Efficient and reliable hp-FEM estimates for quadratic eigenvalue problems and photonic crystal applications

Engström, C.; Giani, S.; Grubišić, L.

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. 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.

Citation

Engström, C., Giani, S., & Grubišić, L. (2016). Efficient and reliable hp-FEM estimates for quadratic eigenvalue problems and photonic crystal applications. Computers and Mathematics with Applications, 72(4), 952-973. https://doi.org/10.1016/j.camwa.2016.06.001

• Journal Article Type: Article
• Acceptance Date: Jun 3, 2016
• Online Publication Date: Jun 22, 2016
• Publication Date: Aug 1, 2016
• Deposit Date: Jun 13, 2016
• Publicly Available Date: Jun 22, 2017
• Journal: Computers and Mathematics with Applications
• Print ISSN: 0898-1221
• Publisher: Elsevier
• Peer Reviewed: Peer Reviewed
• Volume: 72
• Issue: 4
• Pages: 952-973
• DOI: https://doi.org/10.1016/j.camwa.2016.06.001
• Public URL: https://durham-repository.worktribe.com/output/1381139