savoir les sche´mas de discre´tisation des termes de diffusion sur des maillages polygonaux/ polye´driques ge´ne´raux. On pre´sente diffe´rents sche´mas et on discute en de´tail de leurs proprie´te´s nume´riques fondamentales telles que la stabilite´, la consistance et la robustesse. La deuxie`me partie de l'article est consacre´e au controˆle de l'erreur et a`l'adaptivite´pour des proble`mes mode`les en ge´osciences. On pre´sente des estimations a posteriori qui garantissent une borne supe´rieure de l'erreur totale et qui permettent d'identifier les diffe´rentes composantes d'erreur. Ces estimations sont utilise´es pour formuler des crite`res d'arreˆt adaptatifs pour des solveurs line´aires et non line´aires ainsi que pour ajuster le pas de temps et pour raffiner le maillage de fac¸on adaptative. Des essais nume´riques illustrent le caracte`re entie`rement adaptatif de tels algorithmes.Abstract -A Review of Recent Advances in Discretization Methods, a Posteriori Error Analysis, and Adaptive Algorithms for Numerical Modeling in Geosciences -Two research subjects in geosciences which lately underwent significant progress are treated in this review. In the first part, we focus on one key ingredient for the numerical approximation of the Darcy flow problem, namely the discretization of diffusion terms on general polygonal/polyhedral meshes. We present different schemes and discuss in detail their fundamental numerical properties such as stability, consistency, and robustness. The second part of the paper is devoted to error control and adaptivity for model problems in geosciences. We present the available a posteriori estimates guaranteeing the maximal overall error and show how the different error components can be identified. These estimates are used to formulate adaptive stopping criteria for linear and nonlinear solvers, time step choice adjustment, and adaptive mesh refinement. Numerical experiments illustrate such entirely adaptive algorithms.

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Finite volume schemes for diffusion equations: Introduction to and review of modern methods

Cited by 260 publications

References 114 publications

Mimetic finite difference approximation of flows in fractured porous media

Mimetic finite difference approximation of flows in fractured porous media

Positive nonlinear CVFE scheme for degenerate anisotropic Keller-Segel system

A Review of Recent Advances in Discretization Methods,a PosterioriError Analysis, and Adaptive Algorithms for Numerical Modeling in Geosciences

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