Guaranteed a posteriori bounds for eigenvalues and eigenvectors: Multiplicities and clusters
HTML articles powered by AMS MathViewer
- by Eric Cancès, Geneviève Dusson, Yvon Maday, Benjamin Stamm and Martin Vohralík;
- Math. Comp. 89 (2020), 2563-2611
- DOI: https://doi.org/10.1090/mcom/3549
- Published electronically: July 30, 2020
- HTML | PDF | Request permission
Abstract:
This paper presents a posteriori error estimates for conforming numerical approximations of eigenvalue clusters of second-order self-adjoint elliptic linear operators with compact resolvent. Given a cluster of eigenvalues, we estimate the error in the sum of the eigenvalues, as well as the error in the eigenvectors represented through the density matrix, i.e., the orthogonal projector on the associated eigenspace. This allows us to deal with degenerate (multiple) eigenvalues within the framework. All the bounds are valid under the only assumption that the cluster is separated from the surrounding smaller and larger eigenvalues; we show how this assumption can be numerically checked. Our bounds are guaranteed and converge with the same speed as the exact errors. They can be turned into fully computable bounds as soon as an estimate on the dual norm of the residual is available, which is presented in two particular cases: the Laplace eigenvalue problem discretized with conforming finite elements, and a Schrödinger operator with periodic boundary conditions of the form $-\Delta + V$ discretized with planewaves. For these two cases, numerical illustrations are provided on a set of test problems.References
- Randolph E. Bank, Luka Grubišić, and Jeffrey S. Ovall, A framework for robust eigenvalue and eigenvector error estimation and Ritz value convergence enhancement, Appl. Numer. Math. 66 (2013), 1–29. MR 3018645, DOI 10.1016/j.apnum.2012.11.004
- Norman W. Bazley and David W. Fox, Lower bounds for eigenvalues of Schrödinger’s equation, Phys. Rev. (2) 124 (1961), 483–492. MR 142898, DOI 10.1103/PhysRev.124.483
- Daniele Boffi, Ricardo G. Durán, Francesca Gardini, and Lucia Gastaldi, A posteriori error analysis for nonconforming approximation of multiple eigenvalues, Math. Methods Appl. Sci. 40 (2017), no. 2, 350–369. MR 3596542, DOI 10.1002/mma.3452
- Daniele Boffi, Dietmar Gallistl, Francesca Gardini, and Lucia Gastaldi, Optimal convergence of adaptive FEM for eigenvalue clusters in mixed form, Math. Comp. 86 (2017), no. 307, 2213–2237. MR 3647956, DOI 10.1090/mcom/3212
- Andrea Bonito and Alan Demlow, Convergence and optimality of higher-order adaptive finite element methods for eigenvalue clusters, SIAM J. Numer. Anal. 54 (2016), no. 4, 2379–2388. MR 3532806, DOI 10.1137/15M1036877
- Dietrich Braess, Veronika Pillwein, and Joachim Schöberl, Equilibrated residual error estimates are $p$-robust, Comput. Methods Appl. Mech. Engrg. 198 (2009), no. 13-14, 1189–1197. MR 2500243, DOI 10.1016/j.cma.2008.12.010
- Franco Brezzi and Michel Fortin, Mixed and hybrid finite element methods, Springer Series in Computational Mathematics, vol. 15, Springer-Verlag, New York, 1991. MR 1115205, DOI 10.1007/978-1-4612-3172-1
- Eric Cancès, Rachida Chakir, and Yvon Maday, Numerical analysis of the planewave discretization of some orbital-free and Kohn-Sham models, ESAIM Math. Model. Numer. Anal. 46 (2012), no. 2, 341–388. MR 2855646, DOI 10.1051/m2an/2011038
- Eric Cancès, Geneviève Dusson, Yvon Maday, Benjamin Stamm, and Martin Vohralík, Guaranteed and robust a posteriori bounds for Laplace eigenvalues and eigenvectors: conforming approximations, SIAM J. Numer. Anal. 55 (2017), no. 5, 2228–2254. MR 3702871, DOI 10.1137/15M1038633
- Eric Cancès, Geneviève Dusson, Yvon Maday, Benjamin Stamm, and Martin Vohralík, Guaranteed and robust a posteriori bounds for Laplace eigenvalues and eigenvectors: a unified framework, Numer. Math. 140 (2018), no. 4, 1033–1079. MR 3864709, DOI 10.1007/s00211-018-0984-0
- E. Cancés, G. Dusson, Y. Maday, B. Stamm, and M. Vohralík, Post-processing of the planewave approximation of Schrödinger equations. Part I: linear operators, IMA J. Numer. Anal. (2020), DOI 10.1093/imanum/draa044.
