Spatial discretization with Finite Element method
The partial differential equations governing the fluid and solid dynamics are discretized using the finiteelement method. The meshes are made of triangles for twodimensional configurations and of tetrahedra for threedimensional configurations.
The finiteelement method is based on a derivation of the weak formulation of the governing equations. This weak formulation is then discretized using the free software FreeFem++ that enables a spatial discretization of partial differential equations using the Finite Element method. The incompressible NavierStokes equations are discretized using TaylorHood elements: the fluid velocity is discretized with continuous quadratic functions (P2) while the pressure is discretized with continuous linear functions (P1). To discretize the elasticity equations as a firstorder problem in time, the solid velocity field is introduced as an additional variable. The solid displacement and velocity are then discretized with continuous quadratic functions, so that the fluid and solid velocities are dfiscretized by continuous function of similar order at the interface. In the ALE formulation, the extension displacement field, introduced to handle the deformation of the fluid mesh, is also discretized with continuous quadraticelement (P2), so that the solid displacement and extension displacement are discretized at the interface with elements of same order. Two Lagrange multiplier fields are defined on the fluidsolid interface: one to enforce the continuity of the fluid and solid velocities at the fluidsolid interface, and one to enforce the continuity of the solid and extension displacement. These two Lagrange multiplier fields are discretized with continuous linear function (P1) for numerical stability reason.

ALE formulation  Lagrangian decomposition
The spatial discretization of the eigenvalue problem obtained when considering the decomposition of the Lagrangian fluid velocity field is
where the fluid, extension and solid unknows are
and ,are the two Lagrange multipliers discussed above. The fluidstructureextension matrix is composed of the ondiagonal matrices: A_{f} the Jacobian matrix of the fluid equations, A_{s} the discretize operator of the linear elastic equation and A_{e} the discretized extension operator. The offdiagonal matrices couple the fluid, extension and solid equations. In the first line (fluid equation) the matrix C_{fs} is a mass matrix defined on the fluidsolid boundary while the matrix C_{fe} represent the shape derivative of the fluid equation. In the extensnion equation, C_{es }is also a mass matrix defined on the fluidsolid boundary. Interestingly, the matrix on the right handside is not symmetric due to the presence of the matrix B_{fe. }The fluid mesh velocity would be defined as an additional variable, the matrix on the right handside would be symmetric.

ALE formulation  Eulerian decomposition
The decomposition of the Eulerian fluid velocity filed yields to the following discretized eigenvalue problem
where the extension displacement variable has been eliminated. Two new matrices appear in this formulation: the matrix T discretize the transpiration velocity operator on the lfuidsolid boundary while the matrix K_{0} discretizes the addedstiffness terms.