A pure source transfer domain decomposition method for Helmholtz equations in 3D unbounded domain
Abstract
We develop and analyze a three-dimensional pure source transfer domain decomposition method (PSTDDM) for the Helmholtz equation in free space. Extending the two-dimensional PSTDDM proposed by [Du and Wu, J. Sci. Comput., 83(2020),83:68], the proposed method transfers sources successively along the x1, x2, and x3 directions, yielding layer-wise, bar-wise, and block-wise PSTDDMs. The main challenge in the convergence analysis is deriving wavenumber-explicit truncation-error estimates. To address this challenge, we establish inf--sup conditions for the uniaxial perfectly matched layer (UPML) problem under relaxed restrictions on the PML parameters and reformulate the source-transfer operators. We then introduce a positive-definite reconstruction step that produces a globally conforming approximation. In addition, a new quantitative lower bound is established for the imaginary part of the complex PML distance, more easily yielding exponential decay estimates in three dimensions. These lead to wavenumber-explicit convergence estimates for all three PSTDDMs. Numerical experiments demonstrate the effectiveness of the proposed methods both as direct solvers and as preconditioners for GMRES.
Related
- A Helmholtz-Weyl decomposition for Hölder spaces. A nonvariational approach
- An adaptive localized orthogonal decomposition method for Helmholtz problems
- Wavenumber-explicit stability of three-dimensional Cartesian UPMLs with piecewise-constant profiles for Helmholtz and Maxwell equations
- Stability for the inverse source problem of the stochastic fractional Helmholtz equation
- Inverse source problem for the stochastic fractional Helmholtz equation in inhomogeneous media
BibTeX
@misc{bai2026pure,
title = {{A pure source transfer domain decomposition method for Helmholtz equations in 3D unbounded domain}},
author = {Xinran Bai and Yu Du and Haijun Wu},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.19435},
eprint = {2610.19435},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.19435},
}
RIS
TY - UNPB
TI - A pure source transfer domain decomposition method for Helmholtz equations in 3D unbounded domain
AU - Xinran Bai
AU - Yu Du
AU - Haijun Wu
PY - 2026
DA - 2026/10/14
PB - arXiv
DO - 10.48550/arxiv.2610.19435
UR - https://arxiv.org/abs/2610.19435
AB - We develop and analyze a three-dimensional pure source transfer domain decomposition method (PSTDDM) for the Helmholtz equation in free space. Extending the two-dimensional PSTDDM proposed by [Du and Wu, J. Sci. Comput., 83(2020),83:68], the proposed method transfers sources successively along the x_1, x_2, and x_3 directions, yielding layer-wise, bar-wise, and block-wise PSTDDMs. The main challenge in the convergence analysis is deriving wavenumber-explicit truncation-error estimates. To address this challenge, we establish inf--sup conditions for the uniaxial perfectly matched layer (UPML) problem under relaxed restrictions on the PML parameters and reformulate the source-transfer operators. We then introduce a positive-definite reconstruction step that produces a globally conforming approximation. In addition, a new quantitative lower bound is established for the imaginary part of the complex PML distance, more easily yielding exponential decay estimates in three dimensions. These lead to wavenumber-explicit convergence estimates for all three PSTDDMs. Numerical experiments demonstrate the effectiveness of the proposed methods both as direct solvers and as preconditioners for GMRES.
ER -