- Carsten Carstensen and Stefan A. Funken, Fully reliable localized error control in the FEM, SIAM J. Sci. Comput. 21 (1999/00), no. 4, 1465–1484. MR 1742328, DOI 10.1137/S1064827597327486
- Carsten Carstensen and Joscha Gedicke, Guaranteed lower bounds for eigenvalues, Math. Comp. 83 (2014), no. 290, 2605–2629. MR 3246802, DOI 10.1090/S0025-5718-2014-02833-0
- Carsten Carstensen, Joscha Gedicke, and Donsub Rim, Explicit error estimates for Courant, Crouzeix-Raviart and Raviart-Thomas finite element methods, J. Comput. Math. 30 (2012), no. 4, 337–353. MR 2965987, DOI 10.4208/jcm.1108-m3677
- P. G. Ciarlet, The Finite Element Method for Elliptic Problems, Studies in Mathematics and its Applications, vol. 4, North-Holland, Amsterdam, 1978. MR 1115237, DOI 10.1016/S0168-2024(08)70178-4
- Patrick Ciarlet Jr. and Martin Vohralík, Localization of global norms and robust a posteriori error control for transmission problems with sign-changing coefficients, ESAIM Math. Model. Numer. Anal. 52 (2018), no. 5, 2037–2064. MR 3891753, DOI 10.1051/m2an/2018034
- Xiaoying Dai, Lianhua He, and Aihui Zhou, Convergence and quasi-optimal complexity of adaptive finite element computations for multiple eigenvalues, IMA J. Numer. Anal. 35 (2015), no. 4, 1934–1977. MR 3407250, DOI 10.1093/imanum/dru059
- Philippe Destuynder and Brigitte Métivet, Explicit error bounds in a conforming finite element method, Math. Comp. 68 (1999), no. 228, 1379–1396. MR 1648383, DOI 10.1090/S0025-5718-99-01093-5
- Willy Dörfler, A convergent adaptive algorithm for Poisson’s equation, SIAM J. Numer. Anal. 33 (1996), no. 3, 1106–1124. MR 1393904, DOI 10.1137/0733054
- G. Dusson, Post-processing of the planewave approximation of Schrödinger equations. Part II: Kohn–sham models, IMA J Numer. Anal. (2020), DOI 10.1093/imanum/draa052.
