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

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. More specifically, the discrete function space is defined from cell and face unknowns by introducing a suitable piecewise
affine reconstruction on a pyramidal subdivision of the original mesh.

This subdivision is fictitious in the sense that the original mesh is the only one that needs to be manipulated by the end-user. Two applications are considered in which the discrete space plays an important role, namely (i) the design of a locking-free primal (as opposed to mixed) method for quasi-incompressible linear elasticity on
general polygonal meshes; (ii) the design of an inf-sup stable method for the Stokes equations on general polygonal or polyhedral meshes.
In both cases, the relation between the proposed method and classical finite volume as well as finite element methods on standard meshes is
investigated. Finally, it is shown how similar ideas can be exploited to mimic key properties of the lowest-order Raviart-Thomas space on
general polygonal or polyhedral meshes.

contatto:
luca.formaggia@polimi.it; marco.verani@polimi.it