  In this paper, we introduce quadratic and cubic polynomial enrichments of the
classical Crouzeix--Raviart finite element, with the aim of constructing
accurate approximations in such enriched elements. To achieve this goal, we
respectively add three and seven weighted line integrals as enriched degrees of
freedom. For each case, we present a necessary and sufficient condition under
which these augmented elements are well-defined. For illustration purposes, we
then use a general approach to define two-parameter families of admissible
degrees of freedom. Additionally, we provide explicit expressions for the
associated basis functions and subsequently introduce new quadratic and cubic
approximation operators based on the proposed admissible elements. The
efficiency of the enriched methods is compared to the triangular
Crouzeix--Raviart element. As expected, the numerical results exhibit a
significant improvement, confirming the effectiveness of the developed
enrichment strategy.
