Wave propagation problems governed by the Helmholtz equation remain among the most challenging in scientific computing, due to their indefinite nature. Domain decomposition methods with spectral coarse spaces have emerged as some of the most effective preconditioners, yet their theoretical guarantees often lag behind practical performance. In this work, we introduce and analyse the $\Delta_k$-GenEO coarse space within the two-level additive Schwarz preconditioners for heterogeneous Helmholtz problems. This is an adaptation of the $\Delta$-GenEO coarse space. Our results sharpen the $k$-explicit conditions for GMRES convergence, reducing the restrictions on the subdomain size and eigenvalue threshold. This narrows the long-standing gap between pessimistic theory and empirical evidence, and reveals why GenEO spaces based on SPD (symmetric positive definite) eigenvalue problems remain surprisingly effective despite their apparent limitations. Numerical experiments confirm the theory, demonstrating scalability, robustness to heterogeneity for low to moderate frequencies (while experiencing limitations in the high frequency cases), and significantly milder coarse-space growth than conservative estimates predict.

翻译：由Helmholtz方程控制的波传播问题因其不定性，仍然是科学计算中最具挑战性的问题之一。具有谱粗空间的区域分解方法已成为最有效的预条件子之一，但其理论保证往往滞后于实际性能。本文针对异质Helmholtz问题，在两层加性Schwarz预条件子框架内引入并分析了$\Delta_k$-GenEO粗空间，这是对$\Delta$-GenEO粗空间的适应性改进。我们的结果锐化了GMRES收敛的$k$显式条件，降低了对子域尺寸和特征值阈值的限制，从而缩小了悲观理论与经验证据之间长期存在的差距，并揭示了基于SPD（对称正定）特征值问题的GenEO空间尽管存在明显局限性，却仍能保持惊人有效性的原因。数值实验验证了该理论，证明了其在低至中频情况下的可扩展性、对异质性的鲁棒性（但在高频情况下存在局限性），以及粗空间增长显著低于保守估计预测的程度。