Conforming hierarchical vector elements

This paper deals with the construction of hierarchical basis (scalar and vector) functions for solving Maxwell´s equations. A unified approach based on the de Rham diagram links the basis functions for potential (0 form), fields (1 forms), flux densities (2 forms), and charge density (3 form). Although, the construction of such hierarchical bases can be non-unique, the approach taken here sought to mimic the Helmholtz decomposition theorem by explicitly forming basis functions for the s of corresponding operators. In 1 forms, this is done through tree-cotree splitting of edge elements, for higher order terms come directly from taking the gradient of hierarchical scalar basis functions. Similarly, for 2 forms, the splitting of the lowest order Whitney 2 forms can be developed along the same concept, called the generalized tree-cotree splitting. It is the aim in this paper to outline the procedure which lies at the heart of hierarchical bases that are suitable for multigrid-type solvers, such as additive and multiplicative Schwarz methods.