## Preprint 2016-31

## Ana Alonso-Rodriguez, Jessika Camaño, Rodolfo Rodríguez, Alberto Valli, Pablo Venegas:

### Finite element approximation of the spectrum of the curl operator in a multiply-connected domain

### Abstract:

In this paper we are concerned with two topics: the formulation and analysis of the eigenvalue problem for the curl operator in a multiply-connected domain and its numerical approximation by means of finite elements. We prove that the curl operator is self-adjoint on suitable Hilbert spaces, all of them being contained in the space of vector fields for which the normal component of its curl vanishes on the boundary. Additional conditions must be imposed when the physical domain is not topologically trivial: we show that a viable choice is the vanishing of the line integrals of the vector field on suitable homological cycles lying on the boundary. A saddle-point variational formulation is devised and analyzed and a finite element numerical scheme is proposed. It is proved that eigenvalues and eigenfunctions are efficiently approximated and some numerical results are presented in order to test the performance of the method.

This preprint gave rise to the following definitive publication(s):

**Ana ALONSO-RODRIGUEZ, Jessika CAMAñO, Rodolfo RODRíGUEZ, Alberto VALLI, Pablo VENEGAS: ***Finite element approximation of the spectrum of the curl operator in a multiply-connected domain*. Foundations of Computational Mathematics, vol. 18, 6, pp. 1493-1533, (2018).