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A multiscale hp-FEM for 2D photonic crystal band

by H. Brandsmeier and K. Schmidt and Ch. Schwab

(Report number 2010-12)

Abstract
A Multiscale generalized $hp$-Finite Element Method (MSFEM) for time harmonic wave propagation in bands of locally periodic media of large, but finite extent, e.g., photonic crystal (PhC) bands, is presented. The method distinguishes itself by its size robustness, i.e., to achieve a prescribed error its computational effort does not depend on the number of periods. The proposed method shows this property for general incident fields, including plane waves incident at a certain angle to the infinite crystal surface, and at frequencies in and outside of the bandgap of the PhC. The proposed MSFEM is based on a precomputed problem adapted multiscale basis. This basis incorporates a set of complex Bloch modes, the eigenfunctions of the infinite PhC, which are modulated by macroscopic piecewise polynomials on a macroscopic FE mesh in the finite size photonic crystal domain of interest. The multiscale basis is shown to be efficient for finite PhC bands of any size, provided that boundary effects are resolved with a simple macroscopic boundary layer mesh. The MSFEM, constructed by combing the multiscale basis inside the crystal with some exterior discretisation, is interpreted as particular instance of generalised (gFEM) or extended (XFEM) finite element method. For the rapid evaluation of the matrix entries we introduce a size robust algorithm for integrals of quasi-periodic micro functions and polynomial macro functions. Size robustness of the present MSFEM in both, the number of basis functions and the computation time, is verified in extensive numerical experiments.

Keywords: Finite Photonic Crystals, Multiscale FEM, Fast Quadrature of quasi-periodic functions.

@Techreport{BSS10_426,
  author = {H. Brandsmeier and K. Schmidt and Ch. Schwab},
  title = {A multiscale hp-FEM for 2D photonic crystal band},
  institution = {Seminar for Applied Mathematics, ETH Z{\"u}rich},
  number = {2010-12},
  address = {Switzerland},
  url = {https://www.sam.math.ethz.ch/sam_reports/reports_final/reports2010/2010-12.pdf },
  year = {2010}
}

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