## Preprint 2011-24

## Gabriel N. Gatica, Norbert Heuer, Salim Meddahi:

### Coupling of mixed finite element and stabilized boundary element methods for a fluid-solid interaction problem in 3D

### Abstract:

We introduce and analyze a Mixed-FEM and BEM coupling for a three-dimensional fluid-solid interaction problem. The media are governed by the acoustic and elastodynamic equations in time-harmonic regime coupled with adequate transmission conditions posed on the interface between the two media. We employ a dual-mixed variational formulation in the solid, in which the Cauchy stress tensor and the rotation are the only unknowns, and use the exterior acoustic problem to deduce a nonlocal boundary condition for this problem. Actually, the variational formulation in the solid is completed with boundary integral equations relating the Cauchy data of the acoustic problem on the coupling interface. Both the trace and the normal derivative of the pressure appear as boundary variables in the global FEM-BEM formulation and the pressure in the exterior domain may be recovered by means of an integral representation formula. A crucial point in our formulation is the stabilization technique introduced by Hiptmair and co-authors to avoid the well known instability issue appearing in the BEM treatment of the exterior Helmholtz problem. The main novelty of this formulation, with respect to a previous approach, consists in reducing the computational domain to the solid media and providing a more accurate treatment of the far field effect. We show that a suitable decomposition of the space of stresses allows the application of the Babuska-Brezzi theory and the Fredholm alternative for concluding the solvability of the whole coupled problem. The unknowns of the solid are then approximated by the Arnold-Falk-Winther finite element of order $1$, which yields a conforming Galerkin scheme. The stability and convergence of the discrete method relies on a stable decomposition of the finite element space used to approximate the stress and also on a classical result on conforming Galerkin approximations for Fredholm operators of index zero.

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

**Gabriel N. GATICA, Norbert HEUER, Salim MEDDAHI: ***Coupling of mixed finite element and stabilized boundary element methods for a fluid-solid interaction problem in 3D*. Numerical Methods for Partial Differential Equations, vol. 30, 4, pp. 1211-1233, (2014).