- Alexandre Ern and Martin Vohralík, Polynomial-degree-robust a posteriori estimates in a unified setting for conforming, nonconforming, discontinuous Galerkin, and mixed discretizations, SIAM J. Numer. Anal. 53 (2015), no. 2, 1058–1081. MR 3335498, DOI 10.1137/130950100
- Alexandre Ern and Martin Vohralík, Stable broken $H^1$ and $H(\textrm {div})$ polynomial extensions for polynomial-degree-robust potential and flux reconstruction in three space dimensions, Math. Comp. 89 (2020), no. 322, 551–594. MR 4044442, DOI 10.1090/mcom/3482
- George E. Forsythe, Asymptotic lower bounds for the fundamental frequency of convex membranes, Pacific J. Math. 5 (1955), 691–702. MR 73048, DOI 10.2140/pjm.1955.5.691
- Dietmar Gallistl, Adaptive nonconforming finite element approximation of eigenvalue clusters, Comput. Methods Appl. Math. 14 (2014), no. 4, 509–535. MR 3259027, DOI 10.1515/cmam-2014-0020
- Dietmar Gallistl, An optimal adaptive FEM for eigenvalue clusters, Numer. Math. 130 (2015), no. 3, 467–496. MR 3347459, DOI 10.1007/s00211-014-0671-8
- Stefano Giani and Edward J. C. Hall, An a posteriori error estimator for $hp$-adaptive discontinuous Galerkin methods for elliptic eigenvalue problems, Math. Models Methods Appl. Sci. 22 (2012), no. 10, 1250030, 35. MR 2974168, DOI 10.1142/S0218202512500303
- P. Grisvard, Elliptic problems in nonsmooth domains, Monographs and Studies in Mathematics, vol. 24, Pitman (Advanced Publishing Program), Boston, MA, 1985. MR 775683
- Luka Grubišić and Jeffrey S. Ovall, On estimators for eigenvalue/eigenvector approximations, Math. Comp. 78 (2009), no. 266, 739–770. MR 2476558, DOI 10.1090/S0025-5718-08-02181-9
- Frédéric Hecht, Olivier Pironneau, Jacques Morice, Antoine Le Hyaric, and Kohji Ohtsuka, FreeFem++, http://www.freefem.org/ff++, 2012.
- Jun Hu, Yunqing Huang, and Qun Lin, Lower bounds for eigenvalues of elliptic operators: by nonconforming finite element methods, J. Sci. Comput. 61 (2014), no. 1, 196–221. MR 3254372, DOI 10.1007/s10915-014-9821-5
- Jun Hu, Yunqing Huang, and Quan Shen, The lower/upper bound property of approximate eigenvalues by nonconforming finite element methods for elliptic operators, J. Sci. Comput. 58 (2014), no. 3, 574–591. MR 3163260, DOI 10.1007/s10915-013-9744-6
- Tosio Kato, On the upper and lower bounds of eigenvalues, J. Phys. Soc. Japan 4 (1949), 334–339. MR 38738, DOI 10.1143/JPSJ.4.334
- P. Ladevèze and D. Leguillon, Error estimate procedure in the finite element method and applications, SIAM J. Numer. Anal. 20 (1983), no. 3, 485–509. MR 701093, DOI 10.1137/0720033
- Xuefeng Liu, A framework of verified eigenvalue bounds for self-adjoint differential operators, Appl. Math. Comput. 267 (2015), 341–355. MR 3399052, DOI 10.1016/j.amc.2015.03.048
- Xuefeng Liu and Fumio Kikuchi, Analysis and estimation of error constants for $P_0$ and $P_1$ interpolations over triangular finite elements, J. Math. Sci. Univ. Tokyo 17 (2010), no. 1, 27–78. MR 2676659
- Xuefeng Liu and Shin’ichi Oishi, Verified eigenvalue evaluation for the Laplacian over polygonal domains of arbitrary shape, SIAM J. Numer. Anal. 51 (2013), no. 3, 1634–1654. MR 3061473, DOI 10.1137/120878446
- Xuefeng Liu and Tomáš Vejchodský, Rigorous and fully computable a posteriori error bounds for eigenfunctions, 2019. Preprint arXiv:1904.07903, submitted for publication.
