Generalized finite element methods for quadratic eigenvalue problems
Abstract
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 then 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.
Cite
@article{arxiv.1510.05792,
title = {Generalized finite element methods for quadratic eigenvalue problems},
author = {Axel Målqvist and Daniel Peterseim},
journal= {arXiv preprint arXiv:1510.05792},
year = {2015}
}