Formulae and transformations for simplicial tensorial finite elements via polytopal templates

TL;DR

The paper introduces a unified polytopal-template framework to construct tensor-valued finite elements on simplices, enabling the Regge, HHJ, PS, Hu–Zhang, Hu–Ma–Sun, and GLS families via template tensors tied to simplex polytopes and an underlying $H^1$-conforming scalar space. It develops consistent mappings from reference to physical elements, including non-affine curvatures, and clarifies regularity implications through Reissner–Mindlin plate simulations, highlighting the advantages of minimal regularity and proper boundary treatment. The method demonstrates that tensorial bases can be generated from vectorial templates with appropriate Piola-type transformations, preserving required traces (tangential, normal, or mixed) across interfaces. By decoupling the tensor construction from the scalar subspace, the approach supports higher-order and heterogeneous $p$-refinement and naturally extends to curved or non-simplex domains, offering a flexible, scalable toolkit for complex elasticity and plate problems.

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-Schöberl, Hu-Zhang, Hu-Ma-Sun and Gopalakrishnan-Lederer-Schöberl 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.

Figures (12)

• Figure 1: Vertex base function $n \in \mathcal{V}_i^p(\Omega)$ of the vertex at the origin (a). Edge base function $n \in \mathcal{E}_j^p(\Omega)$ on the left-most edge (b). Face base function $n \in \mathcal{F}_k^p(\Omega)$ on the bottom face (c). Finally, a cell base function $n \in \mathcal{C}_{0123}^p(\Omega)$ (d). The functions belong to $\mathcal{CG}^p(\Omega)$ and are depicted on the reference tetrahedron $\Omega \subset \mathbb{R}^3$. \ref{['fn:ccby']}
• Figure 2: Derivation of template vectors for the remaining edges from the first definition via permutations of the reference triangle using covariant Piola mappings. The depiction exemplifies how the first vertex-edge template vector $\bm{\psi}$ corresponding to the vertex $v_0$ and edge $e_{01}$, is used to derive the vertex-edge template vectors of $v_0$-$e_{02}$ and $v_2$-$e_{12}$. Note that the permutation is always of the original reference triangle. \ref{['fn:ccby']}
• Figure 3: Template vectors for the construction of tangential-continuous base functions on the reference triangle on their corresponding polytope. Each vertex is endowed with two template vectors, one for each of its intersecting edges. Each edge is equipped with two template vectors, one for its tangent and one normal edge-cell vector for the cell. Finally, the cell is endowed with the Cartesian basis. \ref{['fn:ccby']}
• Figure 4: Template vectors for the construction of normal-continuous base functions on the reference triangle on their corresponding polytope. Each vertex is endowed with two template vectors, one for each of its intersecting edges and their respective normals. Each edge is equipped with two template vectors, one for its normal and one tangent edge-cell vector for the cell. Finally, the cell is endowed with the Cartesian basis. \ref{['fn:ccby']}
• Figure 5: Template vectors for the construction of tangential-continuous base functions on the reference tetrahedron on their corresponding polytopes. Only vectors on the visible sides of the tetrahedron are depicted. \ref{['fn:ccby']}

Theorems & Definitions (15)

• Definition 2.1: Polytopes on the reference simplex
• Definition 2.2: Simplex polytopal base functions
• Definition 2.3: Simplex polytopal scalar spaces
• Remark 2.1: Non-hierarchical bases with non-affine mappings
• Definition 3.1: Triangle Regge base functions
• Definition 3.2: Tetrahedral Regge base functions
• Definition 4.1: Hellan--Herrmann--Johnson base functions
• Definition 4.2: Tetrahedral Pechstein--Schöberl base functions
• Definition 4.3: Hu--Zhang base functions
• Definition 4.4: Hu--Ma--Sun base functions