On the Inf-Sup Stabillity of Crouzeix-Raviart Stokes Elements in 3D

04/29/2022

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by   Stefan Sauter, et al.

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We consider non-conforming discretizations of the stationary Stokes equation in three spatial dimensions by Crouzeix-Raviart type elements. The original definition in the seminal paper by M. Crouzeix and P.-A. Raviart in 1973 is implicit and also contains substantial freedom for a concrete choice. In this paper, we introduce canonical Crouzeix-Raviart basis functions in 3D in analogy to the 2D case in a fully explicit way. We prove that this canonical CrouzeixRaviart element for the Stokes equation is inf-sup stable for polynomial degree k =2 (quadratic velocity approximation). We identify spurious pressure modes for the conforming (k; k - 1) 3D Stokes element and show that these are eliminated by using the canonical Crouzeix-Raviart space.