A Domain Decomposition Method for Nonconforming Finite Element Approximations of Eigenvalue Problems

doi:10.1007/s42967-024-00372-3

Communications on Applied Mathematics and Computation ›› 2025, Vol. 7 ›› Issue (2): 606-636.doi: 10.1007/s42967-024-00372-3

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A Domain Decomposition Method for Nonconforming Finite Element Approximations of Eigenvalue Problems

Qigang Liang^1,2, Wei Wang³, Xuejun Xu^1,2

1 School of Mathematical Science, Tongji University, Shanghai 200092, China;
2 Key Laboratory of Intelligent Computing and Applications (Tongji University), Ministry of Education, Shanghai 200092, China;
3 LSEC, Institute of Computational Mathematics and Scientific/Engineering Computing, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing 100190, China

收稿日期:2023-04-23 修回日期:2023-12-29 接受日期:2024-01-06 出版日期:2025-06-20 发布日期:2025-04-21
通讯作者: Xuejun Xu,xxj@lsec.cc.ac.cn;Qigang Liang,qigang_liang@tongji.edu.cn;Wei Wang,ww@lsec.cc.ac.cn E-mail:xxj@lsec.cc.ac.cn;qigang_liang@tongji.edu.cn;ww@lsec.cc.ac.cn
基金资助:
The first author is supported by the China Postdoctoral Science Foundation (No. 2023M742662). The third author is supported by the National Natural Science Foundation of China (Grant Nos. 12071350 and 12331015).

A Domain Decomposition Method for Nonconforming Finite Element Approximations of Eigenvalue Problems

Qigang Liang^1,2, Wei Wang³, Xuejun Xu^1,2

1 School of Mathematical Science, Tongji University, Shanghai 200092, China;
2 Key Laboratory of Intelligent Computing and Applications (Tongji University), Ministry of Education, Shanghai 200092, China;
3 LSEC, Institute of Computational Mathematics and Scientific/Engineering Computing, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing 100190, China

Received:2023-04-23 Revised:2023-12-29 Accepted:2024-01-06 Online:2025-06-20 Published:2025-04-21
Supported by:
The first author is supported by the China Postdoctoral Science Foundation (No. 2023M742662). The third author is supported by the National Natural Science Foundation of China (Grant Nos. 12071350 and 12331015).

摘要/Abstract

摘要： Since the nonconforming finite elements (NFEs) play a significant role in approximating PDE eigenvalues from below, this paper develops a new and parallel two-level preconditioned Jacobi-Davidson (PJD) method for solving the large scale discrete eigenvalue problems resulting from NFE discretization of 2mth (m = 1, 2) order elliptic eigenvalue problems. Combining a spectral projection on the coarse space and an overlapping domain decomposition (DD), a parallel preconditioned system can be solved in each iteration. A rigorous analysis reveals that the convergence rate of our two-level PJD method is optimal and scalable. Numerical results supporting our theory are given.

关键词: PDE eigenvalue problems, Nonconforming finite elements (NFEs), Preconditioned Jacobi-Davidson (PJD) method, Overlapping domain decomposition (DD)

Abstract: Since the nonconforming finite elements (NFEs) play a significant role in approximating PDE eigenvalues from below, this paper develops a new and parallel two-level preconditioned Jacobi-Davidson (PJD) method for solving the large scale discrete eigenvalue problems resulting from NFE discretization of 2mth (m = 1, 2) order elliptic eigenvalue problems. Combining a spectral projection on the coarse space and an overlapping domain decomposition (DD), a parallel preconditioned system can be solved in each iteration. A rigorous analysis reveals that the convergence rate of our two-level PJD method is optimal and scalable. Numerical results supporting our theory are given.

Key words: PDE eigenvalue problems, Nonconforming finite elements (NFEs), Preconditioned Jacobi-Davidson (PJD) method, Overlapping domain decomposition (DD)

中图分类号:

65N30
65N55

Qigang Liang, Wei Wang, Xuejun Xu. A Domain Decomposition Method for Nonconforming Finite Element Approximations of Eigenvalue Problems[J]. Communications on Applied Mathematics and Computation, 2025, 7(2): 606-636.

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A Domain Decomposition Method for Nonconforming Finite Element Approximations of Eigenvalue Problems

A Domain Decomposition Method for Nonconforming Finite Element Approximations of Eigenvalue Problems

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