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

doi:10.1007/s42967-024-00372-3

Abstract

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)

CLC Number:

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.

TrendMD

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)

[1]	Xiu Ye, Shangyou Zhang. Constructing a CDG Finite Element with Order Two Superconvergence on Rectangular Meshes [J]. Communications on Applied Mathematics and Computation, 2025, 7(2): 411-425.
[2]	Shangyou Zhang. Conforming P₃ Divergence-Free Finite Elements for the Stokes Equations on Subquadrilateral Triangular Meshes [J]. Communications on Applied Mathematics and Computation, 2025, 7(2): 426-441.
[3]	Shangyou Zhang, Zhimin Zhang. A Locking-Free and Reduction-Free Conforming Finite Element Method for the Reissner-Mindlin Plate on Rectangular Meshes [J]. Communications on Applied Mathematics and Computation, 2025, 7(2): 470-484.
[4]	Xiaoying Dai, Yunyun Du, Fang Liu, Aihui Zhou. An Augmented Two-Scale Finite Element Method for Eigenvalue Problems [J]. Communications on Applied Mathematics and Computation, 2025, 7(2): 663-688.
[5]	Jonas Zeifang, Andrea Beck. A Low Mach Number IMEX Flux Splitting for the Level Set Ghost Fluid Method [J]. Communications on Applied Mathematics and Computation, 2023, 5(2): 722-750.
[6]	Yanping Lin, Xiu Ye, Shangyou Zhang. A Mixed Finite-Element Method on Polytopal Mesh [J]. Communications on Applied Mathematics and Computation, 2022, 4(4): 1374-1385.
[7]	Mahboub Baccouch. Convergence and Superconvergence of the Local Discontinuous Galerkin Method for Semilinear Second-Order Elliptic Problems on Cartesian Grids [J]. Communications on Applied Mathematics and Computation, 2022, 4(2): 437-476.
[8]	Gang Chen, Bernardo Cockburn, John R. Singler, Yangwen Zhang. Superconvergent Interpolatory HDG Methods for Reaction Difusion Equations II: HHO-Inspired Methods [J]. Communications on Applied Mathematics and Computation, 2022, 4(2): 477-499.
[9]	Xiaobing Feng, Thomas Lewis, Aaron Rapp. Dual-Wind Discontinuous Galerkin Methods for Stationary Hamilton-Jacobi Equations and Regularized Hamilton-Jacobi Equations [J]. Communications on Applied Mathematics and Computation, 2022, 4(2): 563-596.
[10]	Andreas Dedner, Tristan Pryer. Discontinuous Galerkin Methods for a Class of Nonvariational Problems [J]. Communications on Applied Mathematics and Computation, 2022, 4(2): 634-656.
[11]	Will Pazner, Tzanio Kolev. Uniform Subspace Correction Preconditioners for Discontinuous Galerkin Methods with hp-Refnement [J]. Communications on Applied Mathematics and Computation, 2022, 4(2): 697-727.
[12]	Vít Dolejší, Filip Roskovec. Goal-Oriented Anisotropic hp-Adaptive Discontinuous Galerkin Method for the Euler Equations [J]. Communications on Applied Mathematics and Computation, 2022, 4(1): 143-179.
[13]	Lina Zhao, Ming Fai Lam, Eric Chung. A Uniformly Robust Staggered DG Method for the Unsteady Darcy-Forchheimer-Brinkman Problem [J]. Communications on Applied Mathematics and Computation, 2022, 4(1): 205-226.
[14]	Aycil Cesmelioglu, Sander Rhebergen. A Compatible Embedded-Hybridized Discontinuous Galerkin Method for the Stokes-Darcy-Transport Problem [J]. Communications on Applied Mathematics and Computation, 2022, 4(1): 293-318.
[15]	Ahmed Al-Taweel, Yinlin Dong, Saqib Hussain, Xiaoshen Wang. A Weak Galerkin Harmonic Finite Element Method for Laplace Equation [J]. Communications on Applied Mathematics and Computation, 2021, 3(3): 527-544.