We introduce a unified method for constructing the basis functions of a wide variety of partially continuous tensor-valued finite elements on simplices using polytopal templates. These finite element spaces are essential for achieving well-posed discretisations of mixed formulations of partial differential equations that involve tensor-valued functions, such as the Hellinger-Reissner formulation of linear elasticity. In our proposed polytopal template method, the basis functions are constructed from template tensors associated with the geometric polytopes (vertices, edges, faces etc.) of the reference simplex and any scalar-valued $H^1$-conforming finite element space. From this starting point we can construct the Regge, Hellan-Herrmann-Johnson, Pechstein-Sch\"oberl, Hu-Zhang, Hu-Ma-Sun and Gopalakrishnan-Lederer-Sch\"oberl elements. Because the Hu-Zhang element and the Hu-Ma-Sun element cannot be mapped from the reference simplex to a physical simplex via standard double Piola mappings, we also demonstrate that the polytopal template tensors can be used to define a consistent mapping from a reference simplex even to a non-affine simplex in the physical mesh. Finally, we discuss the implications of element regularity with two numerical examples for the Reissner-Mindlin plate problem.

翻译：我们提出一种统一方法，通过多面体模板构建单纯形上多种部分连续张量值有限元的基函数。这些有限元空间对于实现涉及张量值函数的偏微分方程混合形式（如线弹性理论的Hellinger-Reissner公式）的适定离散至关重要。在所提议的多面体模板方法中，基函数由与参考单纯形的几何多面体（顶点、边、面等）相关的模板张量以及任意标量值$H^1$协调有限元空间共同构造。基于此框架，可构建Regge、Hellan-Herrmann-Johnson、Pechstein-Schöberl、Hu-Zhang、Hu-Ma-Sun和Gopalakrishnan-Lederer-Schöberl等单元。由于Hu-Zhang单元和Hu-Ma-Sun单元无法通过标准双Piola映射从参考单纯形映射至物理单纯形，我们进一步证明多面体模板张量可用于定义从参考单纯形到物理网格中非仿射单纯形的一致映射。最后，通过两个数值算例（针对Reissner-Mindlin板问题）讨论单元正则性的影响。