Coupled Finite And Boundary Element Methods for Fluid-Solid Interaction Eigenvalue Problems

We analyze the approximation of a vibro-acoustic eigenvalue problem for an elastic body which is submerged in a compressible inviscid fluid in R3. As a model, the time-harmonic elastodynamic and the Helmholtz equation are used and are coupled in a strong sense via the standard transmission conditions on the interface between the solid and the fluid. Our approach is based on a coupling of the field equations for the solid with boundary integral equations for the fluid. The coupled formulation of the eigenvalue problem leads to a nonlinear eigenvalue problem with respect to the eigenvalue parameter since the frequency occurs nonlinearly in the used boundary integral operators for the Helmholtz equation. The nonlinear eigenvalue problem and its Galerkin discretization are analyzed within the framework of eigenvalue problems for Fredholm operator-valued functions where convergence is shown and error estimates are given. For the numerical solution of the discretized nonlinear matrix eigenvalue problem, the contour integral method is a reliable method which is demonstrated by some numerical examples.