A Physics-Driven Multiscale Method for High-Frequency Helmholtz Equations
Abstract
We develop and analyze a physics-driven multiscale method for model reduction of the Helmholtz equation at high frequency. The method constructs a reduced approximation space from local wave responses within a mixed finite element formulation with velocity elimination. These responses are computed by solving Helmholtz problems with boundary excitations on oversampled regions, giving snapshots that incorporate the oscillatory behavior and propagation characteristics of the underlying medium. Local generalized eigenvalue problems then select modes from this physically informed space to form a reduced global system with sparse interelement coupling. To establish stability and convergence of this reduction, we relate the local spectral approximation to approximation of the global adjoint problem. Starting from an inf--sup estimate for the fine-grid formulation, this argument yields a multiscale inf--sup condition governed by fine-grid resolution, oversampling, and the first omitted local eigenvalues. The resulting stability establishes well-posedness and quasi-optimality of the multiscale approximation in a discrete energy norm. Together with local spectral estimates, it gives an a priori error bound that explicitly quantifies the roles of the coarse and fine mesh sizes, oversampling depth, spectral truncation, and wavenumber. This analysis provides a quantitative link between local space reduction and the stability and accuracy of the global approximation. Numerical experiments in two and three dimensions demonstrate accurate wavefield approximation with substantial reductions in global dimension and solution cost for homogeneous and strongly heterogeneous media, including perfectly matched layer truncations.
Related
- Robust edge multiscale space based hybrid Schwarz preconditioner for high-frequency Helmholtz problems
- Discrete Edge Multiscale Space based Hybrid Schwarz Preconditioner for High-Frequency Helmholtz Problems
- Cartesian PML truncation for the Helmholtz equation is exponentially accurate at high frequency
- Wavenumber-explicit stability of three-dimensional Cartesian UPMLs with piecewise-constant profiles for Helmholtz and Maxwell equations
- Efficient C1 Bernstein Quasi-Trefftz Discretization for Heterogeneous High-Frequency Helmholtz Problems
BibTeX
@misc{chen2026physicsdriven,
title = {{A Physics-Driven Multiscale Method for High-Frequency Helmholtz Equations}},
author = {Huangxin Chen and Jianhui Chen and Shubin Fu and Zhiyu Tan},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.03144},
eprint = {2610.03144},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.03144},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - A Physics-Driven Multiscale Method for High-Frequency Helmholtz Equations
AU - Huangxin Chen
AU - Jianhui Chen
AU - Shubin Fu
AU - Zhiyu Tan
PY - 2026
DA - 2026/10/02
PB - arXiv
DO - 10.48550/arxiv.2610.03144
UR - https://arxiv.org/abs/2610.03144
AB - We develop and analyze a physics-driven multiscale method for model reduction of the Helmholtz equation at high frequency. The method constructs a reduced approximation space from local wave responses within a mixed finite element formulation with velocity elimination. These responses are computed by solving Helmholtz problems with boundary excitations on oversampled regions, giving snapshots that incorporate the oscillatory behavior and propagation characteristics of the underlying medium. Local generalized eigenvalue problems then select modes from this physically informed space to form a reduced global system with sparse interelement coupling. To establish stability and convergence of this reduction, we relate the local spectral approximation to approximation of the global adjoint problem. Starting from an inf--sup estimate for the fine-grid formulation, this argument yields a multiscale inf--sup condition governed by fine-grid resolution, oversampling, and the first omitted local eigenvalues. The resulting stability establishes well-posedness and quasi-optimality of the multiscale approximation in a discrete energy norm. Together with local spectral estimates, it gives an a priori error bound that explicitly quantifies the roles of the coarse and fine mesh sizes, oversampling depth, spectral truncation, and wavenumber. This analysis provides a quantitative link between local space reduction and the stability and accuracy of the global approximation. Numerical experiments in two and three dimensions demonstrate accurate wavefield approximation with substantial reductions in global dimension and solution cost for homogeneous and strongly heterogeneous media, including perfectly matched layer truncations.
ER -