Efficient and reliable hp-FEM estimates for quadratic eigenvalue problems and photonic crystal applications
(2016) In Computers and Mathematics with Applications 72(4). p.952-973- Abstract
We present a-posteriori analysis of higher order finite element approximations (hp-FEM) for quadratic Fredholm-valued operator functions. Residual estimates for approximations of the algebraic eigenspaces are derived and we reduce the analysis of the estimator to the analysis of an associated boundary value problem. For the reasons of robustness we also consider approximations of the associated invariant pairs. We show that our estimator inherits the efficiency and reliability properties of the underlying boundary value estimator. As a model problem we consider spectral problems arising in analysis of photonic crystals. In particular, we present an example where a targeted family of eigenvalues cannot be guaranteed to be semisimple.... (More)
We present a-posteriori analysis of higher order finite element approximations (hp-FEM) for quadratic Fredholm-valued operator functions. Residual estimates for approximations of the algebraic eigenspaces are derived and we reduce the analysis of the estimator to the analysis of an associated boundary value problem. For the reasons of robustness we also consider approximations of the associated invariant pairs. We show that our estimator inherits the efficiency and reliability properties of the underlying boundary value estimator. As a model problem we consider spectral problems arising in analysis of photonic crystals. In particular, we present an example where a targeted family of eigenvalues cannot be guaranteed to be semisimple. Numerical experiments with hp-FEM show the predicted convergence rates. The measured effectivities of the estimator compare favorably with the performance of the same estimator on the associated boundary value problem. We also present a benchmark estimator, based on the dual weighted residual (DWR) approach, which is more expensive to compute but whose measured effectivities are close to one.
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- author
- Engström, Christian LU ; Giani, Stefano and Grubišić, Luka
- publishing date
- 2016-08-01
- type
- Contribution to journal
- publication status
- published
- subject
- keywords
- Invariant pairs, Nonlinear eigenvalue problems, Numerical methods
- in
- Computers and Mathematics with Applications
- volume
- 72
- issue
- 4
- pages
- 22 pages
- publisher
- Elsevier
- external identifiers
-
- scopus:84992530233
- ISSN
- 0898-1221
- DOI
- 10.1016/j.camwa.2016.06.001
- language
- English
- LU publication?
- no
- additional info
- Funding Information: The work of L.G. has been supported by the Croatian Science Foundation Grant Number 9345 . C.E. gratefully acknowledges the support of the Swedish Research Council Grant Number 621-2012-3863 under the project grant Spectral analysis and approximation theory for a class of operator functions. Publisher Copyright: © 2016 Elsevier Ltd
- id
- 092b7b2d-d6e6-40a3-a1cd-a71181cac1e8
- date added to LUP
- 2023-03-24 11:09:22
- date last changed
- 2023-03-24 14:02:40
@article{092b7b2d-d6e6-40a3-a1cd-a71181cac1e8,
abstract = {{<p>We present a-posteriori analysis of higher order finite element approximations (hp-FEM) for quadratic Fredholm-valued operator functions. Residual estimates for approximations of the algebraic eigenspaces are derived and we reduce the analysis of the estimator to the analysis of an associated boundary value problem. For the reasons of robustness we also consider approximations of the associated invariant pairs. We show that our estimator inherits the efficiency and reliability properties of the underlying boundary value estimator. As a model problem we consider spectral problems arising in analysis of photonic crystals. In particular, we present an example where a targeted family of eigenvalues cannot be guaranteed to be semisimple. Numerical experiments with hp-FEM show the predicted convergence rates. The measured effectivities of the estimator compare favorably with the performance of the same estimator on the associated boundary value problem. We also present a benchmark estimator, based on the dual weighted residual (DWR) approach, which is more expensive to compute but whose measured effectivities are close to one.</p>}},
author = {{Engström, Christian and Giani, Stefano and Grubišić, Luka}},
issn = {{0898-1221}},
keywords = {{Invariant pairs; Nonlinear eigenvalue problems; Numerical methods}},
language = {{eng}},
month = {{08}},
number = {{4}},
pages = {{952--973}},
publisher = {{Elsevier}},
series = {{Computers and Mathematics with Applications}},
title = {{Efficient and reliable hp-FEM estimates for quadratic eigenvalue problems and photonic crystal applications}},
url = {{http://dx.doi.org/10.1016/j.camwa.2016.06.001}},
doi = {{10.1016/j.camwa.2016.06.001}},
volume = {{72}},
year = {{2016}},
}