Dra. Camaño-Valenzuela, Jessika

In this paper we are concerned with two topics: the formulation and analysis of the eigenvalue problem for the curlcurl operator in a multiply connected domain and its numerical approximation by means of finite elements. We prove that the curlcurl operator is self-adjoint on suitable Hilbert spaces, all of them being contained in the space for which curlvv⋅nn=0curl⁡vv⋅nn=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 vvvv 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.

In the published article, Figure 5 corresponds to an eigenfunction associated not with the first smallest positive eigenvalue. A correct eigenfunction of the latter is depicted in Fig. 1 here. Note that this eigenfunction is axisymmetric, as can be seen from Fig. 2 where its radial, azimuthal and vertical components are plotted on different meridian sections.

We propose and analyze an efficient algorithm for the computation of a basis of the space of divergence-free Raviart–Thomas finite elements. The algorithm is based on graph techniques. The key point is to realize that, with very natural degrees of freedom for fields in the space of Raviart–Thomas finite elements of degree r+1r+1 and for elements of the space of discontinuous piecewise polynomial functions of degree r≥0r≥0, the matrix associated with the divergence operator is the incidence matrix of a particular graph. By choosing a spanning tree of this graph, it is possible to identify an invertible square submatrix of the divergence matrix and to compute easily the moments of a field in the space of Raviart–Thomas finite elements with assigned divergence. This approach extends to finite elements of high degree the method introduced by Alotto and Perugia (Calcolo 36:233–248, 1999) for finite elements of degree one. The analyzed approach is used to construct a basis of the space of divergence-free Raviart–Thomas finite elements. The numerical tests show that the performance of the algorithm depends neither on the topology of the domain nor or the polynomial degree r.