A Multigrid Discretization of Discontinuous Galerkin Method for the Stokes Eigenvalue Problem

In this paper, based on the velocity-pressure formulation of the Stokes eigenvalue problem in $d$-dimensional case $(d=2,3),$ we propose a multigrid discretization of discontinuous Galerkin method using $\mathbb{P}_k−\mathbb{P}_k−1$ element $(k≥1)$ and prove its a priori error estimate. We also 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 is efficient and can achieve the optimal convergence order $\mathcal{O}(do f^{ \frac{−2k}{d}} ).$