  We present a priori and a posteriori error analysis of a Virtual Element
Method (VEM) to approximate the vibration frequencies and modes of an elastic
solid. We analyze a variational formulation relying only on the solid
displacement and propose an $H^1({\Omega})$-conforming discretization by means
of VEM. Under standard assumptions on the computational domain, we show that
the resulting scheme provides a correct approximation of the spectrum and prove
an optimal order error estimate for the eigenfunctions and a double order for
the eigenvalues. Since, the VEM has the advantage of using general polygonal
meshes, which allows implementing efficiently mesh refinement strategies, we
also introduce a residual-type a posteriori error estimator and prove its
reliability and efficiency. We use the corresponding error estimator to drive
an adaptive scheme. Finally, we report the results of a couple of numerical
tests that allow us to assess the performance of this approach.
