How to Choose the Shift in the Shifted Laplace Preconditioner for the Helmholtz Equation Combined with Deflation

Lahaye, Domenico; Vuik, C.

doi:10.1007/978-3-319-28832-1_4

Cited by 7 publications

(9 citation statements)

References 35 publications

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“…and, in the same way, c n s ( , dτ * ) = c n s ( , τ * ), where we associate s min with s 1 (k = 1), and s max with s n s (k = n s ), cf. (21) in Theorem 4.1(iv). Thus, s 1 and s n s scale with d as the interval boundaries do.…”

Section: Theorem 41 (Optimal Seed Frequency For Preconditioned Shifted Gmres) Let {Smentioning

confidence: 91%

“…Here, we use n s = 20 frequencies equally spaced within the interval [1,9] Hz. The imaginary parts that belong to the extreme frequencies are equal as imposed in (21), but not equal to zero. (For interpretation of the colors in the figure(s), the reader is referred to the web version of this article.…”

Section: Theorem 41 (Optimal Seed Frequency For Preconditioned Shifted Gmres) Let {Smentioning

confidence: 99%

“…Moreover, note that we can not include directly frequency-dependent Sommerfeld boundary conditions in the preconditioner (2), as recommended in [23]. When (2) is applied inexactly using a multi-grid algorithm [29] or deflation [21], the choice of τ is usually combined with damping such that the multi-grid solver works well, cf. [12,19] for an analysis in the Helmholtz case.…”

Section: Introductionmentioning

confidence: 99%

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An efficient two-level preconditioner for multi-frequency wave propagation problems

Baumann

Gijzen

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Section: Theorem 41 (Optimal Seed Frequency For Preconditioned Shifted Gmres) Let {Smentioning

confidence: 91%

Section: Theorem 41 (Optimal Seed Frequency For Preconditioned Shifted Gmres) Let {Smentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

An efficient two-level preconditioner for multi-frequency wave propagation problems

Baumann

Gijzen

2019

Applied Numerical Mathematics

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“…For the Helmholtz problem, this includes parallel direct solvers, such as [4,5], and preconditioned iterative methods that utilise multigrids, such as [6,7], or the so-called "shifted Laplace" approach [8,9]. This latter (complex) shifted Laplace preconditioner has seen much interest into its practical use [10] and further developments through deflation techniques, most recently in [11,12].…”

Section: Introductionmentioning

confidence: 99%

Can DtN and GenEO Coarse Spaces Be Sufficiently Robust for Heterogeneous Helmholtz Problems?

Bootland

Dolean

2022

MCA

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Numerical solutions of heterogeneous Helmholtz problems present various computational challenges, with descriptive theory remaining out of reach for many popular approaches. Robustness and scalability are key for practical and reliable solvers in large-scale applications, especially for large wave number problems. In this work, we explore the use of a GenEO-type coarse space to build a two-level additive Schwarz method applicable to highly indefinite Helmholtz problems. Through a range of numerical tests on a 2D model problem, discretised by finite elements on pollution-free meshes, we observe robust convergence, iteration counts that do not increase with the wave number, and good scalability of our approach. We further provide results showing a favourable comparison with the DtN coarse space. Our numerical study shows promise that our solver methodology can be effective for challenging heterogeneous applications.

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“…For the Helmholtz problem this includes parallel direct solvers, such as [32,54], and preconditioned iterative methods that utilise multigrid, such as [12,41], or the so-called "shifted Laplace" approach [23,24]. This latter (complex-)shifted Laplace preconditioner has seen much interest into its practical use [14] and further developments through deflation techniques, most recently in [20,45].…”

mentioning

confidence: 99%

Can DtN and GenEO coarse spaces be sufficiently robust for heterogeneous Helmholtz problems?

Bootland,

Dolean

2022

Preprint

View full text Add to dashboard Cite

Numerical solution of heterogeneous Helmholtz problems presents various computational challenges, with descriptive theory remaining out of reach for many popular approaches. Robustness and scalability are key for practical and reliable solvers in large-scale applications, especially for large wave number problems. In this work we explore the use of a GenEO-type coarse space to build a two-level additive Schwarz method applicable to highly indefinite Helmholtz problems. Through a range of numerical tests on a 2D model problem, discretised by finite elements on pollution-free meshes, we observe robust, wave number independent convergence and scalability of our approach. We further provide results showing a favourable comparison with the DtN coarse space. Our numerical study shows promise that our solver methodology can be effective for challenging heterogeneous applications.

show abstract

How to Choose the Shift in the Shifted Laplace Preconditioner for the Helmholtz Equation Combined with Deflation

Cited by 7 publications

References 35 publications

An efficient two-level preconditioner for multi-frequency wave propagation problems

An efficient two-level preconditioner for multi-frequency wave propagation problems

Can DtN and GenEO Coarse Spaces Be Sufficiently Robust for Heterogeneous Helmholtz Problems?

Can DtN and GenEO coarse spaces be sufficiently robust for heterogeneous Helmholtz problems?

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