Can Symmetric Positive Definite (SPD) coarse spaces perform well for indefinite Helmholtz problems?
For researchers in domain decomposition methods for wave propagation, this work narrows the gap between pessimistic theory and empirical evidence for indefinite Helmholtz problems.
The paper introduces the Δ_k-GenEO coarse space for two-level additive Schwarz preconditioners for heterogeneous Helmholtz problems, sharpening k-explicit GMRES convergence conditions and reducing restrictions on subdomain size and eigenvalue threshold. Numerical experiments confirm scalability and robustness for low to moderate frequencies, with milder coarse-space growth than conservative estimates.
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 $Î_k$-GenEO coarse space within the two-level additive Schwarz preconditioners for heterogeneous Helmholtz problems. This is an adaptation of the $Î$-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.