## Section: New Results

### Advances in high order mixed finite elements for Maxwell's equations

Participant : Francesca Rapetti.

The implementation of high order curl- or div-conforming finite element spaces is quite delicate, especially in the three-dimensional case. I have worked on an implementation strategy, which has been applied in the open source finite element software FreeFem++. In particular, I have used the inverse of a generalized Vandermonde matrix to build a basis of generators in duality with the degrees of freedom, which then provides in FreeFem++ an easy-to-use but powerful interpolation operator. With Marcella Bonazzoli, now at the Inria Team DEFI in Saclay, I have carefully addressed the problem of applying the same Vandermonde matrix to possibly differently oriented tetrahedra of the mesh over the computational domain. [17]

High order mixed finite element spaces generally lack natural choices of bases but they do have spanning families. I have worked on these FEs for simplicial meshes and proven theoretically their effectiveness. I have also commented on some aspects of a new set of degrees of freedom, the so-called weights on the small simplices, to represent discrete functions in these spaces [11].