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Spectral element methods for eigenvalue problems based on domain decomposition. (English) Zbl 1496.65228

The authors are concerned with efficient methods to solve elliptic PDE eigenvalue problems by applying the spectral element discretization and domain decomposition techniques. In fact, they are focusing only on the Laplacian eigenvalue problem. Thus, they provide a rough introduction to the overlapping domain decomposition method and consider two types of methods to solve the discrete eigenvalue problems: the non-shifted scheme, represented by the preconditioned gradient method, and the shifted scheme. A convergence analysis in 1D, 2D and 3D cases is carried out along with the construction of discrete partition of unity function. It is shown that the convergence rate of eigenpairs could be independent of the number of spectral elements, the degrees of polynomials, and the number of subdomains. Some numerical experiments for all three cases mentioned above are displayed.
Reviewer’s remark. The errors computed in this paper do not seem to depend on the order \(k\) of the eigenpair in the ascending sequence of eigenvalues (see Lemma 2.1). This is a result in total contradiction with the enormous amount of numerical results that exist.

MSC:

65N35 Spectral, collocation and related methods for boundary value problems involving PDEs
65N25 Numerical methods for eigenvalue problems for boundary value problems involving PDEs
65N55 Multigrid methods; domain decomposition for boundary value problems involving PDEs
65N12 Stability and convergence of numerical methods for boundary value problems involving PDEs
65Y05 Parallel numerical computation

Software:

