Crouzeix-Raviart triangular elements are inf-sup stable

Carstensen, Carsten; Sauter, Stefan

doi:10.1090/mcom/3742

Crouzeix-Raviart triangular elements are inf-sup stable
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by Carsten Carstensen and Stefan A. Sauter;

Math. Comp. 91 (2022), 2041-2057

DOI: https://doi.org/10.1090/mcom/3742

Published electronically: June 8, 2022

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Abstract:

The Crouzeix-Raviart triangular finite elements are $\inf$-$\sup$ stable for the Stokes equations for any mesh with at least one interior vertex. This result affirms a conjecture of Crouzeix-Falk from 1989 for $p=3$. Our proof applies to any odd degree $p\ge 3$ and concludes the overall stability analysis: Crouzeix-Raviart triangular finite elements of degree $p$ in two dimensions and the piecewise polynomials of degree $p-1$ with vanishing integral form a stable Stokes pair for all positive integers $p$.

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