A Hybrid Finite Volume Extension of the Crouzeix-Raviart Space To General Meshes: Applications To Quasi-Incompressible Linear Elasticity and Stokes Flows With Large Irrotational Body Forces

Abstract eng:
We introduce a discrete space on general polygonal or polyhedral meshes which mimics two important properties of the Crouzeix-Raviart space, namely the continuity of mean values at interfaces and the element-wise conservation of the mean value of the gradient for the interpolate of a function. The figure below presents one example of the kind of meshes we consider, featuring all polygonal elements between the triangle and the hexagon. The construction of the space borrows ideas from both cell centered Galerkin [2] and hybrid finite volume [3] 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 enduser. Two applications are considered in which the discrete space plays an important role, namely (i) the approximation of the linear elasticity equations in the quasi-incompressible limit, where we design a locking-free primal (as opposed to mixed) method on general polygonal meshes, inspired by the work of Brenner and Sung [1]; (ii) the discretization of Stokes flows on general polygonal or polyhedral meshes, where we propose an inf-sup stable scheme which yields an approximation of the velocity field unaffected by the presence of large irrotational body forces.

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