Discrete Edge Multiscale Space based Hybrid Schwarz Preconditioner for High-Frequency Helmholtz Problems
Abstract
In this work, we develop a novel hybrid Schwarz method, termed as edge multiscale space based hybrid Schwarz (EMs-HS), for solving the Helmholtz problem with large wavenumbers. The problem is discretized using H1-conforming nodal finite element methods on meshes of size h decreasing faster than k−3/2 such that the discretization error can be bounded by the data uniformly with respect to k. EMs-HS consists of a one-level Schwarz preconditioner (RAS-imp) and a coarse solver in a multiplicative way. The RAS-imp preconditioner solves local problems on overlapping subdomains with impedance boundary conditions in parallel, and combines the local solutions using partition of unity. On the PDE level, this coarse space corresponds to an edge multiscale space proposed in {[S. Fu, G. Li, R. Craster, and S. Guenneau, \emph{Multiscale Model. Simul.}, 19(4):1684-1709, 2021]}. The approximation properties of this coarse space to the discrete solutions from the primal and adjoint problems are derived by a novel conforming Galerkin finite element method based upon a nonstandard variational formulation. This work represents the first theoretical analysis of an edge multiscale space at the discrete level. We prove that the EMs-HS preconditioner leads to a convergent fixed-point iteration uniformly for large wavenumbers, by rigorously analyzing the approximation properties of the coarse space to the global Helmholtz harmonic extension part and to the solution of the adjoint problem. We provide extensive numerical results on the sharpness of the theoretical findings and also demonstrate the method on challenging heterogeneous models.
Related
- Robust edge multiscale space based hybrid Schwarz preconditioner for high-frequency Helmholtz problems
- A Massively Parallel Three-Grid Preconditioner for the High-Frequency Helmholtz Equation
- A Physics-Driven Multiscale Method for High-Frequency Helmholtz Equations
- Efficient C1 Bernstein Quasi-Trefftz Discretization for Heterogeneous High-Frequency Helmholtz Problems
- Frequency-explicit convergence analysis of a multiscale finite element method for highly heterogeneous scattering problems
BibTeX
@misc{fu2026discrete,
title = {{Discrete Edge Multiscale Space based Hybrid Schwarz Preconditioner for High-Frequency Helmholtz Problems}},
author = {Shubin Fu and Shihua Gong and Guanglian Li and Yueqi Wang},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.19354},
eprint = {2610.19354},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.19354},
}
RIS
TY - UNPB
TI - Discrete Edge Multiscale Space based Hybrid Schwarz Preconditioner for High-Frequency Helmholtz Problems
AU - Shubin Fu
AU - Shihua Gong
AU - Guanglian Li
AU - Yueqi Wang
PY - 2026
DA - 2026/10/14
PB - arXiv
DO - 10.48550/arxiv.2610.19354
UR - https://arxiv.org/abs/2610.19354
AB - In this work, we develop a novel hybrid Schwarz method, termed as edge multiscale space based hybrid Schwarz (EMs-HS), for solving the Helmholtz problem with large wavenumbers. The problem is discretized using H^1-conforming nodal finite element methods on meshes of size h decreasing faster than k^−3/2 such that the discretization error can be bounded by the data uniformly with respect to k. EMs-HS consists of a one-level Schwarz preconditioner (RAS-imp) and a coarse solver in a multiplicative way. The RAS-imp preconditioner solves local problems on overlapping subdomains with impedance boundary conditions in parallel, and combines the local solutions using partition of unity. On the PDE level, this coarse space corresponds to an edge multiscale space proposed in {[S. Fu, G. Li, R. Craster, and S. Guenneau, \emph{Multiscale Model. Simul.}, 19(4):1684-1709, 2021]}. The approximation properties of this coarse space to the discrete solutions from the primal and adjoint problems are derived by a novel conforming Galerkin finite element method based upon a nonstandard variational formulation. This work represents the first theoretical analysis of an edge multiscale space at the discrete level. We prove that the EMs-HS preconditioner leads to a convergent fixed-point iteration uniformly for large wavenumbers, by rigorously analyzing the approximation properties of the coarse space to the global Helmholtz harmonic extension part and to the solution of the adjoint problem. We provide extensive numerical results on the sharpness of the theoretical findings and also demonstrate the method on challenging heterogeneous models.
ER -