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An efficient adaptive multigrid method for the elasticity eigenvalue problem

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Abstract

This paper aims to introduce a novel adaptive multigrid method for the elasticity eigenvalue problem. Different from the developing adaptive algorithms for the elasticity eigenvalue problem, the proposed approach transforms the elasticity eigenvalue problem into a series of boundary value problems in the adaptive spaces and some small-scale elasticity eigenvalue problems in a low-dimensional space. As our algorithm avoids solving large-scale elasticity eigenvalue problems, which is time-consuming, and the boundary value problem can be solved efficiently by the adaptive multigrid method, our algorithm can evidently improve the overall solving efficiency for the elasticity eigenvalue problem. Meanwhile, we present a rigorous theoretical analysis of the convergence and optimal complexity. Finally, some numerical experiments are presented to validate the theoretical conclusions and verify the numerical efficiency of our approach.

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Acknowledgements

We thank both referees for their valuable comments and helpful suggestions which improved this paper. The work of F. Xu was partially supported by NSFC 11801021. The work of Q. Huang was partially supported by NSFC 11971047. The work of M. Xie was partially supported by NSFC 12001402.

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Authors and Affiliations

  1. Beijing Institute for Scientific and Engineering Computing, Faculty of Science, Beijing University of Technology, Beijing, 100124, China

    Fei Xu & Qiumei Huang

  2. Center for Applied Mathematics, Tianjin University, Tianjin, 300072, China

    Manting Xie

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  1. Fei Xu
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Correspondence to Manting Xie.

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Xu, F., Huang, Q. & Xie, M. An efficient adaptive multigrid method for the elasticity eigenvalue problem. Bit Numer Math 62, 2005–2033 (2022). https://doi.org/10.1007/s10543-022-00939-7

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