A two-level preconditioned Helmholtz-Jacobi-Davidson method for the Maxwell eigenvalue problem
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Abstract:
In this paper, based on a domain decomposition method, we propose an efficient two-level preconditioned Helmholtz-Jacobi-Davidson (PHJD) method for solving the algebraic eigenvalue problem resulting from the edge element approximation of the Maxwell eigenvalue problem. In order to eliminate the components in orthogonal complement space of the eigenvalue, we shall solve a parallel preconditioned system and a Helmholtz projection system together in fine space. After one coarse space correction in each iteration and minimizing the Rayleigh quotient in a small dimensional Davidson space, we finally get the error reduction of this two-level PHJD method as $\gamma =c(H)(1-C\frac {\delta ^{2}}{H^{2}})$, where $C$ is a constant independent of the mesh size $h$ and the diameter of subdomains $H$, $\delta$ is the overlapping size among the subdomains, and $c(H)$ decreasing monotonically to $1$ as $H\to 0$, which means the greater the number of subdomains, the better the convergence rate. Numerical results supporting our theory are given.References
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Additional Information
- Qigang Liang
- Affiliation: School of Mathematical Science, Tongji University, Shanghai 200092, People’s Republic of China
- ORCID: 0000-0001-8955-4920
- Email: qigang_liang@tongji.edu.cn
- Xuejun Xu
- Affiliation: School of Mathematical Science, Tongji University, Shanghai 200092, China; and Institute of Computational Mathematics, AMSS, Chinese Academy of Sciences, Beijing 100190, People’s Republic of China
- MR Author ID: 365400
- Email: xxj@lsec.cc.ac.cn
- Received by editor(s): June 30, 2020
- Received by editor(s) in revised form: April 14, 2021
- Published electronically: November 17, 2021
- Additional Notes: The authors were supported by National Natural Science Foundation of China (Grant Nos. 12071350, 11871272), Shanghai Municipal Science and Technology Major Project No. 2021SHZDZX0100, and Science and Technology Commission of Shanghai Municipality.
The second author is the corresponding author. - © Copyright 2021 American Mathematical Society
- Journal: Math. Comp. 91 (2022), 623-657
- MSC (2020): Primary 65N30, 65N55
- DOI: https://doi.org/10.1090/mcom/3702
- MathSciNet review: 4379971