$H(\textrm{div})$-conforming Finite Element Tensors with Constraints

A unified construction of $H(\textrm{div})$-conforming finite element tensors, including vector element, symmetric matrix element, traceless 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. Each tensor at a sub-simplex is decomposed into tangential and normal components. The tangential component forms the bubble function space, while the normal component characterizes the trace. Some degrees of freedom can be redistributed to $(n-1)$-dimensional faces. The developed finite element spaces are $H(\textrm{div})$-conforming and satisfy the discrete inf-sup condition. Intrinsic bases of the constraint tensor space are also established.