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An extension of the Crouzeix-Raviart space to general meshes with application to quasi-incompressible linear elasticity and Stokes flow

Authors: Daniele A. Di Pietro and Simon Lemaire
Journal: Math. Comp. 84 (2015), 1-31
MSC (2010): Primary 65N08, 65N30; Secondary 74B05, 76D07
Published electronically: August 4, 2014
MathSciNet review: 3266951
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Abstract: In this work we introduce a discrete functional space on general polygonal or polyhedral meshes which mimics two important properties of the standard Crouzeix-Raviart space, namely the continuity of mean values at interfaces and the existence of an interpolator which preserves the mean value of the gradient inside each element. The construction borrows ideas from both Cell Centered Galerkin and Hybrid Finite Volume methods. The discrete function space is defined from cell and face unknowns by introducing a suitable piecewise affine reconstruction on a (fictitious) pyramidal subdivision of the original mesh. Two applications are considered in which the discrete space plays an important role, namely
\begin{inparaenum}[(i)]\item the design of a locking-free primal (as opposed to ... ... a Helmholtz decomposition of the right-hand side is available. \end{inparaenum}
The relation between the proposed methods and classical finite volume and finite element schemes on standard meshes is investigated. Finally, similar ideas are exploited to mimic key properties of the lowest-order Raviart-Thomas space on general polygonal or polyhedral meshes.

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Additional Information

Daniele A. Di Pietro
Affiliation: Université Montpellier 2, I3M, 34057 Montpellier CEDEX 5, France

Simon Lemaire
Affiliation: IFP Énergies nouvelles, Department of Applied Mathematics, 1 & 4 avenue de Bois-Préau, 92852 Rueil-Malmaison CEDEX, France

Received by editor(s): November 19, 2012
Received by editor(s) in revised form: June 5, 2013
Published electronically: August 4, 2014
Article copyright: © Copyright 2014 American Mathematical Society

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