We construct direct serendipity finite elements on general cuboidal hexahedra, which are H1-conforming and optimally approximate to any order. The new finite elements are direct in that the shape functions are directly defined on the physical element. Moreover, they are serendipity by possessing a minimal number of degrees of freedom satisfying the conformity requirement. Their shape function spaces consist of polynomials plus (generally nonpolynomial) supplemental functions, where the polynomials are included for the approximation property and supplements are added to achieve H1-conformity. The finite elements are fully constructive. The shape function spaces of higher order r≥3 are developed first, and then the lower order spaces are constructed as subspaces of the third order space. Under a shape regularity assumption, and a mild restriction on the choice of supplemental functions, we develop the convergence properties of the new direct serendipity finite elements. Numerical results with different choices of supplements are compared on two mesh sequences, one regularly distorted and the other one randomly distorted. They all possess a convergence rate that aligns with the theory, while a slight difference lies in their performance.

中文翻译：

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长方体六面体上的直接巧凑边点有限元


我们在一般的长方体六面体上构建直接巧凑有限元，这些有限元符合 H1 并且最接近任何阶次。新的有限元是直接的，因为形函数直接在物理元上定义。此外，它们通过拥有满足一致性要求的最小自由度数来获得偶然性。它们的形状函数空间由多项式和（通常是非多项式）补充函数组成，其中多项式包括近似属性，并添加补充以实现 H1 一致性。有限元是完全构造的。首先开发高阶 r≥3 的形状函数空间，然后将低阶空间构造为三阶空间的子空间。在形状规则性假设和对补充函数选择的温和限制下，我们开发了新的直接巧凑边点有限元的收敛特性。在两个网格序列上比较了不同补充剂选择的数值结果，一个规则扭曲，另一个随机扭曲。它们都具有与理论一致的收敛率，而它们的性能略有不同。