A Multigrid Discretization of Discontinuous Galerkin Method for the Stokes Eigenvalue Problem (Communications in Computational Physics, 2023)
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 ofdiscontinuous Galerkin method using $\mathbb{P}_k−\mathbb{P}_{k−1}$ element $(k≥1)$ and prove its a priorierror estimate. We also give the a posteriori error estimators for approximate eigenpairs, prove their reliability and efficiency for eigenfunctions, and also analyze theirreliability for eigenvalues. We implement adaptive calculation, and the numerical results confirm our theoretical predictions and show that our method is efficient and canachieve the optimal convergence order $\mathcal{O}(do f^{ \frac{−2k}{d}} ).$