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Computation of the band structure of two-dimensional Photonic Crystals with hp Finite Elements

by K. Schmidt and P. Kauf

(Report number 2007-05)

Abstract
The band structure of 2D photonic crystals and their eigenmodes can be efficiently computed with the finite element method (FEM). For second order elliptic boundary value problems with piecewise analytic coefficients it is known that the solution converges extremly fast, i.e. exponentially, when using p-FEM for smooth and hp-FEM for polygonal interfaces and boundaries. In this article we discretise the variational eigenvalue problems for the transverse electric (TE) and transverse magnetic (TM) modes in scalar variables with quasi-periodic boundary conditions by means of p- and hp-FEM. Our computations show exponential convergence of the numerical eigenvalues for smooth and polygonal lines of discontinuity of dielectric material properties.

Keywords: hp-FEM, exponential convergence, corner singularities, photonic crystals, photonic band structure, quasi-periodic boundary condition

@Techreport{SK07_366,
  author = {K. Schmidt and P. Kauf},
  title = {Computation of the band structure of two-dimensional Photonic Crystals with hp Finite Elements},
  institution = {Seminar for Applied Mathematics, ETH Z{\"u}rich},
  number = {2007-05},
  address = {Switzerland},
  url = {https://www.sam.math.ethz.ch/sam_reports/reports_final/reports2007/2007-05.pdf },
  year = {2007}
}

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