Two general operations are proposed on finite element differential complexes on cubical meshes that can be used to construct and analyze sequences of “nonstandard” convergent methods. The first operation, called DoF-transfer, moves edge degrees of freedom to vertices in a way that reduces global degrees of freedom while increasing continuity order at vertices. The second operation, called serendipity, eliminates interior bubble functions and degrees of freedom locally on each element without affecting edge degrees of freedom. These operations can be used independently or in tandem to create nonstandard complexes that incorporate Hermite, Adini and “trimmed-Adini” elements. The resulting elements lead to convergent non-conforming methods for problems requiring stronger regularity and satisfy a discrete Korn inequality. Potential benefits of applying these elements to Stokes, biharmonic and elasticity problems are discussed.

中文翻译：

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立方网格上的非标准有限元 de Rham 复形

在立方网格上的有限元微分复形上提出了两种通用操作，可用于构造和分析“非标准”收敛方法的序列。第一个操作称为 DoF-transfer，它以一种降低全局自由度同时增加顶点连续性顺序的方式将边缘自由度移动到顶点。第二个操作称为 serendipity，在不影响边缘自由度的情况下，消除每个元素上局部的内部气泡函数和自由度。这些操作可以独立使用或串联使用，以创建包含 Hermite、Adini 和“trimmed-Adini”元素的非标准复合体。对于需要更强正则性和满足离散 Korn 不等式的问题，由此产生的元素导致收敛的非一致性方法。