A quasi-optimal non-overlapping domain decomposition algorithm for the Helmholtz equation

Boubendir, Yassine; Antoine, Xavier; Geuzaine, Christophe

doi:10.1016/j.jcp.2011.08.007

Cited by 119 publications

(190 citation statements)

References 38 publications

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“…We have the constraint that its null-space must be equal to {0}. It is well known that this choice influences the rate of convergence [5,7], and it was shown in [19] that the optimum is obtained for S being the exterior Dirichlet-to-Neumann (DtN) map D of the complement of the subdomain Ω \ Ω i , defined on a boundary Σ as:…”

Section: Description Of the Algorithmmentioning

confidence: 99%

“…[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%

“…The solution produced by these sources inside the individual subdomains must match the restriction of the solution of the full problem on the subdomains. Such a formulation falls into the framework of Schwarz methods [2,3,4,5]; recent improvements in the approximation of the transmission operators have made its convergence rate little sensitive to the wavenumber and discretization density [6,7].…”

Section: Introductionmentioning

confidence: 99%

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

Vion

Geuzaine

2014

Journal of Computational Physics

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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: Description Of the Algorithmmentioning

confidence: 99%

Section: Dirichlet-to-neumann Mapmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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

Vion

Geuzaine

2014

Journal of Computational Physics

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104

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“…Finally the solution of the interface problem serves as the right-handside of each local problem. This method has been applied in many domains like mechanics [19,20], acoustic wave propagation [21][22][23], and in electromagnetism [24][25][26][27][28][29][30][31][32]. For example, related DDM methods have been developed for simulating the interactions of photonic crystals with electromagnetic waves [33,34].…”

Section: Introductionmentioning

confidence: 99%

“…This avoids the appearing of spurious solutions. This boundary condition can also be seen as a crude approximation of a transparency operator and many efforts have been done for optimizing the coefficients arising in this boundary condition, but only when the interfaces between subdomains are plane [23,30,31,35].…”

Section: Introductionmentioning

confidence: 99%

Scattered Field Computation With an Extended Feti-Dpem2 Method

Voznyuk¹,

Tortel²,

Litman³

2013

PIER

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Abstract-Due to the increasing number of applications in engineering design and optimization, more and more attention has been paid to full-wave simulations based on computational electromagnetics. In particular, the finite-element method (FEM) is well suited for problems involving inhomogeneous and arbitrary shaped objects. Unfortunately, solving large-scale electromagnetic problems with FEM may be time consuming. A numerical scheme, called the dual-primal finite element tearing and interconnecting method (FETI-DPEM2), distinguishes itself through the partioning on the computation domain into non-overlapping subdomains where incomplete solutions of the electrical field are evaluated independently. Next, all the subdomains are "glued" together using a modified Robin-type transmission condition along each common internal interface, apart from the corner points where a simple Neumann-type boundary condition is imposed. We propose an extension of the FETI-DPEM2 method where we impose a Robin type boundary conditions at each interface point, even at the corner points. We have implemented this Extended FETI-DPEM2 method in a bidimensional configuration while computing the field scattered by a set of heterogeneous, eventually anistropic, scatterers. The results presented here will assert the efficiency of the proposed method with respect to the classical FETI-DPEM2 method, whatever the mesh partition is arbitrary defined.

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A two‐level domain decomposition method with accurate interface conditions for the Helmholtz problem

Astaneh

Guddati

2015

Numerical Meth Engineering

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Summary A new and efficient two‐level, non‐overlapping domain decomposition (DD) method is developed for the Helmholtz equation in the two Lagrange multiplier framework. The transmission conditions are designed by utilizing perfectly matched discrete layers (PMDLs), which are a more accurate representation of the exterior Dirichlet‐to‐Neumann map than the polynomial approximations used in the optimized Schwarz method. Another important ingredient affecting the convergence of a DD method, namely, the coarse space augmentation, is also revisited. Specifically, the widely successful approach based on plane waves is modified to that based on interface waves, defined directly on the subdomain boundaries, hence ensuring linear independence and facilitating the estimation of the optimal size for the coarse problem. The effectiveness of both PMDL‐based transmission conditions and interface‐wave‐based coarse space augmentation is illustrated with an array of numerical experiments that include comprehensive scalability studies with respect to frequency, mesh size and the number of subdomains. Copyright © 2015 John Wiley & Sons, Ltd.

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A quasi-optimal non-overlapping domain decomposition algorithm for the Helmholtz equation

Cited by 119 publications

References 38 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

Scattered Field Computation With an Extended Feti-Dpem2 Method

A two‐level domain decomposition method with accurate interface conditions for the Helmholtz problem

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