Systematic construction and performance comparison for higher order hierarchical tetrahedral vector elements

Nedelec's functional spaces of H
₁
(curl) hierarchical tetrahedral vector elements are analyzed systematically by using the method combined Nedelec constraints and complete polynomials. The relations between various vector elements and the functional spaces are validated. The results of a numerical experiment that investigates the resonant problem of a rectangular cavity, compare the performance of different hierarchical vector elements systematically (such as calculated accuracy, condition numbers, selectivity of the facet related basis functions), which are also applied to the eigen-solution of inhomogeneously-fllled cavities. The methods can be extended to the analysis of ultra higher order vector elements with any type effectively.