Generalized finite element methods for quadratic eigenvalue problems
Målqvist, Axel1 ; Peterseim, Daniel2
1 Department of Mathematics, Chalmers University of Technology and University of Gothenburg, Sweden.
2 Insitute for Numerical Simulation, Rheinische Friedrich-Wilhelms-Universität, Bonn, Germany.
ESAIM: Mathematical Modelling and Numerical Analysis , Tome 51 (2017) no. 1, pp. 147-163.

We consider a large-scale quadratic eigenvalue problem (QEP), formulated using P1 finite elements on a fine scale reference mesh. This model describes damped vibrations in a structural mechanical system. In particular we focus on problems with rapid material data variation, e.g., composite materials. We construct a low dimensional generalized finite element (GFE) space based on the localized orthogonal decomposition (LOD) technique. The construction involves the (parallel) solution of independent localized linear Poisson-type problems. The GFE space is used to compress the large-scale algebraic QEP to a much smaller one with a similar modeling accuracy. The small scale QEP can then be solved by standard techniques at a significantly reduced computational cost. We prove convergence with rate for the proposed method and numerical experiments confirm our theoretical findings.

Reçu le :
Accepté le :
DOI : 10.1051/m2an/2016019
Classification : 65N30, 65N25, 65N15
Mots-clés : Quadratic eigenvalue problem, finite element, localized orthogonal decomposition
Målqvist, Axel 1 ; Peterseim, Daniel 2

1 Department of Mathematics, Chalmers University of Technology and University of Gothenburg, Sweden.
2 Insitute for Numerical Simulation, Rheinische Friedrich-Wilhelms-Universität, Bonn, Germany. 
@article{M2AN_2017__51_1_147_0,
     author = {M\r{a}lqvist, Axel and Peterseim, Daniel},
     title = {Generalized finite element methods for quadratic eigenvalue problems},
     journal = {ESAIM: Mathematical Modelling and Numerical Analysis },
     pages = {147--163},
     publisher = {EDP-Sciences},
     volume = {51},
     number = {1},
     year = {2017},
     doi = {10.1051/m2an/2016019},
     mrnumber = {3601004},
     zbl = {1360.65270},
     language = {en},
     url = {http://archive.numdam.org/articles/10.1051/m2an/2016019/}
}
TY  - JOUR
AU  - Målqvist, Axel
AU  - Peterseim, Daniel
TI  - Generalized finite element methods for quadratic eigenvalue problems
JO  - ESAIM: Mathematical Modelling and Numerical Analysis 
PY  - 2017
SP  - 147
EP  - 163
VL  - 51
IS  - 1
PB  - EDP-Sciences
UR  - http://archive.numdam.org/articles/10.1051/m2an/2016019/
DO  - 10.1051/m2an/2016019
LA  - en
ID  - M2AN_2017__51_1_147_0
ER  - 
%0 Journal Article
%A Målqvist, Axel
%A Peterseim, Daniel
%T Generalized finite element methods for quadratic eigenvalue problems
%J ESAIM: Mathematical Modelling and Numerical Analysis 
%D 2017
%P 147-163
%V 51
%N 1
%I EDP-Sciences
%U http://archive.numdam.org/articles/10.1051/m2an/2016019/
%R 10.1051/m2an/2016019
%G en
%F M2AN_2017__51_1_147_0
Målqvist, Axel; Peterseim, Daniel. Generalized finite element methods for quadratic eigenvalue problems. ESAIM: Mathematical Modelling and Numerical Analysis , Tome 51 (2017) no. 1, pp. 147-163. doi : 10.1051/m2an/2016019. http://archive.numdam.org/articles/10.1051/m2an/2016019/

I. Babuška and J. Osborn, Eigenvalue problems. In Handbook of numerical analysis. Vol. II of Handb. Numer. Anal. North-Holland, Amsterdam (1991) 641–787. | MR | Zbl

I. Babuška and J. Osborn, Can a finite element method perform arbitrary badly? Math. Comp. 69 (1999) 443–462. | DOI | MR | Zbl

A. Bermúdez, R. G. Durán, R. Rodríguez and J. Solomin, Finite element analysis of a quadratic eigenvalue problem arising in dissipative acoustics. SIAM J. Numer. Anal. 38 (2000) 267–291. | DOI | MR | Zbl

T. Betcke, N. Higham, V. Mehrmann and C. Schröder, NLEVP: A Collection of Nonlinear Eigenvalue Problems. ACM Trans. Math. Softw. 39 (2013) 7:1–7:28. | DOI | MR | Zbl

D. Boffi, Finite element approximation of eigenvalue problems. Acta Numer. 19 (2010) 1–120. | DOI | MR | Zbl

D. Brown and D. Peterseim, A multiscale method for porous microstructures. Preprint (2014). | arXiv | MR

C. Carstensen and R. Verfürth, Edge residuals dominate a posteriori error estimates for low order finite element methods. SIAM J. Numer. Anal. 36 (1999) 1571–1587. | DOI | MR | Zbl

A.K. Chopra, Title Dynamics of Structures: Theory and Applications to Earthquake Engineering, 4th edition. Prentice Hall (2012).

