In this paper, the stabilized finite element approximation of the Stokes
eigenvalue problems is considered for both the two-field
(displacement-pressure) and the three-field (stress-displacement-pressure)
formulations. The method presented is based on a subgrid scale concept, and
depends on the approximation of the unresolvable scales of the continuous
solution. In general, subgrid scale techniques consist in the addition of a
residual based term to the basic Galerkin formulation. The application of a
standard residual based stabilization method to a linear eigenvalue problem
leads to a quadratic eigenvalue problem in discrete form which is physically
inconvenient. As a distinguished feature of the present study, we take the
space of the unresolved subscales orthogonal to the finite element space, which
promises a remedy to the above mentioned complication. In essence, we put
forward that only if the orthogonal projection is used, the residual is
simplified and the use of term by term stabilization is allowed. Thus, we do
not need to put the whole residual in the formulation, and the linear
eigenproblem form is recovered properly. We prove that the method applied is
convergent, and present the error estimates for the eigenvalues and the
eigenfunctions. We report several numerical tests in order to illustrate that
the theoretical results are validated