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

Abstract

Citation for published item:ingstr¤ omD gF nd qiniD F nd qrui § si¡ D vF @PHITA 9i0ient nd relile hpEpiw estimtes for qudrti eigenvlue prolems nd photoni rystl pplitionsF9D gomputers nd mthemtis with pplitionsFD UP @RAF ppF WSPEWUQF Further information on publisher's website: Use policyThe full-text may be used and/or reproduced, and given to third parties in any format or medium, without prior permission or charge, for personal research or study, educational, or not-for-prot purposes provided that:• a full bibliographic reference is made to the original source • a link is made to the metadata record in DRO • the full-text is not changed in any way The full-text must not be sold in any format or medium without the formal permission of the copyright holders.Please consult the full DRO policy for further details. AbstractWe 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.

Computation of scattering resonances in absorptive and dispersive media with applications to metal-dielectric nano-structures

“…to propose an a-priori hp-strategy for non-selfadjoint eigenvalue problems with piecewise constant coefficients based on an element-wise application of (i). The a-priori strategy for enriching the finite element space developed in this paper can in principle also be combined with an a-posteriori based strategy such as [27,28].…”

On spurious solutions in finite element approximations of resonances in open systems

“…Then, approximation theory of linear non-selfadjoint operators can be applied [34,15]. In particular, the results in this section show that the a-posteriori error estimations in [18] can be applied to the DtN-formulation of the resonance problem.…”

“…(iii) Use hp-adaptivity, e.g. a technique similar to [41] for PML and [18] for DtN, to reduce the errors in the target eigenvalues. (iv) Check that the new eigenpars significantly reduce the residual in the Lippmann-Schwinger equation.…”

Solving elliptic eigenproblems with adaptive multimesh hp-FEM

“…Conventional adaptive FE algorithms solve for n eigenpairs simultaneously, calling a generalized eigensolver after each mesh refinement step [7,5]. The calls to the generalized eigensolver on the sequence of meshes are all independent making it difficult to track eigenpairs through the sequence of meshes.…”