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Resolvent Convergence and Patch Approximation for Subwavelength Guided Modes in Non-Periodic Systems of High-Contrast Resonators

by H. Ammari and B. Miao and J. Qiu

(Report number 2026-19)

Abstract
This paper develops, analyzes, and validates a fast algorithm for computing guided modes within bent interfaces and non-periodic defects in high-contrast resonator crystals, where the Floquet--Bloch theory is not applicable. We first establish the resolvent convergence of the governing continuous operator to the discrete capacitance operator. This result rigorously justifies the reduction of the continuous spectral problem to a discrete eigenvalue problem. Then, we develop a truncation scheme of the discrete operator, named the patch approximation, and derive a rigorous error estimate for the patch approximation. Finally, we validate the accuracy and efficiency of our scheme through various examples. Our framework provides a general, computationally efficient, and rigorously justified approach to simulate guided modes in non-periodic systems of high-contrast resonators.

Keywords: Metamaterial, subwavelength waveguiding, bent interface, bent defect, subwavelength frequencies, capacitance operator, resolvent convergence, patch approximation, error approximation

@Techreport{AMQ26_1174,
  author = {H. Ammari and B. Miao and J. Qiu},
  title = {Resolvent Convergence and Patch Approximation for Subwavelength Guided Modes in Non-Periodic Systems of High-Contrast Resonators},
  institution = {Seminar for Applied Mathematics, ETH Z{\"u}rich},
  number = {2026-19},
  address = {Switzerland},
  url = {https://www.sam.math.ethz.ch/sam_reports/reports_final/reports2026/2026-19.pdf },
  year = {2026}
}

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