High-order discretized ACMS method for the simulation of finite-size two-dimensional photonic crystals

Abstract

The computational complexity and efficiency of the approximate mode component synthesis (ACMS) method is investigated for the two-dimensional heterogeneous Helmholtz equations, aiming at the simulation of large but finite-size photonic crystals. The ACMS method is a Galerkin method that relies on a non-overlapping domain decomposition and special basis functions defined based on the domain decomposition. While, in previous works, the ACMS method was realized using first-order finite elements, we use an underlying hp-finite element method. We study the accuracy of the ACMS method for different wavenumbers, domain decompositions, and discretization parameters. Moreover, the computational complexity of the method is investigated theoretically and compared with computing times for an implementation based on the open source software package NGSolve. The numerical results indicate that, for relevant wavenumber regimes, the size of the resulting linear systems for the ACMS method remains moderate, such that sparse direct solvers are a reasonable choice. Moreover, the ACMS method exhibits only a weak dependence on the selected domain decomposition, allowing for greater flexibility in its choice. Additionally, the numerical results show that the error of the ACMS method achieves the predicted convergence rate for increasing wavenumbers. Finally, to display the versatility of the implementation, the results of simulations of large but finite-size photonic crystals with defects are presented.

Figures (14)

• Figure 1: Left: Sparsity pattern of $\mathbb{S}_{A}$ for $I_{\mathcal{E}} = 32$ and $J = 36$ domains. The bandwidth is approximately $b_A \sim 3\sqrt{J} I_{\mathcal{E}}$, hence $b_A \sim 384$. Right: dof numbering of a crystal with $J = 4$ and $I_{\mathcal{E}} = 2$.
• Figure 2: Timings and reference lines for varying number of edge modes $I_{\mathcal{E}}$ (left) and number of sub domains $J$ (right).
• Figure 3: In \ref{['fig:CircularDomain']}, the domain $\Omega$ and the domain decomposition $\mathcal{D}$. In \ref{['fig:CircularDomain_Mesh']}, the corresponding coarse finite element mesh with $h=0.1$. The vertices in $\mathcal{V}$ are marked with dots.
• Figure 4: Example in \ref{['test:numex1']}, on a circular domain with decomposition $\mathcal{D}$ depicted in \ref{['fig:CircularDomain']}. The $L^2$--relative error $\| u - u^{h,p}_A \|_{L^2(\Omega)}/\| u\|_{L^2(\Omega)}$ is computed against the exact solution for a fixed mesh size and number of edge modes. We choose $h = 0.025$ and $I_{\mathcal{E}} = 32$ (left), $h = 0.0125$ and $I_{\mathcal{E}} = 64$ (middle), $h=0.00625$ and $I_{\mathcal{E}} = 128$ (right). Test with increasing degree of approximation $p=1,2,3,4,5$ (horizontal axis), for different wavenumber values $\kappa = 16 , \ 32, \ 64 , \ 128$. In the blue region, the values are highlighted for a better comparison with \ref{['fig:Circle_Convergence_Full']}.
• Figure 5: Example in \ref{['test:numex1']}, on a circular domain with decomposition $\mathcal{D}$ depicted in \ref{['fig:CircularDomain']}. The $L^2$--relative error $\| u - u^{h,p}_A \|_{L^2(\Omega)}/\| u\|_{L^2(\Omega)}$ is computed against the exact solution for mesh size $h = 0.025$ and for different wavenumber values $\kappa = 16 , \ 32 , \ 64 , \ 128$. The modes considered are $I_{\mathcal{E}} = 32$ for degree $p=1$, $I_{\mathcal{E}} = 64$ for degree $p=2,3$ and $I_{\mathcal{E}} = 128$ for degree $p=4,5$. In the blue region, the values are highlighted for a better comparison with \ref{['fig:Circle_Convergence_Saturation']}.

Theorems & Definitions (6)

• Remark 1
• Remark 2
• Remark 3
• Remark 4
• Remark 5
• Remark 6