Workshop on Numerical Methods and Analysis in CFD - Abstract

Sauter, Stefan

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

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 our talk, 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 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 (k,k-1) 3D Stokes element and show that these are eliminated by using the canonical Crouzeix-Raviart space.