An abstract analysis framework for nonconforming approximations of the single phase Darcy equation

An abstra t analysis framework for non onforming approximations of diusion problems on general meshes Léo Agélas ?1 , Daniele A. Di Pietro †1 , Robert Eymard ‡2 , and Roland Masson 1 1 IFP, 1 & 4 av. du Bois-Préau 92852 Rueil-Malmaison Cedex (Fran e) 2 Université Paris-Est Marne-la-Vallée 5 bd. Des artes F-77454 Champs-sur-Marne Marne-la-Vallée Cedex 2 (Fran e) February 11, 2010 Abstra t In this work we propose a unied analysis framework en ompassing a wide range of non onforming dis retizations of anisotropi heterogeneous diusion operators on general meshes. The analysis relies on two dis rete fun tion analyti tools for pie ewise polynomial spa es, namely a dis rete Sobolev-Poin aré inequality and a dis rete Relli h theorem. The onver- gen e requirements are grouped into seven hypotheses, ea h of them har- a terizing one salient ingredient of the analysis. Finite volume s hemes as well as the most ommon dis ontinuous Galerkin methods are shown to t in the analysis. A new nite volume ell- entered method is also introdu ed. 1 Introdu tion Several methods have been developed through the years to solve the single phase Dar y equation, often of non- onforming type.

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• methods


• marne marne


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• finite volume


• fun tion


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• sobolev-poin aré inequality


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