Massively parallel Schwarz methods for the high-frequency Helmholtz equation with O(k) convergence
Abstract
We develop a one-level restricted additive Schwarz solver with perfectly matched layer transmission conditions (RAS--PML) for high-frequency Helmholtz systems. For problems with O(kd) unknowns, d=2,3, the solver uses Cartesian decompositions and O(kd) concurrent local solves for a single right-hand side, where k is the wavenumber. Starting from the convergence theory for fixed decompositions established in~\texttt{arXiv:2404.02156}, we address the growing cost {as k increases} of local solves through a nested RAS--PML construction. Under suitable stability and contraction assumptions, we prove that sufficiently accurate recursive local solves preserve convergence of the outer iteration. We also prove that a single inner iteration at each non-terminal level yields a correction algebraically equivalent to one-level RAS--PML, provided that the hierarchy-induced finest local problems and composite restriction and weighted extension operators are retained. This equivalence provides a hierarchical interpretation and permits a direct implementation without recursive transfers. We assess the one-level solver using PML and overlap widths of order O(k−1log k). Constant-coefficient experiments cover 300≤ k≤19200 in two dimensions and 60≤ k≤780 in three dimensions, reaching 1.47×109 and 4.75×108 complex unknowns, respectively. Richardson iteration counts grow approximately linearly with k over these ranges, while total parallel runtime, including setup and solution, exhibits near-linear growth at the larger tested wavenumbers. Further experiments assess the effects of mesh refinement and heterogeneous media.
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BibTeX
@misc{xie2026massively,
title = {{Massively parallel Schwarz methods for the high-frequency Helmholtz equation with O(k) convergence}},
author = {Yan Xie and Yue Yu and Shihua Gong and Ivan G. Graham and Chen-song Zhang},
howpublished = {arXiv},
year = {2026},
month = nov,
doi = {10.48550/arxiv.2611.01075},
eprint = {2611.01075},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2611.01075},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Massively parallel Schwarz methods for the high-frequency Helmholtz equation with O(k) convergence
AU - Yan Xie
AU - Yue Yu
AU - Shihua Gong
AU - Ivan G. Graham
AU - Chen-song Zhang
PY - 2026
DA - 2026/11/01
PB - arXiv
DO - 10.48550/arxiv.2611.01075
UR - https://arxiv.org/abs/2611.01075
AB - We develop a one-level restricted additive Schwarz solver with perfectly matched layer transmission conditions (RAS--PML) for high-frequency Helmholtz systems. For problems with O(k^d) unknowns, d=2,3, the solver uses Cartesian decompositions and O(k^d) concurrent local solves for a single right-hand side, where k is the wavenumber. Starting from the convergence theory for fixed decompositions established in~\texttt{arXiv:2404.02156}, we address the growing cost {as k increases} of local solves through a nested RAS--PML construction. Under suitable stability and contraction assumptions, we prove that sufficiently accurate recursive local solves preserve convergence of the outer iteration. We also prove that a single inner iteration at each non-terminal level yields a correction algebraically equivalent to one-level RAS--PML, provided that the hierarchy-induced finest local problems and composite restriction and weighted extension operators are retained. This equivalence provides a hierarchical interpretation and permits a direct implementation without recursive transfers. We assess the one-level solver using PML and overlap widths of order O(k^−1log k). Constant-coefficient experiments cover 300≤ k≤19200 in two dimensions and 60≤ k≤780 in three dimensions, reaching 1.47×10^9 and 4.75×10^8 complex unknowns, respectively. Richardson iteration counts grow approximately linearly with k over these ranges, while total parallel runtime, including setup and solution, exhibits near-linear growth at the larger tested wavenumbers. Further experiments assess the effects of mesh refinement and heterogeneous media.
ER -