Corners and stable optimized domain decomposition methods for the Helmholtz problem

Després, Bruno; Nicolopoulos, Anouk; Thierry, Bertrand

doi:10.1007/s00211-021-01251-2

Cited by 15 publications

(22 citation statements)

References 29 publications

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“…This operator Π loc should thus be understood as the exchange operator that is used in the rest of the literature on Optimized Schwarz Methods. This is indeed the operator considered in [4,17,16,18,10,31,9,25,32,23,15,14,12,11,5,20,33,2] for enforcing coupling between neighbouring subdomains. Expression (44) can be re-arranged so as to take a more symetric form…”

Section: Bounds On the Trace Operatormentioning

confidence: 99%

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Non-self adjoint impedance in Generalized Optimized Schwarz Methods

Claeys¹

2021

Preprint

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We present a convergence theory for Optimized Schwarz Methods that rely on a nonlocal exchange operator and covers the case of coercive possibly non-self-adjoint impedance operators. This analysis also naturally deals with the presence of cross-points in subdomain partitions of arbitrary shape. In the particular case of self-adjoint impedance, we recover the theory proposed in [Claeys & Parolin,2021].

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Section: Bounds On the Trace Operatormentioning

confidence: 99%

“…Now we can introduce a so called scattering operator. The following result is inspired by [16,Lemma 6]. Lemma 5.2.…”

Section: Scattering Operatormentioning

confidence: 99%

Non-self adjoint impedance in Generalized Optimized Schwarz Methods

Claeys¹

2021

Preprint

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“…Although a rate of convergence was not provided, when it first appeared this work inspired huge interest in non-overlapping methods which continues today and a recent review can be found in the introduction to [13]. Indeed the results of [16], [4] have recently been extended to higher-order boundary conditions in [17]. Furthermore, there has been much interest in handling cross points in non-overlapping domain decomposition methods, e.g., at the PDE level in [11] and at the discrete level in [13] and [49].…”

Section: Literature Reviewmentioning

confidence: 99%

Convergence of parallel overlapping domain decomposition methods for the Helmholtz equation

Gong¹,

Gander²,

Graham³

et al. 2021

Preprint

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We analyse parallel overlapping Schwarz domain decomposition methods for the Helmholtz equation, where the subdomain problems satisfy first-order absorbing (impedance) transmission conditions, and exchange of information between subdomains is achieved using a partition of unity. We provide a novel analysis of this method at the PDE level (without discretization). First, we formulate the method as a fixed point iteration, and show (in dimensions 1,2,3) that it is well-defined in a tensor product of appropriate local function spaces, each with L 2 impedance boundary data. We then obtain a bound on the norm of the fixed point operator in terms of the local norms of certain impedance-to-impedance maps arising from local interactions between subdomains. These bounds provide conditions under which (some power of) the fixed point operator is a contraction. In 2-d, for rectangular domains and strip-wise domain decompositions (with each subdomain only overlapping its immediate neighbours), we present two techniques for verifying the assumptions on the impedance-to-impedance maps that ensure power contractivity of the fixed point operator. The first is through semiclassical analysis, which gives rigorous estimates valid as the frequency tends to infinity. At least for a model case with two subdomains, these results verify the required assumptions for sufficiently large overlap. For more realistic domain decompositions, we directly compute the norms of the impedance-to-impedance maps by solving certain canonical (local) eigenvalue problems. We give numerical experiments that illustrate the theory. These also show that the iterative method remains convergent and/or provides a good preconditioner in cases not covered by the theory, including for general domain decompositions, such as those obtained via automatic graph-partitioning software.

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“…On the other side, when we consider much larger domains (for example via the use of a separate network of nodes rather than with cables) and that the number of nodes is limited, we must switch to iterative or hybrid methods of domain decomposition type. The methods we develop are not only motivated by the current trend in seismic imaging, meaning the development of sparse node devices (OBN) for data acquisition in the oil industry [4], but in the last decades, since the seminal work of Desprès [9], they have become the method of choice when solving the discretised Helmholtz equations.…”

Section: Why the Time-harmonic Problem In Mid And High Frequency Is Hardmentioning

confidence: 99%

Several ways to achieve robustness when solving wave propagation problems

Bootland,

Dolean,

Jolivet

et al. 2021

Preprint

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Corners and stable optimized domain decomposition methods for the Helmholtz problem

Cited by 15 publications

References 29 publications

Non-self adjoint impedance in Generalized Optimized Schwarz Methods

Non-self adjoint impedance in Generalized Optimized Schwarz Methods

Convergence of parallel overlapping domain decomposition methods for the Helmholtz equation

Several ways to achieve robustness when solving wave propagation problems

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