A non-overlapping domain decomposition method for the exterior Helmholtz problem

Bourdonnaye, Armel de La; Farhat, Charbel; Macedo, Antonini; Magoulès, Frédéric; Roux, François‐Xavier

doi:10.1090/conm/218/03001

Cited by 54 publications

(65 citation statements)

References 5 publications

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“…[3]. A great variety of techniques based on local transmission conditions have thus been proposed over the years: these include the class of FETI-H methods [31,32,10,33], the optimized Schwarz approach [6], the evanescent modes damping algorithm [34,35,36] and the Padé-localized square-root operator [7]. All these local transmission conditions can be seen as approximations of the exact DtN operator; the better the related impedance operators approximate the exact DtN operator on all the modes of the solution, the better the convergence properties of the resulting DDM.…”

Section: Dirichlet-to-neumann Mapmentioning

confidence: 99%

Double sweep preconditioner for optimized Schwarz methods applied to the Helmholtz problem

Vion

Geuzaine

2014

Journal of Computational Physics

104

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This paper presents a preconditioner for non-overlapping Schwarz methods applied to the Helmholtz problem. Starting from a simple analytic example, we show how such a preconditioner can be designed by approximating the inverse of the iteration operator for a layered partitioning of the domain. The preconditioner works by propagating information globally by concurrently sweeping in both directions over the subdomains, and can be interpreted as a coarse grid for the domain decomposition method. The resulting algorithm is shown to converge very fast, independently of the number of subdomains and frequency. The preconditioner has the advantage that, like the original Schwarz algorithm, it can be implemented as a matrix-free routine, with no additional preprocessing.

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Section: Dirichlet-to-neumann Mapmentioning

confidence: 99%

Double sweep preconditioner for optimized Schwarz methods applied to the Helmholtz problem

Vion

Geuzaine

2014

Journal of Computational Physics

104

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“…in [9,5,7,10,6,2]. The name optimized Schwarz methods was introduced in [13] to denote the class of Schwarz methods with improved transmission conditions that has been developed over the previous years in [4,17,20]; for an up to date historical review, and complete results for the positive definite case, see [15].…”

Section: Introductionmentioning

confidence: 99%

An optimized Schwarz method with two‐sided Robin transmission conditions for the Helmholtz equation

Gander¹,

Halpern

Magoulès

2007

Numerical Methods in Fluids

103

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SUMMARYOptimized Schwarz methods are working like classical Schwarz methods, but they are exchanging physically more valuable information between subdomains and hence have better convergence behavior. The new transmission conditions include also derivative information, not just function values, and optimized Schwarz methods can be used without overlap. In this paper, we present a new optimized Schwarz method without overlap in the 2D case, which uses a different Robin condition for neighboring subdomains at their common interface, and which we call two-sided Robin condition. We optimize the parameters in the Robin conditions and show that for a fixed frequency ω, an asymptotic convergence factor of 1 − O(h 1 4 ) in the mesh parameter h can be achieved. If the frequency is related to the mesh parameter h, h = O( 1 ω γ ) for γ ≥ 1, then the optimized asymptotic convergence factor is 1 − O(ω 1−2γ 8 ). We illustrate our analysis with 2D numerical experiments.

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“…The classical Schwarz algorithm is not effective for Helmholtz problems because the convergence mechanism of the Schwarz algorithm works only for the evanescent modes, not for the propagative ones. Nevertheless, the Schwarz algorithm has been applied to Helmholtz problems by adding a relatively fine coarse mesh for the propagative modes in [3] and changing the transmission conditions from Dirichlet in the classical Schwarz case to Robin, as first done in [9], and then in [8], [1], [7], [22], [2], [21], [6]. We study in this paper the influence of the transmission conditions on the Schwarz algorithm for the Helmholtz equation.…”

mentioning

confidence: 99%

Optimized Schwarz Methods without Overlap for the Helmholtz Equation

Gander¹,

Magoulès²,

Nataf³

2002

SIAM J. Sci. Comput.

Self Cite

319

315

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Abstract. The classical Schwarz method is a domain decomposition method to solve elliptic partial differential equations in parallel. Convergence is achieved through overlap of the subdomains. We study in this paper a variant of the Schwarz method which converges without overlap for the Helmholtz equation. We show that the key ingredients for such an algorithm are the transmission conditions. We derive optimal transmission conditions which lead to convergence of the algorithm in a finite number of steps. These conditions are, however, nonlocal in nature, and we introduce local approximations which we optimize for performance of the Schwarz method. This leads to an algorithm in the class of optimized Schwarz methods. We present an asymptotic analysis of the optimized Schwarz method for two types of transmission conditions, Robin conditions and transmission conditions with second order tangential derivatives. Numerical results illustrate the effectiveness of the optimized Schwarz method on a model problem and on a problem from industry.Key words. optimized Schwarz methods, domain decomposition, preconditioner, iterative parallel methods, acoustics AMS subject classifications. 65F10, 65N22PII. S10648275013870121. Introduction. The classical Schwarz algorithm has a long history. It was invented by Schwarz more than a century ago [25] to prove existence and uniqueness of solutions to Laplace's equation on irregular domains. Schwarz decomposed the irregular domain into overlapping regular ones and formulated an iteration which used only solutions on regular domains and which converged to a unique solution on the irregular domain. A century later the Schwarz method was proposed as a computational method by Miller in [23], but it was only with the advent of parallel computers that the Schwarz method really gained popularity and was analyzed in depth both at the continuous level (see, for example, [17] [6]. We study in this paper the influence of the transmission conditions on the Schwarz algorithm for the Helmholtz equation. We derive optimal transmission conditions which lead to the best possible convergence of the Schwarz algorithm and which do not require overlap to be effective as in [12]. These optimal

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A non-overlapping domain decomposition method for the exterior Helmholtz problem

Cited by 54 publications

References 5 publications

Double sweep preconditioner for optimized Schwarz methods applied to the Helmholtz problem

Double sweep preconditioner for optimized Schwarz methods applied to the Helmholtz problem

An optimized Schwarz method with two‐sided Robin transmission conditions for the Helmholtz equation

Optimized Schwarz Methods without Overlap for the Helmholtz Equation

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