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Volume 34, Issue 5
A Multigrid Discretization of Discontinuous Galerkin Method for the Stokes Eigenvalue Problem

Ling Ling Sun, Hai Bi & Yidu Yang

DOI: 10.4208/cicp.OA-2023-0027

Commun. Comput. Phys., 34 (2023), pp. 1391-1419.

Published online: 2023-12

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  • Abstract

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 $\mathcal{O}(do f^{ \frac{−2k}{d}} ).$

  • Keywords

Stokes eigenvalue problem, discontinuous Galerkin method, multigrid discretizations, adaptive algorithm.

  • AMS Subject Headings

65N25, 65N30

  • Copyright

COPYRIGHT: © Global Science Press

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@Article{CiCP-34-1391, author = {Sun , Ling LingBi , Hai and Yang , Yidu}, title = {A Multigrid Discretization of Discontinuous Galerkin Method for the Stokes Eigenvalue Problem}, journal = {Communications in Computational Physics}, year = {2023}, volume = {34}, number = {5}, pages = {1391--1419}, abstract = {

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 $\mathcal{O}(do f^{ \frac{−2k}{d}} ).$

}, issn = {1991-7120}, doi = {https://doi.org/10.4208/cicp.OA-2023-0027}, url = {http://global-sci.org/intro/article_detail/cicp/22257.html} }
TY - JOUR T1 - A Multigrid Discretization of Discontinuous Galerkin Method for the Stokes Eigenvalue Problem AU - Sun , Ling Ling AU - Bi , Hai AU - Yang , Yidu JO - Communications in Computational Physics VL - 5 SP - 1391 EP - 1419 PY - 2023 DA - 2023/12 SN - 34 DO - http://doi.org/10.4208/cicp.OA-2023-0027 UR - https://global-sci.org/intro/article_detail/cicp/22257.html KW - Stokes eigenvalue problem, discontinuous Galerkin method, multigrid discretizations, adaptive algorithm. AB -

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 $\mathcal{O}(do f^{ \frac{−2k}{d}} ).$

Ling Ling Sun, Hai Bi & Yidu Yang. (2023). A Multigrid Discretization of Discontinuous Galerkin Method for the Stokes Eigenvalue Problem. Communications in Computational Physics. 34 (5). 1391-1419. doi:10.4208/cicp.OA-2023-0027
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