Thesis

Robust spectral coarse spaces for indefinite Helmholtz-type problems

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Awarding institution
  • University of Strathclyde
Date of award
  • 2026
Thesis identifier
  • T17981
Person Identifier (Local)
  • 202176965
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Abstract
  • High–frequency Helmholtz problems remain exceptionally difficult for both direct and iterative solvers. Finite element discretisations produce large, indefinite and often non-normal linear systems; standard preconditioners typically lose robustness as the wavenumber and the number of subdomains increase. Domain Decomposition Methods (DDMs) address these issues by splitting the global problem into local subproblems solved in parallel by direct methods. While DDMs may be used as iterative solvers, they have proven to be effective preconditioners within Krylov methods, where the use of an appropriate coarse space can significantly affect convergence and scalability. This thesis develops scalable two–level preconditioners tailored to the Helmholtz problem. After establishing the mathematical background, we overview classical DDMs. The core contributions then focus on spectral coarse spaces. First, we revisit the GenEO coarse space (Δ-GenEO) and introduce an augmented variant, Δk-GenEO, obtained by modifying the local generalised eigenproblems to incorporate frequency information. We analyse eigenvalue selection strategies and the trade–off between coarse dimension and robustness. Second, we propose a new Helmholtz adapted space, Hk-GenEO, constructed from the indefinite Helmholtz problem. As previous methods have been based on positive (semi-) definite problems, we first had to develop the underlying theory to prove the required bounds for the method. We use numerical simulations to demonstrate the methods’ effectiveness. We finish this work with a systematic comparative numerical study of newly developed coarse space designs in the GenEO family on various homogeneous and heterogeneous problems.
Advisor / supervisor
  • Langer, Matthias
  • Dolean, Victorita
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DOI

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