Convergent adaptive finite element methods for photonic \ud crystal applications

Article English OPEN
Giani, Stefano
  • Publisher: World Scientific

We prove the convergence of an adaptive finite element method for computing the band structure of 2D \ud periodic photonic crystals with or without compact defects in both the TM and TE polarization cases. These \ud eigenvalue problems involve non-coercive elliptic operators with discontinuous coefficients. The error analysis \ud extends the theory of convergence of adaptive methods for elliptic eigenvalue problems to photonic crystal \ud problems, and in particular deals with various complications which arise essentially from the lack of coercivity \ud of the elliptic operator with discontinuous coefficients. We prove the convergence of the adaptive method in \ud an oscillation-free way and with no extra assumptions on the initial mesh, beside the conformity and shape \ud regularity. Also we present and prove the convergence of an adaptive method to compute efficiently an entire \ud band in the spectrum. This method is guaranteed to converge to the correct global maximum and minimum \ud of the band, which is a very useful piece of information in practice. Our numerical results cover both the cases \ud of periodic structures with and without compact defects.
  • References (26)
    26 references, page 1 of 3

    [1] H. Ammari and F. Santosa, Guided waves in a photonic bandgap structure with a line defect, SIAM J. Appl. Math. 64 (2004) 2018-2033.

    [2] N. W. Ashcroft, N. D. Mermin Solid State Physics, (Brooks/Cole, 1976).

    [3] W. Axmann and P. Kuchment, An efficient finite element method for computing spectra of photonic and acoustic band-gap materials, J. Comput. Physics 150 (1999) 468-481.

    [4] C. Bernardi and R.Verfu¨rth, Adaptive finite element methods for elliptic equations with nonsmooth coefficients, Numer. Math. 85 (2000) 579-608.

    [5] D. Boffi, M. Conforti and L. Gastaldi, Modified edge finite elements for photonic crystals, Numer. Math. 105 (2006) 249-266.

    [6] Y. Cao, Z. Hou and Y. Liu, Convergence problem of plane-wave expansion method for photonic crystals, Physics Letters A 327 (2004) 247-253.

    [7] C. Carstensen and J. Gedicke, An oscillation-free adaptive FEM for symmetric eigenvalue problems, Preprint 489, DFG Research Center Matheon, Strasse des 17.Juni 136, D-10623 Berlin, 2008.

    [8] S. J. Cox and D. C. Dobson, Maximizing band gaps in two-dimensional photonic crystals, SIAM J. Appl. Math. 59 (1999) 2108-2120.

    [9] X. Dai, J. Xu and A. Zhou, Convergence and optimal complexity of adaptive finite element eigenvalue computations, Numer. Math. 110 (2008) 313-355.

    [10] D. C. Dobson, An Efficient Method for Band Structure Calculations in 2D Photonic Crystals, J. Comp. Phys. 149 (1999) 363-376.

  • Metrics
    0
    views in OpenAIRE
    0
    views in local repository
    35
    downloads in local repository

    The information is available from the following content providers:

    From Number Of Views Number Of Downloads
    Nottingham ePrints - IRUS-UK 0 35
Share - Bookmark