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Generalized finite element methods for quadratic eigenvalue problems

Journal article
Authors Axel Målqvist
Daniel Peterseim
Published in ESAIM: Mathematical Modelling and Numerical Analysis
Volume 51
Issue 1
Pages 147-163
ISSN 0764-583X
Publication year 2017
Published at Department of Mathematical Sciences, Applied Mathematics and Statistics
Pages 147-163
Language en
Links https://doi.org/10.1051/m2an/201601...
Keywords Finite element, Localized orthogonal decomposition, Quadratic eigenvalue problem
Subject categories Computational Mathematics, Applied mathematics

Abstract

© EDP Sciences, SMAI 2016. 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.

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