Abstract
We consider discretizations of the stationary Stokes equation in three spatial dimensions by non-conforming Crouzeix-Raviart elements. The original definition in the seminal paper by M. Crouzeix and P.-A. Raviart in 1973 [Rev. Française Automat. Informat. Recherche Opérationnelle Sér. Rouge 7 (1973), pp. 33–75] is implicit and also contains substantial freedom for a concrete choice. In this paper, we introduce basic Crouzeix-Raviart spaces in 3D in analogy to the 2D case in a fully explicit way. We prove that this basic Crouzeix-Raviart element for the Stokes equation is inf-sup stable for polynomial degree k = 2 k=2 (quadratic velocity approximation). We identify spurious pressure modes for the conforming ( k , k − 1 ) \left ( k,k-1\right ) 3D Stokes element and show that these are eliminated by using the basic Crouzeix-Raviart space.
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