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

• Published in: Mathematics of computation 2015-01, Vol.84 (291), p.1-31
• Main Authors: Di Pietro, Daniele A., Lemaire, Simon
• Format: Article
• Language: English
• Subjects: Mathematics; Numerical Analysis; Research article
• ISSN: 0025-5718; 1088-6842
• DOI: 10.1090/S0025-5718-2014-02861-5

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 \inparaenum[(i)] the design of a locking-free primal (as opposed to mixed) method for quasi-incompressible planar elasticity on general polygonal meshes; the design of an inf-sup stable method for the Stokes equations on general polygonal or polyhedral meshes. In this context, we also propose a general modification, applicable to any suitable discretization, which guarantees that the velocity approximation is unaffected by the presence of large irrotational body forces provided a Helmholtz decomposition of the right-hand side is available. \endinparaenum 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.