Domain Decomposition and Additive Schwarz Techniques in the Solution of a TE Model of the Scattering by an Electrically Deep Cavity

Balin, Nolwenn; Bendali, Abderrahmane; Collino, Francis

doi:10.1007/3-540-26825-1_11

Cited by 6 publications

(8 citation statements)

References 5 publications

(4 reference statements)

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“…The Schwarz' preconditioner, as formulated in (11), has been used in earlier works for the computation of the Helmholtz problem in bounded domains (see [29,30]) and for the scattering waves (see [10,13,14]). …”

Section: Iterative Methods With Schwarz' Preconditionermentioning

confidence: 99%

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Convergence bounds of GMRES with Schwarz' preconditioner for the scattering problem

Belgacem

Gmati

Jelassi

2009

Numerical Meth Engineering

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SUMMARYWe consider the Jacobi preconditioner of the GMRES method introduced by Liu and Jin for the scattering problem (IEEE Trans. Ante. Prop. 2002; 50:132-140). We explain why it is a particular form of the Schwarz' preconditioner with a complete overlap and specific transmission conditions. So far, a superlinear convergence has been predicted by the general theory without any additional indication on the convergence rates. Here, we establish error bounds that provide accurate convergence rates in two and three dimensions. Courant-Weyl's min-max principle applied to some kernel operators together with some polynomial approximation estimates are the milestones for the proofs.

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Section: Iterative Methods With Schwarz' Preconditionermentioning

confidence: 99%

“…We refer to [2,9,11,13] for such a numerical study. We focus on whether we have a faster convergence than the expected superlinearity.…”

Section: Numerical Computationsmentioning

confidence: 99%

Convergence bounds of GMRES with Schwarz' preconditioner for the scattering problem

Belgacem

Gmati

Jelassi

2009

Numerical Meth Engineering

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“…These can be classified as nonoverlapping methods, if subdomains are interconnected only by their boundaries, for instance finite element tearing interconnecting (FETI) [7][8][9], and overlapping methods, if the subdomains have overlapping parts, which are also known as Schwarz methods [10][11][12][13][14]. At present there are many electromagnetic applications of the DD approach; mainly devoted to the solution on very large and complex problems as finite arrays [15], finite electromagnetic or photonic band gap structures [16], or very large open problems [17].…”

Section: Introductionmentioning

confidence: 99%

A domain decomposition technique for finite element based parametric sweep and tolerance analyses of microwave passive devices

Guarnieri¹,

Pelosi

Rossi

et al. 2008

Int J RF and Microwave Comp Aid Eng

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A domain decomposition approach is here applied to the finite element solution of a multiport waveguide passive device. The approach allows separating the problem in multiple, coupled subproblems which can be solved individually. By appropriately defining one of these subdomains as containing all the possible variations to be studied it is hence possible to restrict the tolerance analysis to this latter, smaller domain. Numerical results showing the gain in computing time are presented.

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“…The idea of using Robin-to-Robin maps as robust alternative to the more popular Dirichlet to Neumann maps can be traced back to the work [22] where it was used to good effect for calculations involving periodic waveguides containing defects/perturbations; see also [15] for a more recent application to computation of guided modes in photonic crystal waveguides. The ideas of using Schur complements for solution of DDM for wave propagation problems was presented in [5] in the context of scattering by deep cavities. The Schur complement elimination procedure that is central to our algorithm is equivalent to a hierarchical merging of the subdomains Robin-to-Robin maps to compute the global interior Robin-to-Robin map of the domain that contains inside the cloud of scatterers.…”

Section: Introductionmentioning

confidence: 99%

Schur complement domain decomposition methods for the solution of multiple scattering problems

Pedneault

Turc

Boubendir

2017

IMA Journal of Applied Mathematics

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We present a Schur complement Domain Decomposition (DD) algorithm for the solution of frequency domain multiple scattering problems. Just as in the classical DD methods we (1) enclose the ensemble of scatterers in a domain bounded by an artificial boundary, (2) we subdivide this domain into a collection of nonoverlapping subdomains so that the boundaries of the subdomains do not intersect any of the scatterers, and (3) we connect the solutions of the subproblems via Robin boundary conditions matching on the common interfaces between subdomains. We use subdomain Robin-to-Robin maps to recast the DD problem as a sparse linear system whose unknown consists of Robin data on the interfaces between subdomains-two unknowns per interface. The Robin-to-Robin maps are computed in terms of well-conditioned boundary integral operators. Unlike classical DD, we do not reformulate the Domain Decomposition problem in the form a fixed point iteration, but rather we solve the ensuing linear system by Gaussian elimination of the unknowns corresponding to inner interfaces between subdomains via Schur complements. Once all the unknowns corresponding to inner subdomains interfaces have been eliminated, we solve a much smaller linear system involving unknowns on the inner and outer artificial boundary. We present numerical evidence that our Schur complement DD algorithm can produce accurate solutions of very large multiple scattering problems that are out of reach for other existing approaches.

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Domain Decomposition and Additive Schwarz Techniques in the Solution of a TE Model of the Scattering by an Electrically Deep Cavity

Cited by 6 publications

References 5 publications

Convergence bounds of GMRES with Schwarz' preconditioner for the scattering problem

Convergence bounds of GMRES with Schwarz' preconditioner for the scattering problem

A domain decomposition technique for finite element based parametric sweep and tolerance analyses of microwave passive devices

Schur complement domain decomposition methods for the solution of multiple scattering problems

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