Interior penalty discontinuous Galerkin methods for the velocity-pressure formulation of the Stokes spectral problem

In this paper, we analyze discontinuous Galerkin methods based in the interior penalty method in order to approximate the eigenvalues and eigenfunctions of the Stokes eigenvalue problem. The considered methods in this work are based in discontinuous polynomials approximations for the velocity field and the pressure fluctuation in two and three dimensions. The methods under consideration are symmetric and nonsymmetric, leading to variations on the associated matrices and, hence, on the computation of the eigenvalues and eigenfunctions where real and complex results may appear, depending on the choice of the method. We derive a convergence result and error estimates for the proposed methods, together with a rigorous computational analysis of the effects of the stabilization parameter in the appearance of spurious modes when the spectrum is computed, when symmetric and nonsymmetric methods are performed.