Shifted Laplacian RAS Solvers for the Helmholtz Equation

Kimn, Jung‐Han; Sarkis, Marcus

doi:10.1007/978-3-642-35275-1_16

Cited by 9 publications

(10 citation statements)

References 10 publications

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“…There are also a few results on overlapping domain decomposition methods e.g. [41], [30], [31], with the latter explicitly using absorption; these demonstrated the potential of the methods analysed in this paper. Finally, we note that [44] introduces a new sweeping-style method for the Helmholtz equation, and also contains a good literature review of both domain-decomposition and sweeping-style methods.…”

Section: 3mentioning

confidence: 55%

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Domain decomposition preconditioning for high-frequency Helmholtz problems with absorption

2017

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Abstract. In this paper we give new results on domain decomposition preconditioners for GM-RES when computing piecewise-linear finite-element approximations of the Helmholtz equation −∆u − (k 2 + iε)u = f , with absorption parameter ε ∈ R. Multigrid approximations of this equation with ε = 0 are commonly used as preconditioners for the pure Helmholtz case (ε = 0). However a rigorous theory for such (so-called "shifted Laplace") preconditioners, either for the pure Helmholtz equation, or even the absorptive equation (ε = 0), is still missing. We present a new theory for the absorptive equation that provides rates of convergence for (left-or right-) preconditioned GMRES, via estimates of the norm and field of values of the preconditioned matrix. This theory uses a k-and ε-explicit coercivity result for the underlying sesquilinear form and shows, for example, that if |ε| ∼ k 2 , then classical overlapping additive Schwarz will perform optimally for the absorptive problem, provided the subdomain and coarse mesh diameters are carefully chosen. Extensive numerical experiments are given that support the theoretical results. While the theory applies to a certain weighted variant of GMRES, the experiments for both weighted and classical GMRES give comparable results. The theory for the absorptive case gives insight into how its domain decomposition approximations perform as preconditioners for the pure Helmholtz case ε = 0. At the end of the paper we propose a (scalable) multilevel preconditioner for the pure Helmholtz problem that has an empirical computation time complexity of about O(n 4/3 ) for solving finite element systems of size n = O(k 3 ), where we have chosen the mesh diameter h ∼ k −3/2 to avoid the pollution effect. Experiments on problems with h ∼ k −1 , i.e. a fixed number of grid points per wavelength, are also given.

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Section: 3mentioning

confidence: 55%

“…The second variant is the Restrictive Additive Schwarz (RAS) preconditioner, which is well-known in the literature [3], [31]. Here to define the local operator, for each j ∈ I h , choose a single ℓ = ℓ(j) with the property that x j ∈ Ω ℓ(j) .…”

Section: Numerical Experimentsmentioning

confidence: 99%

Domain decomposition preconditioning for high-frequency Helmholtz problems with absorption

2017

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“…Other uses of the shifted Laplacian preconditioner include its use with ε ∼ k 2 in the context of domain decomposition methods in [30], and its use with ε ∼ k in the sweeping preconditioner of Enquist and Ying in [13] (these authors consider preconditioning the Helmholtz equation with k replaced by k + iδ with δ ∼ 1, and this corresponds to choosing ε ∼ k). Finally we note that solving the problem with absorption by preconditioning with the inverse of the Laplacian (i.e., aiming to achieve (P2) with ε = 0) has been investigated in [24,25].…”

Section: Previous Work On the Shifted Laplacian Preconditionermentioning

confidence: 99%

Applying GMRES to the Helmholtz equation with shifted Laplacian preconditioning: what is the largest shift for which wavenumber-independent convergence is guaranteed?

2015

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There has been much recent research on preconditioning discretisations of the Helmholtz operator + k 2 (subject to suitable boundary conditions) using a discrete version of the so-called "shifted Laplacian" + (k 2 + iε) for some ε > 0. This is motivated by the fact that, as ε increases, the shifted problem becomes easier to solve iteratively. Despite many numerical investigations, there has been no rigorous analysis of how to chose the shift. In this paper, we focus on the question of how large ε can be so that the shifted problem provides a preconditioner that leads to k-independent convergence of GMRES, and our main result is a sufficient condition on ε for this property to hold. This result holds for finite element discretisations of both the interior impedance problem and the sound-soft scattering problem (with the radiation condition in the latter problem imposed as a far-field impedance boundary condition). Note that we do not address the important question of how large ε should be so that the preconditioner can easily be inverted by standard iterative methods.Mathematics Subject Classification 35J05 · 65F08 · 65F10 · 65N30 · 78A45

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“…One advantage of DD over multigrid in this context is that "wave-based" components such as impedance or PML boundary conditions on the subdomains can more-easily be incorporated into DD preconditioners (see §1.5 below). The numerical experiments in [41] and [42] (following earlier experiments in [52] and [53]) show that additive Schwarz DD preconditioners for A can perform well for k ξ k 2 if impedance boundary conditions are used on the subdomain problems, instead of Dirichlet ones, and these experiments are backed up by analysis in [43] that shows that Property (ii) can be satisfied in some situations with |ξ| ∼ k 1+β for β small. 1.4.…”

Section: 2mentioning

confidence: 85%

Domain decomposition preconditioning for the high-frequency time-harmonic Maxwell equations with absorption

et al. 2019

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This paper rigorously analyses preconditioners for the time-harmonic Maxwell equations with absorption, where the PDE is discretised using curl-conforming finite-element methods of fixed, arbitrary order and the preconditioner is constructed using Additive Schwarz domain decomposition methods. The theory developed here shows that if the absorption is large enough, and if the subdomain and coarse mesh diameters and overlap are chosen appropriately, then the classical two-level overlapping Additive Schwarz preconditioner (with PEC boundary conditions on the subdomains) performs optimally -in the sense that GMRES converges in a wavenumberindependent number of iterations -for the problem with absorption. An important feature of the theory is that it allows the coarse space to be built from low-order elements even if the PDE is discretised using high-order elements. It also shows that additive methods with minimal overlap can be robust. Numerical experiments are given that illustrate the theory and its dependence on various parameters. These experiments motivate some extensions of the preconditioners which have better robustness for problems with less absorption, including the propagative case. At the end of the paper we illustrate the performance of these on two substantial applications; the first (a problem with absorption arising from medical imaging) shows the empirical robustness of the preconditioner against heterogeneity, and the second (scattering by a COBRA cavity) shows good scalability of the preconditioner with up to 3,000 processors.

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Shifted Laplacian RAS Solvers for the Helmholtz Equation

Cited by 9 publications

References 10 publications

Domain decomposition preconditioning for high-frequency Helmholtz problems with absorption

Domain decomposition preconditioning for high-frequency Helmholtz problems with absorption

Applying GMRES to the Helmholtz equation with shifted Laplacian preconditioning: what is the largest shift for which wavenumber-independent convergence is guaranteed?

Domain decomposition preconditioning for the high-frequency time-harmonic Maxwell equations with absorption

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