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We provide a simple condition, which is both necessary and sucient, that  guarantees the existence of an enriched Crouzeix-Raviart element. Our main  result shows that this latter can be easily expressed in terms of the  approximation error in a multivariate generalized trapezoidal type cubature  formula. Furthermore, we will derive simple explicit formulas for its  associated basis functions, and then prove how to use them to characterize  all admissible added degrees of freedom, that generate well defined enriched  Crouzeix-Raviart elements. We also show that the approximation error using  the proposed enriched element can be written as the error of the  (non-enriched) Crouzeix-Raviart element plus a perturbation that depends on  the enrichment function. Finally, we estimate the approximation error in L2  norm, with explicit constants in both two and three dimensions. A complement  to this result is also given for any dimension.

A simple necessary and sufficient condition for the enrichment of the Crouzeix-Raviart element