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In this paper we present a mathematical and numerical analysis of aneigenvalue problem associated to the elasticity-Stokes equations stated in twoand three dimensions. Both problems are related through the Herrmann pressure.Employing the Babu\v ska--Brezzi theory, it is proved that the resultingcontinuous and discrete variational formulations are well-posed. In particular,the finite element method is based on general inf-sup stables pairs for theStokes system, such that, Taylor--Hood finite elements. By using a generalapproximation theory for compact operators, we obtain optimal order errorestimates for the eigenfunctions and a double order for the eigenvalues. Undermild assumptions, we have that these estimates hold with constants independentof the Lam\'e coefficient $\lambda$. In addition, we carry out the reliabilityand efficiency analysis of a residual-based a posteriori error estimator forthe spectral problem. We report a series of numerical tests in order to assessthe performance of the method and its behavior when the nearly incompressiblecase of elasticity is considered.

Finite element analysis of the nearly incompressible linear elasticity eigenvalue problem with variable coefficients