We introduce and analyze a fully-mixed finite element method for a fluid-solid
interaction problem in 2D. The model consists of an elastic body which is subject to a
given incident wave that travels in the fluid surrounding it. Actually, the fluid is
supposed to occupy an annular region, and hence a Robin boundary condition imitating the
behavior of the scattered field at infinity is imposed on its exterior boundary, which is
located far from the obstacle. The media are governed by the elastodynamic and acoustic
equations in time-harmonic regime, respectively, and the transmission conditions are given
by the equilibrium of forces and the equality of the corresponding normal displacements.
We first apply dual-mixed approaches in both domains, and then employ the governing
equations to eliminate the displacement u of the solid and the pressure p
of the fluid. In addition, since both transmission conditions become essential, they are
enforced weakly by means of two suitable Lagrange multipliers. As a consequence, the
Cauchy stress tensor and the rotation of the solid, together with the gradient of
p and the traces of u and p on the boundary of the
fluid, constitute the unknowns of the coupled problem. Next, we show that suitable
decompositions of the spaces to which the stress and the gradient of p
belong, allow the application of the Babuška–Brezzi theory and the Fredholm alternative
for analyzing the solvability of the resulting continuous formulation. The unknowns of the
solid and the fluid are then approximated by a conforming Galerkin scheme defined in terms
of PEERS elements in the solid, Raviart–Thomas of lowest order in the fluid, and
continuous piecewise linear functions on the boundary. Then, the analysis of the discrete
method relies on a stable decomposition of the corresponding finite element spaces and
also on a classical result on projection methods for Fredholm operators of index zero.
Finally, some numerical results illustrating the theory are presented.