- FuSheng Luo, Qun Lin, and HeHu Xie, Computing the lower and upper bounds of Laplace eigenvalue problem: by combining conforming and nonconforming finite element methods, Sci. China Math. 55 (2012), no. 5, 1069–1082. MR 2912496, DOI 10.1007/s11425-012-4382-2
- W. Prager and J. L. Synge, Approximations in elasticity based on the concept of function space, Quart. Appl. Math. 5 (1947), 241–269. MR 25902, DOI 10.1090/S0033-569X-1947-25902-8
- Michael Reed and Barry Simon, Methods of modern mathematical physics. IV. Analysis of operators, Academic Press [Harcourt Brace Jovanovich, Publishers], New York-London, 1978. MR 493421
- J. E. Roberts and J.-M. Thomas, Mixed and hybrid methods, Handbook of numerical analysis, Vol. II, Handb. Numer. Anal., II, North-Holland, Amsterdam, 1991, pp. 523–639. MR 1115239
- Lloyd N. Trefethen and Timo Betcke, Computed eigenmodes of planar regions, Recent advances in differential equations and mathematical physics, Contemp. Math., vol. 412, Amer. Math. Soc., Providence, RI, 2006, pp. 297–314. MR 2259116, DOI 10.1090/conm/412/07783
- Li Wang, Ludovic Chamoin, Pierre Ladevèze, and Hongzhi Zhong, Computable upper and lower bounds on eigenfrequencies, Comput. Methods Appl. Mech. Engrg. 302 (2016), 27–43. MR 3461103, DOI 10.1016/j.cma.2016.01.002
- H. F. Weinberger, Upper and lower bounds for eigenvalues by finite difference methods, Comm. Pure Appl. Math. 9 (1956), 613–623. MR 84185, DOI 10.1002/cpa.3160090329
Bibliographic Information
- Eric Cancès
- Affiliation: CERMICS, Ecole des Ponts ParisTech, 6 & 8 Av. Pascal, 77455 Marne-la-Vallée, France; and Inria, 2 Rue Simone Iff, 75589 Paris, France
- Email: cances@cermics.enpc.fr
- Geneviève Dusson
- Affiliation: Université Bourgogne Franche-Comté, Laboratoire de Mathématiques de Besançon, UMR CNRS 6623, 16 route de Gray, 25030 Besançon, France
- ORCID: 0000-0002-7160-6064
- Email: genevieve.dusson@math.cnrs.fr
- Yvon Maday
- Affiliation: Sorbonne Université and Université de Paris, CNRS, Laboratoire Jacques-Louis Lions (LJLL), 75005 Paris, France; and Institut Universitaire de France, 75005 Paris, France
- MR Author ID: 117765
- Email: maday@ann.jussieu.fr
- Benjamin Stamm
- Affiliation: Center for Computational Engineering Science, RWTH Aachen University, Aachen, Germany
- MR Author ID: 824171
- ORCID: 0000-0003-3375-483X
- Email: stamm@mathcces.rwth-aachen.de
- Martin Vohralík
- Affiliation: Inria, 2 Rue Simone Iff, 75589 Paris, France; and CERMICS, Ecole des Ponts ParisTech, 6 & 8 Av. Pascal, 77455 Marne-la-Vallée, France
- ORCID: 0000-0002-8838-7689
- Email: martin.vohralik@inria.fr
- Received by editor(s): May 13, 2019
- Received by editor(s) in revised form: March 4, 2020
- Published electronically: July 30, 2020
- Additional Notes: Part of this work has been supported from French state funds managed by the CalSimLab LABEX and the ANR within the “Investissements d’Avenir” program (reference ANR-11-LABX-0037-01). The last author has also received funding from the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation program (Grant Agreement No. 647134 GATIPOR). Part of this work was supported by the French “Investissements d’Avenir” program, project ISITE-BFC (contract ANR-15-IDEX-0003).
The first and third authors acknowledge funding from PICS- CNRS, PHC PROCOPE 2017 (Project No. 37855ZK), and European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme (grant agreement No 810367). - © Copyright 2020 American Mathematical Society
- Journal: Math. Comp. 89 (2020), 2563-2611
- MSC (2010): Primary 35P15, 65N15, 65N25, 65N30
- DOI: https://doi.org/10.1090/mcom/3549
- MathSciNet review: 4136540