## Preprint 2014-22

## Eduardo Lara, Rodolfo Rodríguez, Pablo Venegas:

### Spectral approximation of the curl operator in multiply connected domains

### Abstract:

A numerical scheme based on Nedelec finite elements has been recently introduced to solve the eigenvalue problem for the curl operator in simply connected domains. This topological assumption is not just a technicality, since the eigenvalue problem is ill-posed on multiply connected domains, in the sense that its spectrum is the whole complex plane. However, additional constraints can be added to the eigenvalue problem in order to recover a well-posed problem with a discrete spectrum. Vanishing circulations on each non-bounding cycle of the domain have been chosen as additional constraints in this paper. A mixed weak formulation including a Lagrange multiplier (that turns out to vanish) is introduced and shown to be well-posed. This formulations is discretized by Nedelec elements, while standard finite elements are used for the Lagrange multiplier. Spectral convergence is proved as well as a priori error estimates. It is also shown how to implement this finite element discretization taking care of these additional constraints. Finally, a numerical test to assess the performance of the proposed methods is reported.

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

**Eduardo LARA, Rodolfo RODRíGUEZ, Pablo VENEGAS: ***Spectral approximation of the curl operator in multiply connected domains*. Discrete and Continuous Dynamical Systems - Series S, vol. 9, 1, pp. 235-253, (2016).