Upper spectral bounds and a posteriori error analysis of several mixed finite element approximations for the Stokes eigenvalue problem

This paper discusses conforming mixed finite element approximations for the Stokes eigenvalue problem. Firstly, several mixed finite element identities are proved. Based on these identities, the following new results are given: (1) It is proved that the numerical eigenvalues obtained by mini-element, P (1)-P (1) element and Q (1)-Q (1) element approximate the exact eigenvalues from above. (2) As for the P (1)-P (1), Q (1)-Q (1) and Q (1)-P (0) element eigenvalues, the asymptotically exact a posteriori error indicators are presented. (3) The reliable and efficient a posteriori error estimator proposed by Verfurth is applied to mini-element eigenfunctions. Finally, numerical experiments are carried out to verify the theoretical analysis.