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

TL;DR

This paper introduces a high-order discretized ACMS method using hp-finite elements for efficient simulation of large finite-size 2D photonic crystals, demonstrating favorable accuracy and computational complexity.

Contribution

It extends the ACMS method with high-order discretization and analyzes its accuracy and complexity for simulating photonic crystals with improved flexibility and efficiency.

Findings

Linear systems remain moderate in size for relevant wavenumbers

Weak dependence on domain decomposition enhances flexibility

Achieves predicted convergence rates with increasing wavenumbers

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.