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Can Symmetric Positive Definite (SPD) coarse spaces perform well for indefinite Helmholtz problems?

Victorita Dolean, Mark Fry, Matthias Langer

TL;DR

Can SPD-based GenEO coarse spaces perform well for indefinite Helmholtz problems? This work introduces the Δ_k-GenEO coarse space within a two-level additive Schwarz preconditioner and proves sharper $k$-explicit conditions: $H \lesssim k^{-1}$ and $(1+C_{stab})^{2} k^{2} \lesssim \tau$, reducing previous quadratic and octic dependencies. Numerical experiments on homogeneous and heterogeneous media demonstrate near-$N$-independence of iterations for low to moderate frequencies with milder coarse-space growth than theory predicts, while high-frequency cases require larger $\tau$ and larger coarse spaces. Overall, the results clarify the limits of SPD-based coarse spaces for Helmholtz, explain their practical effectiveness, and motivate development of indefinite/coarse spaces such as $H$-GenEO or $H_k$-GenEO for very large $k$.

Abstract

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 $Δ_k$-GenEO coarse space within the two-level additive Schwarz preconditioners for heterogeneous Helmholtz problems. This is an adaptation of the $Δ$-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.

Can Symmetric Positive Definite (SPD) coarse spaces perform well for indefinite Helmholtz problems?

TL;DR

Can SPD-based GenEO coarse spaces perform well for indefinite Helmholtz problems? This work introduces the Δ_k-GenEO coarse space within a two-level additive Schwarz preconditioner and proves sharper -explicit conditions: and , reducing previous quadratic and octic dependencies. Numerical experiments on homogeneous and heterogeneous media demonstrate near--independence of iterations for low to moderate frequencies with milder coarse-space growth than theory predicts, while high-frequency cases require larger and larger coarse spaces. Overall, the results clarify the limits of SPD-based coarse spaces for Helmholtz, explain their practical effectiveness, and motivate development of indefinite/coarse spaces such as -GenEO or -GenEO for very large .

Abstract

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 -GenEO coarse space within the two-level additive Schwarz preconditioners for heterogeneous Helmholtz problems. This is an adaptation of the -GenEO coarse space. Our results sharpen the -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.
Paper Structure (28 sections, 18 theorems, 113 equations, 5 figures, 2 tables)

This paper contains 28 sections, 18 theorems, 113 equations, 5 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Functional framework and discretisation
  3. Domain decomposition framework
  4. The $\Delta_k$-GenEO coarse space
  5. Statement of the main result and theoretical tools
  6. Coarse space approximation and stable decomposition
  7. Local projector: solvability, basic identities, and stability
  8. Coarse projector $T_0$: solvability and stability
  9. On the size of the coarse space and the $\tau$-bound
  10. Proof of the main result
  11. Numerical results
  12. Implementation Details
  13. Domain partition and overlap
  14. Setup: one-level preconditioner
  15. Setup: two-level ($\Delta_k$-GenEO) coarse space
  16. ...and 13 more sections

Key Result

Lemma 2.4

Suppose that Assumptions ass:A and ass:stab hold. Then there exists $h_0 > 0$ such that, for all $0 < h < h_0$, the discrete problem eq:galerkin admits a unique solution $u_h \in V^h$. Moreover, for any $\varepsilon > 0$, there exists $h_1 = h_1(\varepsilon) > 0$ such that, for all $h < h_1$, and

Figures (5)

  • Figure 1: Influence of the coarse space size (left) and threshold choice (right) on the iteration count for the homogeneous media test case with $k$ = 20 (top) and $k$ = 100 (bottom). The number in brackets indicates the number of subdomains.
  • Figure 2: Influence of the subdomain diameter on the iteration count (left) and coarse space size (right) for the homogeneous media test case $k$ = 20 (top) and $k$ = 100 (bottom). The number in brackets indicates $\tau$ used.
  • Figure 3: Influence of the coarse space size (left) and threshold choice (right) on the iteration count for the heterogeneous media test case with $k$ = 20 (top) and $k$ = 100 (bottom), all with $a_\text{max}(\bm{x}) = 10$. The number in brackets indicates the number of subdomains.
  • Figure 4: Influence of the subdomain diameter on the iteration count (left) and coarse space size (right) for the heterogeneous media test case $k$ = 20 (top) and $k$ = 100 (bottom) , all with $a_\text{max}(\bm{x}) = 10$. The number in brackets indicates $\tau$ used.
  • Figure 5: The heterogeneous function $a(\bm{x})$ within the alternating layers. The shading gives the value of $a(\bm{x})$ with the darkest shade being $a(\bm{x}) = a_{\textup{max}}$, where $a_{\textup{max}} > 1$ is a parameter, and the white taking the value $a_{\textup{min}} = 1$.

Theorems & Definitions (48)

  • Lemma 2.4: Schatz--Wang Schatz:1996:SNE
  • Remark 2.5
  • Lemma 2.6: Friedrichs inequality leoni:2017:FCSS
  • Remark 2.7
  • Definition 2.8: Overlapping subdomains
  • Remark 2.9
  • Proposition 2.10: Bootland et al. Bootland:2022:OSM
  • Definition 2.11: Degrees of freedom, cf. Spillane:2014:ARC
  • Definition 2.12: Partition of unity
  • Definition 2.13: Local spectral problem
  • ...and 38 more