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 $$\mathop {\mathbf {curl}}\nolimits $$ operator in a multiply connected domain and its numerical approximation by means of finite elements. We prove that the $$\mathop {\mathbf {curl}}\nolimits $$ operator is self-adjoint on suitable Hilbert spaces, all of them being contained in the space for which $$\mathop {\mathbf {curl}}\nolimits \varvec{v}\cdot \varvec{n}=0$$ on the boundary. Additional constraints 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 $$\varvec{v}$$ 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 assess the performance of the method.