# American Institute of Mathematical Sciences

February  2016, 9(1): 235-253. doi: 10.3934/dcdss.2016.9.235

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

 1 CI2MA, Departamento de Ingeniería Matemática, Universidad de Concepción, Casilla 160-C, Concepción, Chile, Chile 2 Department of Mathematics, University of Maryland, College Park, MD 20742, United States

Received  September 2014 Revised  February 2015 Published  December 2015

A numerical scheme based on Nédélec 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 in the complement 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 formulation is discretized by Nédélec 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.
Citation: Eduardo Lara, Rodolfo Rodríguez, Pablo Venegas. Spectral approximation of the curl operator in multiply connected domains. Discrete & Continuous Dynamical Systems - S, 2016, 9 (1) : 235-253. doi: 10.3934/dcdss.2016.9.235
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##### References:
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