Picewise uniform bases and energetic approach for discrete constitutive matrices in electromagnetic problems

In the paper we introduce piecewise uniform edge and face vector functions on a simplicial primal cell complex having geometric structure common to Whitney’s vector functions. We also introduce piecewise uniform bases functions in the barycentric dual complex, where the analogous of Whitney’s functions does not exist. By using these piecewise uniform bases functions and by exploiting an energetic approach, we construct symmetric positive definite constitutive matrices for discrete Maxwell’s equations. We also prove that these constitutive matrices are deeply related to those of the finite elements with Whitney’s bases functions.