A multigrid discretization scheme of discontinuous Galerkin method for the Steklov-Lamé eigenproblem

Liangkun Xu; Hai Bi; Liangkun Xu; Hai Bi

doi:10.3934/math.2023727

AIMS Mathematics

2023, Volume 8, Issue 6: 14207-14231. doi: 10.3934/math.2023727

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Research article

A multigrid discretization scheme of discontinuous Galerkin method for the Steklov-Lamé eigenproblem

Liangkun Xu ,
Hai Bi ^,

School of Mathematical Sciences, Guizhou Normal University, Universities Town, Huaxi District, Guiyang, Guizhou, China

Received: 06 January 2023 Revised: 01 April 2023 Accepted: 03 April 2023 Published: 17 April 2023
MSC : 65N25, 65N30

In this paper, for the Steklov-Lamé eigenvalue problem, we propose a multigrid discretization scheme of discontinuous Galerkin method based on the shifted-inverse iteration. Based on the existing a priori error estimates, we give the error estimates for the proposed scheme and prove that the resulting approximations can achieve the optimal convergence order when the mesh sizes fit into some relationships. Finally, we combine the multigrid scheme and adaptive procedure to present some numerical examples which indicate that our scheme are locking-free and efficient for computing Steklov-Lamé eigenvalues.
- Steklov-Lamé eigenvalues,
- discontinuous finite element,
- multigrid discretization,
- shifted-inverse iteration,
- adaptive computation
Citation: Liangkun Xu, Hai Bi. A multigrid discretization scheme of discontinuous Galerkin method for the Steklov-Lamé eigenproblem[J]. AIMS Mathematics, 2023, 8(6): 14207-14231. doi: 10.3934/math.2023727

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Abstract

In this paper, for the Steklov-Lamé eigenvalue problem, we propose a multigrid discretization scheme of discontinuous Galerkin method based on the shifted-inverse iteration. Based on the existing a priori error estimates, we give the error estimates for the proposed scheme and prove that the resulting approximations can achieve the optimal convergence order when the mesh sizes fit into some relationships. Finally, we combine the multigrid scheme and adaptive procedure to present some numerical examples which indicate that our scheme are locking-free and efficient for computing Steklov-Lamé eigenvalues.

References

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