The Crouzeix--Raviart finite element method is widely recognized in the field of finite element analysis due to its nonconforming nature. The main goal of this paper is to present a general strategy for enhancing the Crouzeix--Raviart finite element using quadratic polynomial functions and three additional general degrees of freedom. To achieve this, we present a characterization result on the enriched degrees of freedom, enabling to define a new enriched finite element. This general approach is employed to introduce two distinct admissible families of enriched degrees of freedom. Numerical results demonstrate an enhancement in the accuracy of the proposed method when compared to the standard Crouzeix--Raviart finite element, confirming the effectiveness of the proposed enrichment strategy.

翻译：Crouzeix--Raviart有限元方法因其非协调特性在有限元分析领域广受认可。本文的主要目标是提出一种通用策略，通过使用二次多项式函数和三个附加的一般自由度来强化Crouzeix--Raviart有限元。为此，我们针对强化自由度提出一个刻画性结论，从而能够定义一种新的强化有限元。该通用方法被用于引入两个互异且可接受的强化自由度族。数值结果表明，与标准Crouzeix--Raviart有限元相比，所提方法的精度有所提升，证实了所提强化策略的有效性。