Two one-parameter families of nonconforming enrichments of the Crouzeix-Raviart finite element

In this paper, we introduce two one-parameter families of quadratic polynomial enrichments designed to enhance the accuracy of the classical Crouzeix--Raviart finite element. These enrichments are realized by using weighted line integrals as enriched linear functionals and quadratic polynomial functions as enrichment functions. To validate the effectiveness of our approach, we conduct numerical experiments that confirm the improvement achieved by the proposed method.