JDQZ; JDQR; lobpcg.m
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] J. An and J. Shen, A spectral-element method for transmission eigenvalue problems, J. Sci. Comput., 57 (2013), pp. 670-688. · Zbl 1292.65119
[2] D. N. Arnold, G. David, M. Filoche, D. Jerison, and S. Mayboroda, Computing spectra without solving eigenvalue problems, SIAM J. Sci. Comput., 41 (2019), pp. B69-B92, https://doi.org/10.1137/17M1156721. · Zbl 1420.81009
[3] D. N. Arnold, G. David, M. Filoche, D. Jerison, and S. Mayboroda, Localization of eigenfunctions via an effective potential, Comm. Partial Differential Equations, 44 (2019), pp. 1186-1216. · Zbl 1432.35061
[4] D. N. Arnold, G. David, D. Jerison, S. Mayboroda, and M. Filoche, Effective confining potential of quantum states in disordered media, Phys. Rev. Lett., 116 (2016), 056602.
[5] I. Babuška and J. Osborn, Finite element-Galerkin approximation of the eigenvalues and eigenvectors of selfadjoint problems, Math. Comp., 52 (1989), pp. 275-297. · Zbl 0675.65108
[6] Z. Bai, J. Demmel, J. Dongarra, A. Ruhe, and H. Van der Vorst, Templates for the Solution of Algebraic Eigenvalue Problems: A Practical Guide, Software Environ. Tools 11, SIAM, Philadelphia, 2000, https://doi.org/10.1137/1.9780898719581. · Zbl 0965.65058
[7] C. Bernardi and Y. Maday, Polynomial interpolation results in Sobolev spaces, J. Comput. Appl. Math., 43 (1992), pp. 53-80. · Zbl 0767.41001
[8] S. C. Brenner, A two-level additive Schwarz preconditioner for nonconforming plate elements, Numer. Math., 72 (1996), pp. 419-447. · Zbl 0855.73071
[9] M. A. Casarin, Quasi-optimal Schwarz methods for the conforming spectral element discretization, SIAM J. Numer. Anal., 34 (1997), pp. 2482-2502, https://doi.org/10.1137/S0036142995292281. · Zbl 0889.65123
[10] T. Chan and I. Sharapov, Subspace correction multi-level methods for elliptic eigenvalue problems, Numer. Linear Algebra Appl., 9 (1998), pp. 1-20. · Zbl 1071.65549
[11] J. A. Duersch, M. Shao, C. Yang, and M. Gu, A robust and efficient implementation of LOBPCG, SIAM J. Sci. Comput., 40 (2018), pp. C655-C676, https://doi.org/10.1137/17M1129830. · Zbl 1401.65038
[12] L. Evans, Partial Differential Equations, Grad. Stud. Math. 19, AMS, Providence, RI, 1998. · Zbl 0902.35002
[13] X. Gao, F. Liu, and A. Zhou, Three-scale finite element eigenvalue discretizations, BIT, 48 (2008), pp. 533-562. · Zbl 1157.65061
[14] Q. Gu and W. Gao, Inexact two-grid methods for eigenvalue problems, J. Comput. Math., (2015), pp. 557-575. · Zbl 1349.65602
[15] X. Han, Y. Li, H. Xie, and C. You, Local and parallel finite element algorithm based on multilevel discretization for eigenvalue problems., Int. J. Numer. Anal. Model., 13 (2016), pp. 73-89. · Zbl 1347.65171
[16] X. Hu and X. Cheng, Acceleration of a two-grid method for eigenvalue problems, Math. Comp., 80 (2011), pp. 1287-1301. · Zbl 1232.65141
[17] F. Hwang, Z. Wei, T. Huang, and W. Wang, A parallel additive Schwarz preconditioned Jacobi-Davidson algorithm for polynomial eigenvalue problems in quantum dot simulation, J. Comput. Phys., 229 (2010), pp. 2932-2947. · Zbl 1187.65034
[18] A. V. Knyazev, Preconditioned eigensolvers-an oxymoron?, Electron. Trans. Numer. Anal., 7 (1998), pp. 104-123. · Zbl 1053.65513
[19] A. V. Knyazev, Toward the optimal preconditioned eigensolver: Locally optimal block preconditioned conjugate gradient method, SIAM J. Sci. Comput., 23 (2001), pp. 517-541, https://doi.org/10.1137/S1064827500366124. · Zbl 0992.65028
[20] H. Li, W. Shan, and Z. Zhang, \({C}^1\)-conforming quadrilateral spectral element method for fourth-order equations, Commun. Appl. Math. Comput., 1 (2019), pp. 403-434. · Zbl 1449.65334
[21] H. Li and Z. Zhang, Efficient spectral and spectral element methods for eigenvalue problems of Schrödinger equations with an inverse square potential, SIAM J. Sci. Comput., 39 (2017), pp. A114-A140, https://doi.org/10.1137/16M1069596. · Zbl 1355.65150
[22] Q. Lin and H. Xie, A multi-level correction scheme for eigenvalue problems, Math. Comp., 84 (2015), pp. 71-88. · Zbl 1307.65159
[23] S. Lui, Domain decomposition methods for eigenvalue problems, J. Comput. Appl. Math., 117 (2000), pp. 17-34. · Zbl 0952.65085
[24] S. Maliassov, On the Schwarz alternating method for eigenvalue problems, Russian J. Numer. Anal. Math. Model., 13 (1998), pp. 45-56. · Zbl 0901.65065
[25] B. N. Parlett, The Symmetric Eigenvalue Problem, Classics Appl. Math. 7, SIAM, Philadelphia, 1980, https://doi.org/10.1137/1.9781611971163. · Zbl 0431.65017
[26] L. Pavarino, Additive Schwarz methods for the p-version finite element method, Numer. Math., 66 (1994), pp. 493-515. · Zbl 0791.65083
[27] L. Pavarino, Schwarz methods with local refinement for the p-version finite element method, Numer. Math., 69 (1994), pp. 185-211. · Zbl 0818.65111
[28] A. Schatz and L. Wahlbin, On the quasi-optimality in \(L_{\infty}\) of the \({\mathring{H}}^1\)-projection into finite element spaces, Math. Comp., 38 (1982), pp. 1-22. · Zbl 0483.65006
[29] W. Shan and H. Li, The triangular spectral element method for Stokes eigenvalues, Math. Comp., 86 (2017), pp. 2579-2611. · Zbl 1368.65226
[30] J. Shen, T. Tang, and L. Wang, Spectral Methods. Algorithms, Analysis and Applications, Springer Ser. Comput. Math. 41, Springer-Verlag, New York, 2011. · Zbl 1227.65117
[31] G. L. G. Sleijpen and H. A. Van der Vorst, A Jacobi-Davidson iteration method for linear eigenvalue problems, SIAM Rev., 42 (2000), pp. 267-293, https://doi.org/10.1137/S0036144599363084. · Zbl 0949.65028
[32] X.-C. Tai and M. Espedal, Applications of a space decomposition method to linear and nonlinear elliptic problems, Numer. Methods Partial Differential Equations, 14 (1998), pp. 717-737. · Zbl 0926.65063
[33] X.-C. Tai and M. Espedal, Rate of convergence of some space decomposition methods for linear and nonlinear problems, SIAM J. Numer. Anal., 35 (1998), pp. 1558-1570, https://doi.org/10.1137/S0036142996297461. · Zbl 0915.65063
[34] A. Toselli and O. Widlund, Domain Decomposition Methods-Algorithms and Theory, Springer Ser. Comput. Math. 34, Springer, Berlin, 2005. · Zbl 1069.65138
[35] W. Wang and X. Xu, A two-level overlapping hybrid domain decomposition method for eigenvalue problems, SIAM J. Numer. Anal., 56 (2018), pp. 344-368, https://doi.org/10.1137/16M1088302. · Zbl 1387.65116
[36] W. Wang and X. Xu, On the convergence of a two-level preconditioned Jacobi-Davidson method for eigenvalue problems, Math. Comp., 88 (2019), pp. 2295-2324. · Zbl 1417.65194
[37] J. Xu and A. Zhou, Local and parallel finite element algorithms based on two-grid discretizations, Math. Comp., 69 (2000), pp. 881-909. · Zbl 0948.65122
[38] J. Xu and A. Zhou, A two-grid discretization scheme for eigenvalue problems, Math. Comp., 70 (2001), pp. 17-25. · Zbl 0959.65119
[39] Y. Yang and H. Bi, Two-grid finite element discretization schemes based on shifted-inverse power method for elliptic eigenvalue problems, SIAM J. Numer. Anal., 49 (2011), pp. 1602-1624, https://doi.org/10.1137/100810241. · Zbl 1236.65143
[40] Z. Zhang, How many numerical eigenvalues can we trust?, J. Sci. Comput., 65 (2015), pp. 455-466. · Zbl 1329.65265
[41] T. Zhao, F. Hwang, and X. Cai, Parallel two-level domain decomposition based Jacobi-Davidson algorithms for pyramidal quantum dot simulation, Comput. Phys. Commun., 204 (2016), pp. 74-81. · Zbl 1378.65110
[42] J. Zhou, X. Hu, L. Zhong, S. Shu, and L. Chen, Two-grid methods for Maxwell eigenvalue problems, SIAM J. Numer. Anal., 52 (2014), pp. 2027-2047, https://doi.org/10.1137/130919921. · Zbl 1304.78015
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