A unified construction of $H(\textrm{div})$-conforming finite element tensors, including vector div element, symmetric div matrix element, traceless div matrix element, and in general tensors with linear constraints, is developed in this work. It is based on the geometric decomposition of Lagrange elements into bubble functions on each sub-simplex. Then the tensor at each sub-simplex is decomposed into the tangential and the normal component. The tangential component forms the bubble function space and the normal component characterizes the trace. A deep exploration on boundary degrees of freedom is presented for discovering various finite elements. The developed finite element spaces are $H(\textrm{div})$-conforming and satisfy the discrete inf-sup condition. An explicit basis of the constraint tensor space is also established.

翻译：本文发展了$H(\mathrm{div})$协调有限元张量的统一构造方法，涵盖向量散度元、对称散度矩阵元、无迹散度矩阵元以及具有线性约束的一般张量。该方法基于Lagrange元在每一子单形上的几何分解，将每个子单形上的张量分解为切向分量与法向分量。其中切向分量构成泡泡函数空间，法向分量则刻画迹函数。通过对边界自由度的深入探究，揭示了多种有限元的构造机理。所建立的有限元空间满足$H(\mathrm{div})$协调性并保持离散inf-sup条件。此外，本文还建立了约束张量空间的显式基函数。