On the inf-sup stability of Crouzeix-Raviart Stokes elements in 3D

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