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 with that of the triangular Crouzeix–Raviart element. As expected, the numerical results exhibit a significant improvement, confirming the effectiveness of the developed enrichment strategy.

中文翻译：

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Crouzeix-Raviart 有限元的新二次和三次多项式丰富


在本文中，我们介绍了经典 Crouzeix-Raviart 有限元的二次和三次多项式丰富，目的是在此类丰富的元素中构造精确的近似值。为了实现这一目标，我们分别添加三个和七个加权线积分作为丰富的自由度。对于每种情况，我们提出了一个充分定义这些增强元素的充分必要条件。出于说明目的，我们然后使用通用方法来定义允许自由度的双参数族。此外，我们为相关的基函数提供了显式表达式，并随后基于所提出的允许元素引入了新的二次和三次近似算子。将富集方法的效率与三角形 Crouzeix-Raviart 元素的效率进行比较。正如预期的那样，数值结果显示出显着的改进，证实了所开发的富集策略的有效性。