Formulae and transformations for simplicial tensorial finite elements via polytopal templates

Abstract: 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.