Computation of 3-D electromagnetic field using differential forms based elements and dual formulations

The vector and scalar variables describing electromagnetic fields with different requirements of continuity can be identified to four different degrees of differential forms. The association of differential forms with finite elements leads to a set of differential forms based elements (Whitney elements); they are naturally adapted to the discretization of different vector and scalar variables. With the help of a Tonti diagram, Maxwell equations can be classified by two dual sequences together with the constitutive laws of materials. The application of Whitney elements to the two dual sequences leads to two dual approximation schemes. As an example, two dual formulations for eddy current computation using potential variables and the hybrid finite element—boundary element method are derived, where Whitney 3-D and 2-D elements are employed. A numerical application is given at the end of the paper, where the dual features of the two formulations are reported.