Discontinuous Galerkin Approximations for Computing Electromagnetic Bloch Modes in Photonic Crystals
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- 作者：Zhongjie Lu ; A. Cesmelioglu ; J. J. W. Van der Vegt…
- 关键词：Discontinuous Galerkin methods ; Mixed finite element methods ; Maxwell equations ; Discrete compactness property ; Eigenvalue problems ; Photonic crystals ; Band structure
- 刊名：Journal of Scientific Computing
- 出版年：2017
- 出版时间：February 2017
- 年：2017
- 卷：70
- 期：2
- 页码：922-964
- 全文大小：
- 刊物类别：Mathematics and Statistics
- 刊物主题：Algorithms; Computational Mathematics and Numerical Analysis; Appl.Mathematics/Computational Methods of Engineering; Theoretical, Mathematical and Computational Physics;
- 出版者：Springer US
- ISSN：1573-7691
- 卷排序：70
文摘
We analyze discontinuous Galerkin finite element discretizations of the Maxwell equations with periodic coefficients. These equations are used to model the behavior of light in photonic crystals, which are materials containing a spatially periodic variation of the refractive index commensurate with the wavelength of light. Depending on the geometry, material properties and lattice structure these materials exhibit a photonic band gap in which light of certain frequencies is completely prohibited inside the photonic crystal. By Bloch/Floquet theory, this problem is equivalent to a modified Maxwell eigenvalue problem with periodic boundary conditions, which is discretized with a mixed discontinuous Galerkin (DG) formulation using modified Nédélec basis functions. We also investigate an alternative primal DG interior penalty formulation and compare this method with the mixed DG formulation. To guarantee the non-pollution of the numerical spectrum, we prove a discrete compactness property for the corresponding DG space. The convergence rate of the numerical eigenvalues is twice the minimum of the order of the polynomial basis functions and the regularity of the solution of the Maxwell equations. We present both 2D and 3D numerical examples to verify the convergence rate of the mixed DG method and demonstrate its application to computing the band structure of photonic crystals.