The conation and extrusion techniques were proposed by Bossavit (Math Comput Simul 80:1567–1577, 2010) for constructing $$(m+1)$$ -dimensional Whitney forms on prisms/cones from m-dimensional ones defined on the base shape. We combine the conation and extrusion techniques with the 2D polygonal $$H(\mathrm {div})$$ conforming finite element proposed by Chen and Wang (Math Comput 307:2053–2087, 2017), and construct the lowest-order $$H^1$$ , $$H(\mathrm {curl})$$ and $$H(\mathrm {div})$$ conforming elements on polygon-based prisms and cones. The elements have optimal approximation rates. Despite of the relatively sophisticated theoretical analysis, the construction itself is easy to implement. As an example, we provide a 100-line Matlab code for evaluating the shape functions of $$H^1$$ , $$H(\mathrm {curl})$$ and $$H(\mathrm {div})$$ conforming elements as well as their exterior derivatives on polygon-based cones. Note that all convex and some non-convex 3D polyhedra can be divided into polygon-based cones by connecting the vertices with a chosen interior point. Thus our construction also provides composite elements for all such polyhedra.

中文翻译：

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$$H^1$$, $$H(\mathrm {curl})$$ 和 $$H(\mathrm {div})$$ 基于多边形的棱镜和锥体上的符合元素

波萨维特 (Math Comput Simul 80:1567–1577, 2010) 提出了锥化和挤压技术，用于从定义在基础形状上的 m 维构造棱镜/锥上的 $$(m+1)$$ 维惠特尼形式. 我们将 conation 和挤压技术与 Chen 和 Wang (Math Comput 307:2053–2087, 2017) 提出的符合 2D 多边形 $$H(\mathrm {div})$$ 的有限元相结合，并构造最低阶 $ $H^1$$ 、 $$H(\mathrm {curl})$$ 和 $$H(\mathrm {div})$$ 符合基于多边形的棱镜和锥体上的元素。元素具有最佳逼近率。尽管理论分析相对复杂，但构建本身很容易实现。例如，我们提供了一个 100 行的 Matlab 代码来评估 $$H^1$$ 的形状函数，$$H(\mathrm {curl})$$ 和 $$H(\mathrm {div})$$ 符合元素及其在基于多边形的锥体上的外部导数。请注意，所有凸面和一些非凸面 3D 多面体都可以通过将顶点与选定的内点连接来划分为基于多边形的锥体。因此，我们的构造还为所有此类多面体提供了复合元素。