Communications on Applied Mathematics and Computation >

2025 , Vol. 7 >Issue 2: 606 - 636

DOI: https://doi.org/10.1007/s42967-024-00372-3

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

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  • 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 date: 2023-04-23

  Revised date: 2023-12-29

  Accepted date: 2024-01-06

  Online 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).

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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)

Cite this article

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 . DOI: 10.1007/s42967-024-00372-3

References

1. Adams, R.: Sobolev Spaces, Pure and Applied Mathematics, vol. 65. Academic Press [Harcourt Brace Jovanovich, Publishers], New York (1975)
2. Babuška, I., Osborn, J.: Finite element-Galerkin approximation of the eigenvalues and eigenvectors of selfadjoint problems. Math. Comput. 52(186), 275–297 (1989)
3. Boffi, D.: Finite element approximation of eigenvalue problems. Acta Numer. 19, 1–120 (2010)
4. Brenner, S.: A two-level additive Schwarz preconditioner for nonconforming plate elements. Numer. Math. 72(4), 419–447 (1996)
5. Brenner, S.: Two-level additive Schwarz preconditioners for nonconforming finite element methods. Math. Comput. 65(215), 897–921 (1996)
6. Chen, H., Xie, H., Fei, X.: A full multigrid method for eigenvalue problems. J. Comput. Phys. 322, 747–759 (2016)
7. Evans, L.: Partial Differential Equations, 2nd edn, vol. 19. American Mathematical Society, Providence (2010)
8. Hu, J., Huang, Y.: The correction operator for the canonical interpolation operator of the Adini element and the lower bounds of eigenvalues. Sci. China Math. 55(1), 187–196 (2012)
9. Hu, J., Huang, Y., Lin, Q.: Lower bounds for eigenvalues of elliptic operators: by nonconforming finite element methods. J. Sci. Comput. 61(1), 196–221 (2014)
10. Hu, J., Huang, Y., Shen, Q.: The lower/upper bound property of approximate eigenvalues by nonconforming finite element methods for elliptic operators. J. Sci. Comput. 58(3), 574–591 (2014)
11. Hu, J., Huang, Y., Shen, Q.: Constructing both lower and upper bounds for the eigenvalues of elliptic operators by nonconforming finite element methods. Numer. Math. 131(2), 273–302 (2015)
12. Liang, Q., Wang, W., Xu, X.: A two-level block preconditioned Jacobi-Davidson method for multiple and clustered eigenvalues of elliptic operators. arxiv: 2203. 06327 (2023)
13. Rannacher, R.: Nonconforming finite element methods for eigenvalue problems in linear plate theory. Numer. Math. 33(1), 23–42 (1979)
14. Sarkis, M.: Partition of unity coarse spaces and Schwarz methods with harmonic overlap. In: Recent Developments in Domain Decomposition Methods, pp. 77–94. Springer, Berlin, Heidelberg (2002)
15. Sleijpen, G.L.G., Van der Vorst, H.A.: A Jacobi-Davidson iteration method for linear eigenvalue problems. SIAM Rev. 42(2), 267–293 (2000)
16. Toselli, A., Widlund, O.: Domain decomposition methods-algorithms and theory. In: Springer Series in Computational Mathematics, vol. 34, Springer, Berlin, Heidelberg (2005)
17. Wang, W., Xu, X.J.: A two-level overlapping hybrid domain decomposition method for eigenvalue problems. SIAM J. Numer. Anal. 56(1), 344–368 (2018)
18. Wang, W., Xu, X.J.: On the convergence of a two-level preconditioned Jacobi-Davidson method for eigenvalue problems. Math. Comput. 88(319), 2295–2324 (2019)
19. Xie, H.: A multigrid method for eigenvalue problem. J. Comput. Phys. 274, 550–561 (2014)
20. Xie, H.: A type of multilevel method for the Steklov eigenvalue problem. IMA J. Numer. Anal. 34(2), 592–608 (2014)
21. Xu, F., Xie, H., Zhang, N.: A parallel augmented subspace method for eigenvalue problems. SIAM J. Sci. Comput. 42(5), A2655–A2677 (2020)
22. Xu, J., Zhou, A.: A two-grid discretization scheme for eigenvalue problems. Math. Comput. 70(233), 17–25 (2001)
23. Yang, Y.: A posteriori error estimates in Adini finite element for eigenvalue problems. J. Comput. Math. 18, 413–418 (2000)
24. Yang, Y.: Two-grid discretization schemes of the nonconforming FEM for eigenvalue problems. J. Comput. Math. 27(6), 748–763 (2009)
25. Yang, Y., Bi, H.: Two-grid finite element discretization schemes based on shifted-inverse power method for elliptic eigenvalue problems. SIAM J. Numer. Anal. 49(4), 1602–1624 (2011)
26. Yang, Y., Bi, H., Han, J., Yu. Y.: The shifted-inverse iteration based on the multigrid discretizations for eigenvalue problems. SIAM J. Sci. Comput. 37(6), A2583–A2606 (2015)
27. Yang, Y., Lin, Q., Bi, H., Li, Q.: Eigenvalue approximations from below using Morley elements. Adv. Comput. Math. 36(3), 443–450 (2012)
28. Yang, Y., Zhang, Z., Lin, F.: Eigenvalue approximation from below using non-conforming finite elements. Sci. China Math. 53(1), 137–150 (2010)
29. Zhai, Q., Xie, H., Zhang, R., Zhang, Z.: Acceleration of weak Galerkin methods for the Laplacian eigenvalue problem. J. Sci. Comput. 79(2), 914–934 (2019)
30. Zhao, T., Hwang, F.-N., Cai, X.-C.: A domain decomposition based Jacobi-Davidson algorithm for quantum dot simulation. In: Domain Decomposition Methods in Science and Engineering XXII, pp. 415–423. Springer, New York (2016)
31. Zhao, T., Hwang, F.-N., Cai, X.-C.: Parallel two-level domain decomposition based Jacobi-Davidson algorithms for pyramidal quantum dot simulation. Comput. Phys. Commun. 204, 74–81 (2016)
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