Discontinuous Galerkin Approximations for Computing Electromagnetic Bloch Modes in Photonic Crystals
Discontinuous Galerkin Approximations for Computing Electromagnetic Bloch Modes in Photonic Crystals
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DOI：
10.1007/s10915-016-0270-1
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发表时间：
2016-08
影响因子：
2.5
通讯作者：
Zhong-Lan Lu;A. Çesmelioglu;J. V. D. Vegt;Yan Xu
中科院分区：
文献类型：
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作者：
Zhong-Lan Lu;A. Çesmelioglu;J. V. D. Vegt;Yan Xu
我们分析了具有周期系数的麦克斯韦方程组的间断Galerkin有限元离散。这些方程用于模拟光子晶体中的光的行为，光子晶体是包含与光的波长相称的折射率的空间周期性变化的材料。根据几何形状、材料性质和晶格结构，这些材料表现出光子带隙，其中某些频率的光在光子晶体内被完全禁止。利用Bloch/Floquet理论，将该问题等价为一个具有周期边界条件的修正麦克斯韦特征值问题，并采用修正Nédélec基函数，采用混合间断Galerkin（DG）格式离散。我们还研究了另一种原始DG内部惩罚制定和比较这种方法与混合DG制定。为了保证数值谱的无污染性，我们证明了相应DG空间的离散紧性。数值特征值的收敛速度是多项式基函数阶数和麦克斯韦方程组解的正则性最小值的两倍。我们给出了二维和三维的数值例子来验证混合DG方法的收敛速度，并展示了它在计算光子晶体能带结构中的应用。
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.