发表机构
Xiamen University; Eastern Institute of Technology（厦门大学; 宁波东方理工大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
提出一种物理驱动的多尺度方法，通过局部波响应和特征值选择构建降阶空间，在高频亥姆霍兹方程中实现稳定、准最优的逼近，并在二维和三维实验中显著降低计算成本。
AI 中文摘要
我们开发并分析了一种物理驱动的多尺度方法，用于高频亥姆霍兹方程的模型降阶。该方法在带有速度消除的混合有限元公式中，从局部波响应构造一个降阶逼近空间。这些响应通过在过采样区域上求解具有边界激励的亥姆霍兹问题来计算，得到包含底层介质振荡行为与传播特性的快照。随后，局部广义特征值问题从这一物理信息丰富的空间中选择模态，以形成一个具有稀疏单元间耦合的降阶全局系统。为建立该降阶的稳定性和收敛性，我们将局部谱逼近与全局伴随问题的逼近联系起来。从细网格公式的inf-sup估计出发，这一论证导出了一个由细网格分辨率、过采样和首个被省略的局部特征值控制的多尺度inf-sup条件。由此产生的稳定性在离散能量范数中确立了多尺度逼近的适定性和准最优性。结合局部谱估计，它给出了一个先验误差界，明确量化了粗网格和细网格尺寸、过采样深度、谱截断以及波数的作用。这一分析为局部空间降阶与全局逼近的稳定性和精度之间提供了定量联系。二维和三维数值实验表明，对于均匀和强非均匀介质（包括完美匹配层截断），该方法在全局维度和求解成本大幅降低的情况下，实现了精确的波场逼近。
英文摘要
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.