P.G. Ciarlet, The finite element method for elliptic problems. Vol. 4 of Classics in Applied Mathematics. SIAM, Philadelphia (2002). | MR | Zbl

J. Descloux, N. Nassif and J. Rappaz, On spectral approximation. Part 1: The problem of convergence. RAIRO Anal. Numér. 12 (1978) 97–112. | DOI | Numdam | MR | Zbl

J. Descloux, N. Nassif and J. Rappaz, On spectral approximation. Part 2: Error estimates for the Galerkin method. RAIRO Anal. Numér. 12 (1978) 113–119. | DOI | Numdam | MR | Zbl

C. Engwer, P. Henning, A. Målqvist and D. Peterseim, Efficient implementation of the Localized Orthogonal Decomposition method. Preprint (2016). | arXiv | MR

D. Gallistl and D. Peterseim, Stable multiscale Petrov-Galerkin finite element method for high frequency acoustic scattering. Comp. Meth. Appl. Mech. Eng. 295 (2015) 1–17. | DOI | MR | Zbl

S. Hammarling, C.J. Munro and F. Tisseur, An algorithm for the complete solution of quadratic eigenvalue problems. ACM Trans. Math. Softw. 39 (2013) 18. | DOI | MR | Zbl

P. Henning and D. Peterseim, Oversampling for the multiscale finite element method. Multiscale Model. Simul. 11 (2013) 1149–1175. | DOI | MR | Zbl

P. Henning and A. Målqvist, Localized orthogonal decomposition techniques for boundary value problems. SIAM J. Sci. Comp. 36 (2014) A1609–A1634. | DOI | MR | Zbl

P. Henning, A. Målqvist and D. Peterseim, Two-level discretization techniques for ground state computations of Bose-Einstein condensates. SIAM J. Numer. Anal. 52 (2014) 1525–1550. | DOI | MR | Zbl

P. Henning, P. Morgenstern and D. Peterseim, Multiscale partition of unity. In Meshfree Methods for Partial Differential Equations VII, edited by M. Griebel and M.A. Schweitzer. Vol. 100 of Lect. Notes Comput. Sci. Eng. Springer International Publishing (2015) 185–204. | MR

T.-M. Hwang, W.-W. Lin and V. Mehrmann, Numerical solution of quadratic eigenvalue problems with structure-perserving methods. SIAM J. Sci. Comp. 24 (2003) 1283–1302. | DOI | MR | Zbl

T. Kato, Perturbation theory for linear operators. Springer-Verlag, New York (1976). | MR | Zbl

V. Mehrmann and H. Voss, Nonlinear eigenvalue problems: a challenge for modern eigenvalue methods. GAMM-Mitt. 27 (2004) 121–152. | DOI | MR | Zbl

A. Målqvist and A. Persson, Multiscale techniques for parabolic equations. Preprint (2015). | arXiv | MR

A. Målqvist and D. Peterseim, Localization of elliptic multiscale problems. Math. Comp. 83 (2014) 2583–2603. | DOI | MR | Zbl

A. Målqvist and D. Peterseim, Computation of eigenvalues by numerical upscaling. Numer. Math. 130 (2015) 337–361. | DOI | MR | Zbl

J.E. Osborn, Spectral approximation for compact operators. Math. Comp. 29 (1975) 712–725. | DOI | MR | Zbl

D. Peterseim, Eliminating the pollution effect in Helmholtz problems by local subscale correction. Preprint (2014). | arXiv | MR

D. Peterseim, Variational multiscale stabilization and the exponential decay of fine-scale correctors. Preprint (2015). | arXiv | MR

G. Strang and G. Fix, An analysis of the finite element method. | MR | Zbl

F. Tisseur and K. Meerbergen, A Survey of the Quadratic Eigenvalue Problem. SIAM Rev. 43 (2001) 235–286. | DOI | MR | Zbl

J. Weidmann, Strong operator convergence and spectral theory of ordinary differential operators. Univ. Iagel. Acta Math. 34 (1997) 153–163. | MR | Zbl

Cité par Sources :