Approximation of a structural acoustic vibration problem by hexahedral finite elements

A finite element method to compute elastoacoustic vibration modes in 3D problems on hexahedral meshes is analyzed. It is based on displacement formulations for fluid and solid. In order to avoid spurious modes, the discretization consists of lowest-order hexahedral Raviart-Thomas elements for the former coupled with classical trilinear isoparametric hexahedral elements for the latter. The kinematic constraint is weakly imposed and the meshes on the fluid and solid domains do not need to match on the common interface. Basic interpolation results are proved for the lowest-order hexahedral Raviart-Thomas elements. These results are used to prove convergence of the coupled finite element method, nonexistence of spurious modes, and optimal-order error estimates for eigenfunctions and eigenvalues, under the assumption that the meshes on the fluid domain are asymptotically parallelepiped. Numerical results showing sufficiency and need for this hypothesis are reported.