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.

翻译：本文针对经典Crouzeix-Raviart有限元引入二次和三次多项式富集，旨在构建此类富集单元中的精确逼近。为实现这一目标，我们分别添加三个和七个加权线积分作为富集自由度。针对每种情形，我们给出这些增广单元良定义性的充要条件。为便于说明，我们采用通用方法定义双参数容许自由度族。此外，我们给出关联基函数的显式表达式，并基于所提出的容许单元引入新的二次和三次逼近算子。将富集方法的效率与三角形Crouzeix-Raviart单元进行比较。正如预期，数值结果展现出显著改进，验证了所开发富集策略的有效性。