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Fast convergent PML method for scattering with periodic surfaces: the exceptional case

Ruming Zhang

TL;DR

This work addresses fast convergence of perfectly matched layer (PML) approximations for scattering by periodic surfaces in 2D, focusing on the exceptional case where $2k\\in\\mathbb{N}$. It combines the Floquet-Bloch transform, analytic extensions in the Floquet parameter, and Gauss-Legendre quadrature to discretize the Floquet integral, and derives an error bound that couples quadrature and PML truncation: $\\|u-u_N^\\sigma\\| \\le C\\rho^{-2N} + C|\\sigma|^{-1} N \\exp(-C\\sqrt{k}\\,|\\sigma|\\,N^{-2})$, which yields $\\|u-u_N^\\sigma\\| \\le C\\exp(-c|\\sigma|^{1/3})$ for an appropriate choice $N=O(|\\sigma|^{1/3})$. Numerical experiments on a representative periodic surface corroborate the theory, showing fast, super-algebraic convergence even in the exceptional cases. This extends previous results that excluded half-integer wavenumbers and demonstrates robust, efficient PML-based simulations for periodic-surface scattering in critical regimes.

Abstract

In the author's previous paper (Zhang et al. 2022), exponential convergence was proved for the perfectly matched layers (PML) approximation of scattering problems with periodic surfaces in 2D. However, due to the overlapping of singularities, an exceptional case, i.e., when the wave number is a half integer, has to be excluded in the proof. However, numerical results for these cases still have fast convergence rate and this motivates us to go deeper into these cases. In this paper, we focus on these cases and prove that the fast convergence result for the discretized form. Numerical examples are also presented to support our theoretical results.

Fast convergent PML method for scattering with periodic surfaces: the exceptional case

TL;DR

This work addresses fast convergence of perfectly matched layer (PML) approximations for scattering by periodic surfaces in 2D, focusing on the exceptional case where . It combines the Floquet-Bloch transform, analytic extensions in the Floquet parameter, and Gauss-Legendre quadrature to discretize the Floquet integral, and derives an error bound that couples quadrature and PML truncation: , which yields for an appropriate choice . Numerical experiments on a representative periodic surface corroborate the theory, showing fast, super-algebraic convergence even in the exceptional cases. This extends previous results that excluded half-integer wavenumbers and demonstrates robust, efficient PML-based simulations for periodic-surface scattering in critical regimes.

Abstract

In the author's previous paper (Zhang et al. 2022), exponential convergence was proved for the perfectly matched layers (PML) approximation of scattering problems with periodic surfaces in 2D. However, due to the overlapping of singularities, an exceptional case, i.e., when the wave number is a half integer, has to be excluded in the proof. However, numerical results for these cases still have fast convergence rate and this motivates us to go deeper into these cases. In this paper, we focus on these cases and prove that the fast convergence result for the discretized form. Numerical examples are also presented to support our theoretical results.
Paper Structure (7 sections, 8 theorems, 71 equations, 4 figures, 1 table)

This paper contains 7 sections, 8 theorems, 71 equations, 4 figures, 1 table.

Table of Contents

  1. Introduction
  2. Application of the Floquet-Bloch transform
  3. PML approximation
  4. Discretization and convergence analysis
  5. Gauss-Legendre quadrature rule
  6. Convergence of the PML approximation
  7. Numerical examples

Key Result

Theorem 1

Let ${\mathcal{E}}_{-1,1}^\rho\subset\mathbb{C}$ be the ellipse with foci at $(\pm 1,0)$ and sum of the half-axes $\rho>1$. Let $\zeta : \, [-1,1] \rightarrow \mathbb{C}$ be real analytic with complex analytic extension to ${\mathcal{E}}_{-1,1}^\rho$. Denote by $I$ the integral over $(-1,1)$ with in

Figures (4)

  • Figure 1: Periodic structures: domains and notations.
  • Figure 2: Perfectly matched layers.
  • Figure 3: Left: structure; right: source term.
  • Figure 4: Semi-log plots for relative errors.

Theorems & Definitions (12)

  • Theorem 1: Theorem 5.3.13, Saute2007
  • Theorem 2
  • Lemma 3
  • proof
  • Lemma 4
  • proof
  • Lemma 5
  • Theorem 6
  • proof
  • Theorem 7
  • ...and 2 more