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Kernel compensation method for Maxwell eigenproblem with mimetic finite difference discretization

Chenhao Jin, Yinhua Xia, Yan Xu

TL;DR

This work tackles the challenge of spurious zero modes in Maxwell eigenproblems for 3D photonic crystals by introducing a kernel compensation method built on a compatible mimetic finite difference discretization. By applying a Floquet–Bloch reduction to a shifted curl problem on a unit cube and augmenting the discrete operator with a compensation term $\gamma\mathcal B'\mathcal B$, the authors fill the curl kernel while preserving the nonzero spectrum, with the penalty $\gamma$ shown to be mesh-size independent. They develop a complete MFD formulation that respects the discrete de Rham complex, provide FFT- and multigrid-based preconditioners, and demonstrate robust, scalable performance on isotropic, cubic, BCC, and FCC lattices. The resulting framework enables accurate, efficient band-structure calculations for photonic crystals with strong parallelizability and favorable computational properties. Together, these contributions offer a practical and rigorous tool for high-fidelity simulation of Maxwell eigenproblems in complex periodic media.

Abstract

We present a kernel compensation method for Maxwell eigenproblem for photonic crystals to avoid the infinite-dimensional kernels that cause many difficulties in the calculation of energy gaps. The quasi-periodic problem is first transformed into a periodic one on the cube by the Floquet-Bloch theory. Then the compensation operator is introduced in Maxwell's equation with the shifted curl operator. The discrete problem depends on the compatible discretization of the de Rham complex, which is implemented by the mimetic finite difference method in this paper. We prove that the compensation term exactly fills up the kernel of the original problem and avoids spurious eigenvalues. Also, we propose an efficient preconditioner and its FFT and multigrid solvers, which allow parallel computing. Numerical experiments for different three-dimensional lattices are performed to validate the accuracy and effectiveness of the method.

Kernel compensation method for Maxwell eigenproblem with mimetic finite difference discretization

TL;DR

This work tackles the challenge of spurious zero modes in Maxwell eigenproblems for 3D photonic crystals by introducing a kernel compensation method built on a compatible mimetic finite difference discretization. By applying a Floquet–Bloch reduction to a shifted curl problem on a unit cube and augmenting the discrete operator with a compensation term , the authors fill the curl kernel while preserving the nonzero spectrum, with the penalty shown to be mesh-size independent. They develop a complete MFD formulation that respects the discrete de Rham complex, provide FFT- and multigrid-based preconditioners, and demonstrate robust, scalable performance on isotropic, cubic, BCC, and FCC lattices. The resulting framework enables accurate, efficient band-structure calculations for photonic crystals with strong parallelizability and favorable computational properties. Together, these contributions offer a practical and rigorous tool for high-fidelity simulation of Maxwell eigenproblems in complex periodic media.

Abstract

We present a kernel compensation method for Maxwell eigenproblem for photonic crystals to avoid the infinite-dimensional kernels that cause many difficulties in the calculation of energy gaps. The quasi-periodic problem is first transformed into a periodic one on the cube by the Floquet-Bloch theory. Then the compensation operator is introduced in Maxwell's equation with the shifted curl operator. The discrete problem depends on the compatible discretization of the de Rham complex, which is implemented by the mimetic finite difference method in this paper. We prove that the compensation term exactly fills up the kernel of the original problem and avoids spurious eigenvalues. Also, we propose an efficient preconditioner and its FFT and multigrid solvers, which allow parallel computing. Numerical experiments for different three-dimensional lattices are performed to validate the accuracy and effectiveness of the method.

Paper Structure

This paper contains 27 sections, 8 theorems, 94 equations, 6 figures, 9 tables, 3 algorithms.

Table of Contents

  1. Introduction
  2. Model problem
  3. Kernel compensation method
  4. Eigenvalue distributions
  5. The kernel compensation method
  6. Comparison with finite element method
  7. Compatible finite difference discretization
  8. MFD discretization of function spaces and operators
  9. Grid function spaces
  10. Second order discrete operators
  11. Higher order discretization
  12. Matrix representation
  13. Matrix representation on simple lattices
  14. Matrix representation on general lattices
  15. Derived operators
  16. ...and 12 more sections

Key Result

Lemma 3.1

Assume $M_0$ is a diagonal matrix with positive entries, for $\boldsymbol{u}\in \mathbb{C}^{3N^3}$,

Figures (6)

  • Figure 1: DoFs of scalar and vector grid functions on a single cell
  • Figure 2: Geometric structure of a single lattice and its band gap in Section \ref{['sec:cubic']}. $\varepsilon_1/\varepsilon_0=13.$ Grid size $N=100.$ The ratio of the band gap is 0.075406.
  • Figure 3: Geometric structure of a single lattice and its band gap in Section \ref{['sec:cubic']}. $\varepsilon_1/\varepsilon_0=13.$ Grid size $N=100.$ The ratio of the band gap is 0.21013.
  • Figure 4: Geometric structure of a single lattice with a curved interface and its band gap in Section \ref{['sec:cubic']}. Radii of the center sphere and cylinders are $0.345l$ and $0.11l$. $\varepsilon_1/\varepsilon_0=13.$ Grid size $N=150.$ The ratio of the band gap is 0.14019.
  • Figure 5: Geometric structure of the BCC lattice with a curved interface and its band gap in Sec. \ref{['sec:bcc_fcc']}, $\varepsilon_1/\varepsilon_0=16.$ Grid size $N=150.$ The ratio of the band gap is 0.31745.
  • ...and 1 more figures

Theorems & Definitions (17)

  • Lemma 3.1
  • Proposition 3.2
  • proof
  • Remark 3.3
  • Definition 4.1: Circulant matrix
  • Lemma 4.1
  • proof
  • Proposition 4.2
  • Definition 4.2: Adjoints
  • Proposition 4.3
  • ...and 7 more