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Sharp bounds on Helmholtz impedance-to-impedance maps and application to overlapping domain decomposition

David Lafontaine, Euan A. Spence

Abstract

We prove sharp bounds on certain impedance-to-impedance maps (and their compositions) for the Helmholtz equation with large wavenumber (i.e., at high-frequency) using semiclassical defect measures. The paper [GGGLS] (Gong-Gander-Graham-Lafontaine-Spence, 2022) recently showed that the behaviour of these impedance-to-impedance maps (and their compositions) dictates the convergence of the parallel overlapping Schwarz domain-decomposition method with impedance boundary conditions on the subdomain boundaries. For a model decomposition with two subdomains and sufficiently-large overlap, the results of this paper combined with those in [GGGLS] show that the parallel Schwarz method is power contractive, independent of the wavenumber. For strip-type decompositions with many subdomains, the results of this paper show that the composite impedance-to-impedance maps, in general, behave "badly" with respect to the wavenumber; nevertheless, by proving results about the composite maps applied to a restricted class of data, we give insight into the wavenumber-robustness of the parallel Schwarz method observed in the numerical experiments in [GGGLS].

Sharp bounds on Helmholtz impedance-to-impedance maps and application to overlapping domain decomposition

Abstract

We prove sharp bounds on certain impedance-to-impedance maps (and their compositions) for the Helmholtz equation with large wavenumber (i.e., at high-frequency) using semiclassical defect measures. The paper [GGGLS] (Gong-Gander-Graham-Lafontaine-Spence, 2022) recently showed that the behaviour of these impedance-to-impedance maps (and their compositions) dictates the convergence of the parallel overlapping Schwarz domain-decomposition method with impedance boundary conditions on the subdomain boundaries. For a model decomposition with two subdomains and sufficiently-large overlap, the results of this paper combined with those in [GGGLS] show that the parallel Schwarz method is power contractive, independent of the wavenumber. For strip-type decompositions with many subdomains, the results of this paper show that the composite impedance-to-impedance maps, in general, behave "badly" with respect to the wavenumber; nevertheless, by proving results about the composite maps applied to a restricted class of data, we give insight into the wavenumber-robustness of the parallel Schwarz method observed in the numerical experiments in [GGGLS].
Paper Structure (35 sections, 31 theorems, 257 equations, 3 figures)

This paper contains 35 sections, 31 theorems, 257 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Motivation and outline
  3. Definition of the impedance-to-impedance maps
  4. Upper and lower bounds on the impedance operators
  5. The behaviour of the composite impedance map in (\ref{['eq:model_cell_timp']})
  6. The semiclassical notation and definition of outgoingness
  7. Wellposedness results
  8. Implications of the main results for the parallel overlapping Schwarz method
  9. Definition of the parallel overlapping Schwarz method studied in GoGaGrLaSp:22
  10. Summary of the numerical experiments in GoGaGrLaSp:22 on the performance of the parallel Schwarz method
  11. The error propagation operator $\boldsymbol{\mathcal{T}}$
  12. The goal: proving power contractivity of $\boldsymbol{\mathcal{T}}$
  13. How impedance-to-impedance maps arise in studying $\boldsymbol{\mathcal{T}}^M$ for $M\geq 1$
  14. The impedance-to-impedance map for general decompositions
  15. Specialisation to 2-d strip decompositions
  16. ...and 20 more sections

Key Result

Theorem 1.1

Let ${\mathsf h} , \mathsf {d_l} > 0$ and let $\theta_{\rm max}\in (0,\pi/2)$ be defined by $\;$

Figures (3)

  • Figure 1: The domain $D$, the boundaries $\Gamma_l, \Gamma_r, \Gamma_t$, and $\Gamma_b$, and the interior interface $\Gamma_i$
  • Figure 2: Illustrations of the domain (red) and co-domain (blue) of ${\mathcal{I}}_{\Gamma_{\ell.j'} \to \Gamma_{j,\ell}}$ in 2d (in the case when none of the subdomains $\Omega_j,\Omega_{j'},$ and $\Omega_\ell$ touch $\partial \Omega$)
  • Figure 3: Three overlapping subdomains in the 2-d strip decomposition

Theorems & Definitions (68)

  • Theorem 1.1: Upper and lower bounds for (\ref{['eq:model_cell']})
  • Theorem 1.2: Lower bounds for (\ref{['eq:model_cell_timp']})
  • Theorem 1.3: The composite impedance map
  • Definition 1.4
  • Lemma 1.5
  • Lemma 1.6
  • Remark 2.1
  • Lemma 2.2: Norm on $U_0(\Omega_j)$
  • Definition 2.3: Impedance map
  • Lemma 2.4: Connection between $\boldsymbol{\mathcal{T}}^2$ and the impedance-to-impedance map
  • ...and 58 more