Non-overlapping Schwarz methods with optimized transmission conditions for the Helmholtz equation

Magoulès, Frédéric; Iványi, P.; Topping, B. H. V.

doi:10.1016/j.cma.2004.05.004

Cited by 56 publications

(46 citation statements)

References 6 publications

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“…. .. Table I shows the number of iterations needed for different values of the mesh parameter h for one-sided optimized Robin conditions (see [14,22]), and the new two-sided optimized Robin conditions (see Theorem 4.1), and compares the results with Taylor conditions (i.e. s 12 = s 21 = iω, see [8]) in the case of Krylov acceleration (without, Taylor conditions do not lead to a convergent algorithm, because for all frequencies k > ω, the convergence factor equals 1).…”

Section: A Model Problemmentioning

confidence: 99%

“…Previous optimized Schwarz methods with Robin transmission conditions reduced the number of free parameters by setting s 12 = s 21 . This led in [14,22,21] to an optimized Schwarz method with asymptotic convergence factor…”

Section: Optimization Of the Transmission Conditionsmentioning

confidence: 99%

“…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]. For Helmholtz problems, optimized Schwarz methods were studied and analyzed with one-sided Robin transmission conditions, and second order transmission conditions in [14,22,21]. A different approach using perfectly matched layers was proposed in [27].…”

Section: Introductionmentioning

confidence: 99%

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

Gander¹,

Halpern

Magoulès

2007

Numerical Methods in Fluids

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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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Section: A Model Problemmentioning

confidence: 99%

Section: Optimization Of the Transmission Conditionsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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

Gander¹,

Halpern

Magoulès

2007

Numerical Methods in Fluids

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103

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“…In order to accelerate the convergence of the iterative method (7)- (10), one usually applies a Krylov method (Saad, 1996) to solve directly the interface system formulated in the variables λ 1 and λ 2 , see (Magoulès et al, 2004a). This interface system is obtained by considering (7)-(10) without iteration index k, eliminating u 1 and u 2 from (7) and (8), and then inserting the resulting values for u Γ1 and u Γ2 into (9) and (10), which results in a system in the interface unknowns λ 1 and λ 2 only.…”

Section: Numerical Experimentsmentioning

confidence: 99%

Analysis of Patch Substructuring Methods

Gander¹,

Halpern²,

Magoulès³

et al. 2007

International Journal of Applied Mathematics and Computer Science

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Patch substructuring methods are non-overlapping domain decomposition methods like classical substructuring methods, but they use information from geometric patches reaching into neighboring subdomains, condensated on the interfaces, to enhance the performance of the method, while keeping it non-overlapping. These methods are very convenient to use in practice, but their convergence properties have not been studied yet. We analyze geometric patch substructuring methods for the special case of one patch per interface. We show that this method is equivalent to an overlapping Schwarz method using Neumann transmission conditions. This equivalence is obtained by first studying a new, algebraic patch method, which is equivalent to the classical Schwarz method with Dirichlet transmission conditions and an overlap corresponding to the size of the patches. Our results motivate a new method, the Robin patch method, which is a linear combination of the algebraic and the geometric one, and can be interpreted as an optimized Schwarz method with Robin transmission conditions. This new method has a significantly faster convergence rate than both the algebraic and the geometric one. We complement our results by numerical experiments.

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“…However, the methods also work well in more general settings in many applications; see, for example, [17] for the temperature distribution in an apartment in Montreal and [22,33] for the noise levels in a car compartment. A more general analysis in [31] shows that the asymptotic choice of O(h − 1 2 ) of the Robin parameter (h being the local mesh size) will result in a contraction factor of the form 1 − O(h 1 2 ) for a nonoverlapping Schwarz method.…”

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confidence: 99%

Optimized Schwarz Methods for Circular Domain Decompositions with Overlap

Gander¹,

Xu²

2014

SIAM J. Numer. Anal.

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Abstract. Optimized Schwarz methods are based on transmission conditions between subdomains which are optimized for the problem class that is being solved. Such optimizations have been performed for many different types of partial differential equations, but were almost exclusively based on the assumption of straight interfaces. We study in this paper the influence of curvature on the optimization, and we obtain four interesting new results: first, we show that the curvature does indeed enter the optimized parameters and the contraction factor estimates. Second, we develop an asymptotically accurate approximation technique, based on Turán type inequalities in our case, to solve the much harder optimization problem on the curved interface, and this approximation technique will also be applicable to currently too complex best approximation problems in the area of optimized Schwarz methods. Third, we show that one can obtain transmission conditions from a simple circular model decomposition which have also been found using microlocal analysis but that these are not the best choices for the performance of the optimized Schwarz method. Finally, we find that in the case of curved interfaces, optimized Schwarz methods are not necessarily convergent for all admissible parameters. Our optimization leads, however, to parameter choices that give the same good performance for a circular decomposition as for a straight interface decomposition. We illustrate our analysis with numerical experiments.Key words. optimized Schwarz method, optimized transmission conditions, circular domain decomposition, interface curvature, parallel computation AMS subject classifications. 65N55, 65F10DOI. 10.1137/130946125 1. Introduction. Domain decomposition methods are important techniques for the numerical simulation of large scale physical problems, since they can systematically reduce their complexity by decomposition and lead to efficient parallel solvers. Among the many domain decomposition methods, optimized Schwarz methods, going back to an idea for a nonoverlapping method by Lions [28], have attracted substantial attention over the past decade because their transmission conditions can be adapted to the physics of the underlying problem and thus lead to very efficient methods for hard problems; for an overview and references, see [35,17]. Optimized Schwarz methods are a very active area of research, and they have found their way into many areas of applications: for Helmholtz problems, see [12,6,22], for advection diffusion evolution problems, see [34,20,7,40], for wave equations, see [21,19], for Maxwell problems, see [41,13,38,37,8], and for shallow water and the primitive equations, see [39,3]. Optimized Schwarz methods have also led to new theoretical developments (see, for example, [27,30,29]) and many innovative preconditioners that appeared under different names; see, for example, the sweeping preconditioner in [16,2,23,14], the source transfer method in [11], and also [42], which are among the best iterative

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Non-overlapping Schwarz methods with optimized transmission conditions for the Helmholtz equation

Cited by 56 publications

References 6 publications

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

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

Analysis of Patch Substructuring Methods

Optimized Schwarz Methods for Circular Domain Decompositions with Overlap

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