Abstract
Studying classical wave propagation in periodic high contrast photonic and acoustic media naturally leads to the following spectral problem: -Δu = λεu, where ε(x) (the dielectric constant) is a periodic function that assumes a large value ε near a periodic graph Σ in ℝ2 and is equal to 1 otherwise. High contrast regimes lead to appearence of pseudo-differential operators of the Dirichlet-to-Neumann type on graphs. The paper contains a technique of approximating these pseudo-differential spectral problems by much simpler differential ones that can sometimes be resolved analytically. Numerical experiments show amazing agreement between the spectra of the pseudo-differential and differential problems.
Original language | English (US) |
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Pages (from-to) | 263-290 |
Number of pages | 28 |
Journal | Advances in Computational Mathematics |
Volume | 16 |
Issue number | 2-3 |
DOIs | |
State | Published - 2002 |
Keywords
- Differential operators on graphs
- Dirichlet-to-Neumann map
- Photonic bandgap
- Photonic crystal
- Pseudo-differential operators on graphs
- Spectrum
ASJC Scopus subject areas
- Computational Mathematics
- Applied Mathematics