## Preprint 2019-02

## Ana Alonso-Rodriguez, Jessika Camaño, Eduardo De Los Santos, Rodolfo Rodríguez:

### Divergence-free finite elements for the numerical solution of a hydroelastic vibration problem

### Abstract:

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 displacements for the fluid and the solid, and the pressure of the fluid 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 efficiently approximated and some numerical results are presented in order to assess the performance of the method.

This preprint gave rise to the following definitive publication(s):

**Ana ALONSO-RODRIGUEZ, Jessika CAMAñO, Eduardo DE LOS SANTOS, Rodolfo RODRíGUEZ: ***Divergence-free finite elements for the numerical solution of a hydroelastic vibration problem*. Numerical Methods for Partial Differential Equations, vol. 39, pp. 163-186, (2023).