Dra. Camaño-Valenzuela, Jessika

We construct sets of basis functions of the space of divergence-free finite elements of Raviart–Thomas type in domains of general topology. Two different methods are presented: one using a suitable selection of the curls of Nédélec finite elements, the other based on an efficient algebraic procedure. The first approach looks to be more useful for numerical approximation, as the basis functions have a localized support.

This paper deals with high order Whitney forms. We define a canonical isomorphism between two sets of degrees of freedom. This allows to geometrically localize the classical degrees of freedom, the moments, over the elements of a simplicial mesh. With such a localization, it is thus possible to associate, even with moments, a graph structure relating a field with its potential.

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.