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.