We provide a finite element discretization of $\ell$-form-valued $k$-forms on triangulation in $\mathbb{R}^{n}$ for general $k$, $\ell$ and $n$ and any polynomial degree. The construction generalizes finite element Whitney forms for the de~Rham complex and their higher-order and distributional versions, the Regge finite elements and the Christiansen--Regge elasticity complex, the TDNNS element for symmetric stress tensors, the MCS element for traceless matrix fields, the Hellan--Herrmann--Johnson (HHJ) elements for biharmonic equations, and discrete divdiv and Hessian complexes in [Hu, Lin, and Zhang, 2025]. The construction discretizes the Bernstein--Gelfand--Gelfand (BGG) diagrams. Applications of the construction include discretization of strain and stress tensors in continuum mechanics and metric and curvature tensors in differential geometry in any dimension.

翻译：我们针对一般的$k$、$\ell$和$n$以及任意多项式次数，在$\mathbb{R}^{n}$中的三角剖分上提供了$\ell$-形式值$k$-形式的有限元离散化。该构造推广了de~Rham复形的有限元Whitney形式及其高阶与分布版本、Regge有限元与Christiansen--Regge弹性复形、对称应力张量的TDNNS元、无迹矩阵场的MCS元、双调和方程的Hellan--Herrmann--Johnson（HHJ）元，以及[Hu, Lin, and Zhang, 2025]中的离散divdiv与Hessian复形。该构造离散化了Bernstein--Gelfand--Gelfand（BGG）图。该构造的应用包括任意维数连续介质力学中应变与应力张量的离散化，以及微分几何中度量与曲率张量的离散化。