Arbitrary order hierarchical vector bases for hexahedrons [FEM applications]

We present a clear and general method for constructing hierarchical vector bases of arbitrary polynomial degree for use in the finite element solution of Maxwell's equations. Our focus in this paper is on unstructured hexahedral grids with elements of higher order geometry (i.e. curved elements). Hierarchical bases enable p-refinement methods, where elements in a mesh can have different degrees of approximation, to be easily implemented. This can prove to be quite useful as sections of a computational domain can be selectively refined in order to achieve a greater error tolerance without the cost of refining the entire domain. We present a specific procedure for computing a hierarchical 1-form basis of arbitrary polynomial degree as well as the corresponding hierarchical degrees of freedom.