Two-level hybrid Schwarz preconditioners with preasymptotic piecewise-polynomial coarse spaces for the high-frequency Helmholtz equation
Abstract
We consider two-level hybrid Schwarz domain-decomposition preconditioners for finite element discretizations of the high-frequency Helmholtz equation (with wavenumber k). To date, all existing k-explicit theory for these preconditioners considers coarse spaces in the asymptotic regime, i.e., where the Galerkin solution is k-uniformly quasi-optimal. We consider the situation where the fine and coarse spaces consist, respectively, of piecewise polynomials of degree p on meshes of width h and Hc. We prove results for the coarse space in the preasymptotic regime, under the familiar mesh threshold that (Hc k)2p ρ(k) is sufficiently small, where ρ(k) is the norm of the solution operator (such that ρ(k)∼ k for the Helmholtz equation in free space). We prove that, in this regime, the fixed-point iteration with this preconditioner converges in a number of iterations that grows at most like log k. We give numerical experiments demonstrating this theory. Furthermore, these experiments indicate that our theoretical results are sharp, in that the number of iterations required for these preconditioners to converge with (Hc k)2p ρ(k) ≫ 1 grows rapidly with k.
Related
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BibTeX
@misc{galkowski2026twolevel,
title = {{Two-level hybrid Schwarz preconditioners with preasymptotic piecewise-polynomial coarse spaces for the high-frequency Helmholtz equation}},
author = {Jeffrey Galkowski and Euan A. Spence and Pierre-Henri Tournier},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.11774},
eprint = {2610.11774},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.11774},
}
RIS
TY - UNPB
TI - Two-level hybrid Schwarz preconditioners with preasymptotic piecewise-polynomial coarse spaces for the high-frequency Helmholtz equation
AU - Jeffrey Galkowski
AU - Euan A. Spence
AU - Pierre-Henri Tournier
PY - 2026
DA - 2026/10/08
PB - arXiv
DO - 10.48550/arxiv.2610.11774
UR - https://arxiv.org/abs/2610.11774
AB - We consider two-level hybrid Schwarz domain-decomposition preconditioners for finite element discretizations of the high-frequency Helmholtz equation (with wavenumber k). To date, all existing k-explicit theory for these preconditioners considers coarse spaces in the asymptotic regime, i.e., where the Galerkin solution is k-uniformly quasi-optimal. We consider the situation where the fine and coarse spaces consist, respectively, of piecewise polynomials of degree p on meshes of width h and H_c. We prove results for the coarse space in the preasymptotic regime, under the familiar mesh threshold that (H_c k)^2p ρ(k) is sufficiently small, where ρ(k) is the norm of the solution operator (such that ρ(k)∼ k for the Helmholtz equation in free space). We prove that, in this regime, the fixed-point iteration with this preconditioner converges in a number of iterations that grows at most like log k. We give numerical experiments demonstrating this theory. Furthermore, these experiments indicate that our theoretical results are sharp, in that the number of iterations required for these preconditioners to converge with (H_c k)^2p ρ(k) ≫ 1 grows rapidly with k.
ER -