  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.
