Coupling of mixed finite element and stabilized boundary element methods for a fluid-solid interaction problem in 3D (Q2875713)

The authors investigate the coupling of a mixed finite element method and a boundary element method for a three-dimensional time-harmonic fluid-solid interaction problem. They consider a formulation in which the Cauchy stress tensor and the rotation are the main variables in the elastic structure. The usual pressure formulation is used in the acoustic fluid. The mixed variational formulation in the solid is completed with boundary integral equations relating the Cauchy data of the acoustic problem on the coupling interface. A crucial point in the formulation is the stabilization technique introduced by \textit{A. Moiola} et al. [Z. Angew. Math. Phys. 62, No. 5, 809--837 (2011; Zbl 1263.35070)] to avoid the well-known instability appearing in the boundary element method treatment of the exterior Helmholtz problem. The main novelty of this formulation, with respect to a previous approach of the authors, consists in reducing the computational domain to the solid media and providing a more accurate treatment of the far field effect. It is shown that the continuous problem is well-posed. A conforming Galerkin method based on the lowest-order Arnold-Falk-Winther mixed finite element is proposed. The authors prove that the numerical scheme is convergent with optimal order.