Convergent adaptive finite element methods for photonic crystal applications
We prove the convergence of an adaptive finite element method for computing the band structure of 2D periodic photonic crystals with or without compact defects in both the TM and TE polarization cases. These eigenvalue problems involve non-coercive elliptic operators with discontinuous coefficie...
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| Format: | Article |
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World Scientific
2010
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| Online Access: | https://eprints.nottingham.ac.uk/1418/ |
| _version_ | 1848790602538287104 |
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| author | Giani, Stefano |
| author_facet | Giani, Stefano |
| author_sort | Giani, Stefano |
| building | Nottingham Research Data Repository |
| collection | Online Access |
| description | We prove the convergence of an adaptive finite element method for computing the band structure of 2D
periodic photonic crystals with or without compact defects in both the TM and TE polarization cases. These
eigenvalue problems involve non-coercive elliptic operators with discontinuous coefficients. The error analysis
extends the theory of convergence of adaptive methods for elliptic eigenvalue problems to photonic crystal
problems, and in particular deals with various complications which arise essentially from the lack of coercivity
of the elliptic operator with discontinuous coefficients. We prove the convergence of the adaptive method in
an oscillation-free way and with no extra assumptions on the initial mesh, beside the conformity and shape
regularity. Also we present and prove the convergence of an adaptive method to compute efficiently an entire
band in the spectrum. This method is guaranteed to converge to the correct global maximum and minimum
of the band, which is a very useful piece of information in practice. Our numerical results cover both the cases
of periodic structures with and without compact defects. |
| first_indexed | 2025-11-14T18:15:14Z |
| format | Article |
| id | nottingham-1418 |
| institution | University of Nottingham Malaysia Campus |
| institution_category | Local University |
| last_indexed | 2025-11-14T18:15:14Z |
| publishDate | 2010 |
| publisher | World Scientific |
| recordtype | eprints |
| repository_type | Digital Repository |
| spelling | nottingham-14182020-05-04T20:25:29Z https://eprints.nottingham.ac.uk/1418/ Convergent adaptive finite element methods for photonic crystal applications Giani, Stefano We prove the convergence of an adaptive finite element method for computing the band structure of 2D periodic photonic crystals with or without compact defects in both the TM and TE polarization cases. These eigenvalue problems involve non-coercive elliptic operators with discontinuous coefficients. The error analysis extends the theory of convergence of adaptive methods for elliptic eigenvalue problems to photonic crystal problems, and in particular deals with various complications which arise essentially from the lack of coercivity of the elliptic operator with discontinuous coefficients. We prove the convergence of the adaptive method in an oscillation-free way and with no extra assumptions on the initial mesh, beside the conformity and shape regularity. Also we present and prove the convergence of an adaptive method to compute efficiently an entire band in the spectrum. This method is guaranteed to converge to the correct global maximum and minimum of the band, which is a very useful piece of information in practice. Our numerical results cover both the cases of periodic structures with and without compact defects. World Scientific 2010 Article NonPeerReviewed Giani, Stefano (2010) Convergent adaptive finite element methods for photonic crystal applications. Mathematical Models and Methods in Applied Sciences (M3AS) . ISSN 0218-2025 (Submitted) http://www.worldscinet.com/m3as/mkt/editorial.shtml |
| spellingShingle | Giani, Stefano Convergent adaptive finite element methods for photonic crystal applications |
| title | Convergent adaptive finite element methods for photonic
crystal applications |
| title_full | Convergent adaptive finite element methods for photonic
crystal applications |
| title_fullStr | Convergent adaptive finite element methods for photonic
crystal applications |
| title_full_unstemmed | Convergent adaptive finite element methods for photonic
crystal applications |
| title_short | Convergent adaptive finite element methods for photonic
crystal applications |
| title_sort | convergent adaptive finite element methods for photonic
crystal applications |
| url | https://eprints.nottingham.ac.uk/1418/ https://eprints.nottingham.ac.uk/1418/ |