A posteriori error estimates of mixed discontinuous Galerkin method for a class of Stokes eigenvalue problems

For a class of Stokes eigenvalue problems including the classical Stokes eigenvalue problem and the magnetohydrodynamic Stokes eigenvalue problem a residual type a posteriori error estimate of the mixed discontinuous Galerkin finite element method using ${\mathbb{P}}_k-{\mathbb{P}}_{k-1}$ element $(k\ge 1)$ is studied in this paper. The a posteriori error estimators for approximate eigenpairs are given. The reliability and efficiency of the posteriori error estimator for the eigenfunction are proved and the reliability of the estimator for the eigenvalue is also analyzed. The numerical results are provided to confirm the theoretical predictions and indicate that the method considered in this paper can reach the optimal convergence order $O(do{f}^{\frac{-2k}{d}})$.