Adaptive finite element methods for computing band gaps in photonic crystals

Giani, Stefano; Graham, Ivan G.

doi:10.1007/s00211-011-0425-9

Cited by 36 publications

(40 citation statements)

References 37 publications

(50 reference statements)

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“…However in the DG case the result is not as straightforward as in the continuous Galerkin case [18]. The difficulty raises from the fact that the identity result in Lemma 4.5 holds for the extended form of the operator.…”

Section: Theorem 42 (Reliability For Eigenfunctions)mentioning

confidence: 99%

hp-Adaptive composite discontinuous Galerkin methods for elliptic eigenvalue problems on complicated domains

Giani

2015

Applied Mathematics and Computation

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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-pro t 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. In this paper we develop the a posteriori error estimation of hpadaptive discontinuous Galerkin composite finite element methods (DGFEMs) for the discretization of second-order elliptic eigenvalue problems. DGFEMs allow for the approximation of problems posed on computational domains which may contain local geometric features. The dimension of the underlying composite finite element space is independent of the number of geometric features. This is in contrast with standard finite element methods, as the minimal number of elements needed to represent the underlying domain can be very large and so the dimension of the finite element space. Computable upper bounds on the error for both eigenvalues and eigenfunctions are derived. Numerical experiments highlighting the practical application of the proposed estimators within an automatic hp-adaptive refinement procedure will be presented.

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Section: Theorem 42 (Reliability For Eigenfunctions)mentioning

confidence: 99%

hp-Adaptive composite discontinuous Galerkin methods for elliptic eigenvalue problems on complicated domains

Giani

2015

Applied Mathematics and Computation

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“…This spectrum-shifting trick has been used elsewhere (cf. [16]) for similar theoretical arguments, and we will see below that it has no effect on practical implementation.…”

Section: Theorem 26 Let B(· ·) Be the Any Of The Semi-definite Formentioning

confidence: 76%

Error control for $$hp$$ -adaptive approximations of semi-definite eigenvalue problems

2012

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The nal publication is available at Springer via http://dx.doi.org/10.1007/s00607-012-0260-6. Additional information: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-pro t 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. ERROR CONTROL FOR hp-ADAPTIVE APPROXIMATIONS OF SEMI-DEFINITE EIGENVALUE PROBLEMSSTEFANO GIANI, LUKA GRUBIŠIĆ, AND JEFFREY S. OVALLAbstract. We present reliable a-posteriori error estimates for hp-adaptive finite element approximations of semi-definite eigenvalue/eigenvector problems. Our model problems are motivated by the applications in photonic crystal eigenvalue computations. We present detailed numerical experiments confirming our theory and give several benchmark results which could serve the purpose of numerical testing of other adaptive procedures.

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“…[ does not. According to (14) it is trace type invariant and so it is meaningful to reduce it as a basis independent characteristic of the Riesz spaceŜ m . This is the basis of our basis independent claim for the resolution of eigenvalue multiplicity and for computing a subspace dependent mesh refinement.…”

Section: Be Eigenvectors and Ritz Vectors Which Satisfy Both The Asmentioning

confidence: 99%

“…[3,14]). Such applications are not directly addressed in this work, but many of their inherent computational challenges are captured in our model problem.…”

Section: Introductionmentioning

confidence: 99%

Benchmark results for testing adaptive finite element eigenvalue procedures part 2 (conforming eigenvector and eigenvalue estimates)

Giani

Grubišić

Ovall

2016

Applied Numerical Mathematics

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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-pro t 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 an hp-adaptive continuous Galerkin (hp-CG) method for approximating eigenvalues of elliptic operators, and demonstrate its utility on a collection of benchmark problems having features seen in many important practical applications-for example, high-contrast discontinuous coefficients giving rise to eigenfunctions with reduced regularity. In this continuation of our benchmark study, we concentrate on providing reliability estimates for assessing eigenfunction/invariant subspace error. In particular, we use these estimates to justify the observed robustness of eigenvalue error estimates in the presence of repeated or clustered eigenvalues. We also indicate a means for obtaining efficiency estimates from the available efficiency estimates for the associated boundary value (source) problem. As in the first part of the paper we provide extensive numerical tests for comparison with other high-order methods and also extend the list of analyzed benchmark problems.

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Adaptive finite element methods for computing band gaps in photonic crystals

Cited by 36 publications

References 37 publications

hp-Adaptive composite discontinuous Galerkin methods for elliptic eigenvalue problems on complicated domains

hp-Adaptive composite discontinuous Galerkin methods for elliptic eigenvalue problems on complicated domains

Error control for $$hp$$ -adaptive approximations of semi-definite eigenvalue problems

Benchmark results for testing adaptive finite element eigenvalue procedures part 2 (conforming eigenvector and eigenvalue estimates)

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