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The present paper proposes an inf-sup stable divergence free virtual elementmethod and associated a priori, and a posteriori error analysis to approximatethe eigenvalues and eigenfunctions of the Stokes spectral problem in one shot.For the a priori analysis, we take advantage of the compactness of the solutionoperator to prove convergence of the eigenfunctions and double orderconvergence of eigenvalues. Additionally we also propose an a posterioriestimator of residual type, which we prove is reliable and efficient, in orderto perform adaptive refinements that allow to recover the optimal order ofconvergence for non smooth eigenfunctions. A set of representative numericalexamples investigates such theoretical results.

VEM approximation for the Stokes eigenvalue problem: a priori and a posteriori error analysis