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.