Additive Schwarz Method for Scattering Problems Using the PML Method at Interfaces

Schädle, Achim; Zschiedrich, Lin

doi:10.1007/978-3-540-34469-8_21

Cited by 22 publications

(20 citation statements)

References 10 publications

(14 reference statements)

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“…Last, it is worth pointing out that a sweeping preconditioning approach based on perfectly matched layer (PML) for transmission conditions emerged recently in [21] and led to many other analyses on fast solvers of the Helmholtz equation, e.g., [3,23], which are considered as symmetric Gauss-Seidel preconditioning techniques in GM-RES implementation. In this paper, the Schwarz algorithm with CRBC transmission conditions is examined also as Jacobi-type GMRES preconditioners and a sweeping preconditioning approach with CRBC transmission conditions will be studied elsewhere.…”

Section: Seungil Kim and Hui Zhangmentioning

confidence: 99%

Optimized Schwarz Method with Complete Radiation Transmission Conditions for the Helmholtz Equation in Waveguides

Kim¹,

Zhang²

2015

SIAM J. Numer. Anal.

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In this paper, we present a nonoverlapping optimized Schwarz algorithm with a highorder transmission condition for the Helmholtz equation posed in waveguides. We introduce the new high-order transmission conditions based on the complete radiation boundary conditions (CRBCs) that have been developed for high-order absorbing boundary conditions. We obtain the convergence rate of the algorithm in terms of reflection coefficients of CRBCs, which decrease exponentially with the order of CRBCs. It will be shown that damping parameters involved in the transmission conditions can be selected in an optimal way for enhancing the convergence of the Schwarz algorithm. Also, this algorithm can be employed efficiently for a preconditioner in GMRES implementations based on the substructured form as in [M.1. Introduction. In this paper, we introduce a new high-order transmission condition for the nonoverlapping Schwarz method for the Helmholtz equation posed in waveguides. The Schwarz method [22, 24] is a well-known numerical technique that can provide an efficient solver for treating large scale problems arising from physics and engineering by allowing parallel computations. In developing an optimized Schwarz algorithm without overlap for the Helmholtz equation, it is important to design a highorder transmission condition for communication between neighboring subregions. To do this, we introduce a new transmission condition based on the complete radiation boundary conditions (CRBCs) designed for high-order absorbing boundary conditions for time-harmonic wave propagation problems in [15,16].We consider a nonoverlapping Schwarz algorithm in a waveguide in R d with d = 2 or 3. The model problem is to find the solution u satisfying the Helmholtz equation in a waveguide Ω = {(x, y) ∈ R d : x ∈ R and y ∈ Θ} with Θ a bounded domain in R d−1 with Lipschitz boundary ∂Θ:

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Section: Seungil Kim and Hui Zhangmentioning

confidence: 99%

Optimized Schwarz Method with Complete Radiation Transmission Conditions for the Helmholtz Equation in Waveguides

Kim¹,

Zhang²

2015

SIAM J. Numer. Anal.

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“…In our previous paper [20] the DtN operator was used to state the coupling between the different domains. However the DtN operator must not exist in the periodic setting -the exterior Dirichlet problem is ill-posed in the presence of anomalous modes.…”

Section: Ill-posed Exterior Dirichlet/neumann Boundary Value Problemsmentioning

confidence: 99%

“…In each sub-domain a simplified scattering problem is solved and the scattered field is added to the incoming field for the neighboring domains. We have recently presented an additive Schwarz algorithm for Helmholtz scattering problems with transparent boundary conditions at the domain interfaces [20]. In this publication we used the DtN operator (Dirichlet to Neumann map).…”

Section: Introductionmentioning

confidence: 99%

Domain decomposition method for Maxwell’s equations: Scattering off periodic structures

Schädle

Zschiedrich

Burger

et al. 2007

Journal of Computational Physics

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We present a domain decomposition approach for the computation of the electromagnetic field within periodic structures. We use a Schwarz method with transparent boundary conditions at the interfaces of the domains. Transparent boundary conditions are approximated by the perfectly matched layer method (PML). To cope with Wood anomalies appearing in periodic structures an adaptive strategy to determine optimal PML parameters is developed.We focus on the application to typical EUV lithography line masks. Light propagation within the multi-layer stack of the EUV mask is treated analytically. This results in a drastic reduction of the computational costs and allows for the simulation of next generation lithography masks on a standard personal computer.

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“…In the sequel, different, more advanced techniques have been used; including PML at the interfaces [17][18][19], non-local transmission conditions [20], optimized Schwarz methods [21], and others, e.g. [22].…”

Section: Introductionmentioning

confidence: 99%

A coarse space for heterogeneous Helmholtz problems based on the Dirichlet-to-Neumann operator

Conen

Dolean

Krause

et al. 2014

Journal of Computational and Applied Mathematics

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Additive Schwarz Method for Scattering Problems Using the PML Method at Interfaces

Cited by 22 publications

References 10 publications

Optimized Schwarz Method with Complete Radiation Transmission Conditions for the Helmholtz Equation in Waveguides

Optimized Schwarz Method with Complete Radiation Transmission Conditions for the Helmholtz Equation in Waveguides

Domain decomposition method for Maxwell’s equations: Scattering off periodic structures

A coarse space for heterogeneous Helmholtz problems based on the Dirichlet-to-Neumann operator

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