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. 
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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://www.numdam.org/articles/10.1051/m2an/2016019/

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