Minimal Sets of Unisolvent Weights for High Order Whitney Forms on Simplices

Whitney forms—degree one trimmed polynomials—are a crucial tool for finite element analysis of electromagnetic problem. They not only induce several finite element methods, but they also bear interesting geometrical features. If, on the one hand, features of degree one elements are well understood, when it comes to higher degree elements one is forced to choose between an analytical approach and a geometric one, that is, the duality that holds for the lower degree gets lost. Using tools of finite element exterior calculus, we show a correspondence between the usual basis of a high order Whitney forms space and a subset of the weights, that is, degrees of freedom obtained by integration over subsimplices of the mesh.