Convergence estimates for an higher order optimized Schwarz method for domains with an arbitrary interface

Lui, S. H.

doi:10.1016/j.cam.2010.06.007

Cited by 8 publications

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“…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. Similar results were obtained for higher order transmission conditions [32].…”

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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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confidence: 78%

Optimized Schwarz Methods for Circular Domain Decompositions with Overlap

Gander¹,

Xu²

2014

SIAM J. Numer. Anal.

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“…Optimized Schwarz methods for overlapping circular decompositions have been studied in [28]. For a nonoverlapping optimized Schwarz method with nonstraight interfaces, spectral estimates of Poincaré-Steklov operators confirmed the asymptotic behavior of optimized transmission conditions in the mesh size (see [32,33]), but the precise dependence on the constants and the interface curvature could not be captured. While the overlap in general accelerates the convergence of optimized Schwarz methods [18], nonoverlapping variants have advantages for problems with jumping coefficients [14,15,21], heterogeneous media [34] and also certain interface problems [43].…”

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confidence: 99%

Optimized Schwarz methods with nonoverlapping circular domain decomposition

Gander¹,

2016

Math. Comp.

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While the classical Schwarz method can only be used with overlap, optimized Schwarz methods can also be used without overlap, which can be an advantage when simulating heterogeneous problems, problems with jumping coefficients, or also for independent mesh generation per subdomain. The analysis of nonoverlapping optimized Schwarz methods has so far been restricted to the case of straight interfaces, even though the method has been successfully used with curved interfaces. We close this gap by presenting a rigorous analysis of optimized Schwarz methods for circular domain decompositions. We derive optimized zeroth and second order transmission conditions for a model elliptic operator in two dimensions, and show why the straight interface analysis results, when properly scaled to include the curvature, are also successful for curved interfaces. Our analysis thus complements earlier asymptotic results by Lui for curved interfaces, where the influence of the curvature remained unknown. We illustrate our results with numerical experiments.

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“…Más recientemente se introdujo una nueva formulación de mayor orden de las condiciones de frontera entre los subdominios que produciría tasas de convergencia óptimas (Lui 2010).…”

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Métodos avanzados multiescala núcleo-celdas en geometrías tridimensionales y multigrupos para el cálculo de Reactores de Agua Ligera

Carrascosa¹

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Advanced multi-scale pin-by-pin/core methods in three-dimensional geometries and multigroups for Light Water Reactors calculationsThe relevance of safety in the use of nuclear technology impregnates all the tasks associated to the use of this energy source, beginning with the design phase, exploitation and subsequent dismantling or waste management.In all these steps, computational simulation tools play an essential role as a guide for design, operation supporting or prediction of the reactor materials isotopic evolution.Constant improvements in computational resources from the middle of the 20 th century to this moment as well as the developments in the calculation methods employed, allow tackling the complexity of these situations with greater detail, when this was simply discarded before because of a lack of computational capacity or adequate tools.Present work is centered in the development of a neutronics calculation method for light water reactors based on corrected diffusion theory with a detail to the level of the fuel pin, considering a number of energy groups higher than the traditional fast and thermal groups, and modeling the three-dimensional geometry of the reactor core.

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Convergence estimates for an higher order optimized Schwarz method for domains with an arbitrary interface

Cited by 8 publications

References 18 publications

Optimized Schwarz Methods for Circular Domain Decompositions with Overlap

Optimized Schwarz Methods for Circular Domain Decompositions with Overlap

Optimized Schwarz methods with nonoverlapping circular domain decomposition

Métodos avanzados multiescala núcleo-celdas en geometrías tridimensionales y multigrupos para el cálculo de Reactores de Agua Ligera

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