Geometric Decompositions of $H(\textrm{div})$-conforming Finite Element Tensors, Part I: Vector and Matrix Functions

A unified construction of $H(\textrm{div})$-conforming finite elements, including vector element, symmetric matrix element, and traceless matrix element, is developed in this work. It is based on the geometric decomposition of Lagrange elements into bubble functions on each sub-simplex. Then the vector or matrix at each sub-simplex is decomposed into the tangential and the normal components. The tangential component forms the polynomial bubble function space and the normal component characterizes the trace of div operator. Some degrees of freedom on the normal component can be redistributed facewisely. Discrete inf-sup conditions are verified.