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 for which curl v • n = 0 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 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 test the performance of the method.

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In this paper, we analyze a divergence-free finite element method to solve a fluid-structure interaction spectral problem in the three-dimensional case. The unknowns of the resulting formulation are the fluid and solid displacements and the fluid pressure on the interface separating both media. The resulting mixed eigenvalue problem is approximated by using appropriate basis of the divergence-free lowest order Raviart-Thomas elements for the fluid, piecewise linear elements for the solid and piecewise constant elements for the interface pressure. It is proved that eigenvalues and eigenfunctions are correctly approximated and some numerical results are reported in order to assess the performance of the method.