High order differential form-based elements for the computation of electromagnetic field

The Whitney elements, discrete spaces based on differential forms, have proven their efficiency in electromagnetic field computation. However, they are built only in first order. This paper gives a general description of high order p-form (nodal, edge, facet and volume) elements. Their function spaces and the assignment of degrees of freedom on the simplexes are analyzed. General expressions of the basis functions are given. A comparison of several 2nd order elements is carried out. A procedure for the generation of the hierarchical basis of p-form elements is provided.