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

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.