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Convergence of an approximate method for determining vibra- tional eigenpairs of an elastic solid containing an incompressible uid is exam- ined. The eld variables are solid displacement and uid pressure. We show that in suitable Sobolev spaces a variational formulation exists whose solution eigenvalues and eigenfunctions are identied with those of a compact operator. A nonconforming nite element approximation of this variational problem is described and optimal a priori error estimates are obtained for both the eigen- values and eigenfunctions. Accurate modeling of the interaction between liquid propellants and structures is important to the prediction of dynamic loads in launch vehicles and spacecraft. It is common practice to represent the motion of these systems by superposition of the response of a small number of approximate eigenfunctions, obtained from a nite element representation of an inviscid, incompressible uid coupled to a linear elastic structure. Previous authors have addressed both mathematical and computational aspects of this problem. The existence of real eigenpairs of a two-eld variational formulation, in the presence of a free surface, was established by Berger, Boujot and Ohayon (3) and Boujot (6). However, the Galerkin nite element discretization of uid pressure and structural displacement variables results in a nonsymmetric matrix eigenvalue problem, and direct solution of this nonsymmetric problem results in complex eigenvalues. Solution procedures which eliminate the uid pressure variable at the matrix level by static condensation to the uid boundary have been presented by Coppolino (8) and the text by Morand and Ohayon (13). These methods result in a symmetric matrix eigenvalue problem whose solution results in real eigenvalues. The statically condensed uid representation takes the form of an \added mass matrix". The existence of real eigenpairs of a continuous variational counterpart of the added mass formulation was shown by Bourquin (4), for the special case of a rigid tank with an elastic bottom. Although existence of solutions and computational procedures have been addressed, the rate of convergence of these approximate methods has not been completely established (13). Numerical results for typical vehicles show that, at the

Eigenvalue and eigenfunction error estimates for finite element formulations of linear hydroelasticity