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

Sauter, Stefan; Torres, Céline

doi:10.1090/mcom/3793

On the inf-sup stability of Crouzeix-Raviart Stokes elements in 3D
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Math. Comp. 92 (2023), 1033-1059 Request permission

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 $k=2$ (quadratic velocity approximation). We identify spurious pressure modes for the conforming $\left ( k,k-1\right )$ 3D Stokes element and show that these are eliminated by using the basic Crouzeix-Raviart space.

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