Finite Element Spaces of Double Two-Forms With Polynomial Coefficients
Yakov Berchenko-Kogan, Lily DiPaulo
TL;DR
The paper tackles constructing higher-order finite element spaces for symmetric double 2-forms $\Lambda^{2,2}_{\mathrm{sym}}$ with polynomial coefficients, motivated by elasticity (TDNNS) and numerical relativity (Riemann curvature). It develops a self-contained approach that mirrors Li's Regge element framework but yields two distinct shape-function types, $\beta_{ijk}$ and $\gamma_{ijkl}$, associated with triangles and tetrahedra, respectively. A key theoretical result is the decomposition $\Lambda^{2,2}_{\mathrm{sym}} = \Lambda_0^{2,2} \oplus \Lambda^4$, with $\Lambda_0^{2,2}$ encoding the Bianchi identity constraints, and a geometrically decomposed basis for constant and polynomial coefficient spaces $\mathcal{P}_r\Lambda^{2,2}_0$. The authors provide explicit DOF counts per face and construct extension operators to assemble a global $\mathcal{P}_r\Lambda^{2,2}_0(\mathcal{T})$, enabling practical higher-order discretizations for applications in elasticity and numerical geometry/relativity.
Abstract
We develop finite element spaces of symmetric tensor products of two-forms with polynomial coefficients. In three dimensions, these give higher order finite element spaces of matrix fields with normal-normal continuity, which have applications to the TDNNS method for elasticity, for example. In general dimension, these spaces can be used to represent the Riemann curvature tensor in numerical relativity. In many ways, our methods parallel Li's work generalizing Regge calculus to higher order, as Regge elements can be thought of as symmetric tensor products of one-forms. However, whereas the constant coefficient Regge space has one shape function per edge, the constant coefficient space of double-forms in our paper has one shape function per triangle and two shape functions per tetrahedron, so we must address the fact that there are shape functions of two different types. Like Li, we obtain an explicit geometrically decomposed basis of shape functions.