A posteriori error estimates of mixed discontinuous Galerkin method for the Stokes eigenvalue problem

In this paper, for the Stokes eigenvalue problem in $d$-dimensional case $(d=2,3)$, we present an a posteriori error estimate of residual type of the mixed discontinuous Galerkin finite element method using $P_k-P_{k-1}$ element $(k\ge 1)$. We give the a posteriori error estimators for approximate eigenpairs, prove their reliability and efficiency for eigenfunctions, and also analyze their reliability for eigenvalues. We implement adaptive calculation, and the numerical results confirm our theoretical predictions and show that our method can achieve the optimal convergence order $O(do{f}^{-2k\mathrm{/}d})$.