Optimized Schwarz method with complete radiation transmission conditions for the Helmholtz equation in waveguides

Seungil Kim; Hui Zhang

doi:10.1137/140980491

Optimized Schwarz method with complete radiation transmission conditions for the Helmholtz equation in waveguides

Seungil Kim, Hui Zhang

Research output: Contribution to journal › Article › peer-review

16 Citations (Scopus)

Abstract

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. J. Gander, F. Magoules, and F. Nataf, SIAM J. Sci. Comput., 24 (2002), pp. 38-60]. Finally, numerical examples confirming the theory will be presented.

Original language	English
Pages (from-to)	1537-1558
Number of pages	22
Journal	SIAM Journal on Numerical Analysis
Volume	53
Issue number	3
DOIs	https://doi.org/10.1137/140980491
Publication status	Published - 2015
Externally published	Yes

Keywords

Complete radiation boundary
Condition
Helmholtz equation
Nonoverlapping Schwarz method
Waveguide

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10.1137/140980491

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title = "Optimized Schwarz method with complete radiation transmission conditions for the Helmholtz equation in waveguides",

abstract = "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. J. Gander, F. Magoules, and F. Nataf, SIAM J. Sci. Comput., 24 (2002), pp. 38-60]. Finally, numerical examples confirming the theory will be presented.",

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author = "Seungil Kim and Hui Zhang",

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year = "2015",

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pages = "1537--1558",

journal = "SIAM Journal on Numerical Analysis",

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T1 - Optimized Schwarz method with complete radiation transmission conditions for the Helmholtz equation in waveguides

AU - Kim, Seungil

AU - Zhang, Hui

N1 - Publisher Copyright: © by SIAM.

PY - 2015

Y1 - 2015

N2 - 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. J. Gander, F. Magoules, and F. Nataf, SIAM J. Sci. Comput., 24 (2002), pp. 38-60]. Finally, numerical examples confirming the theory will be presented.

AB - 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. J. Gander, F. Magoules, and F. Nataf, SIAM J. Sci. Comput., 24 (2002), pp. 38-60]. Finally, numerical examples confirming the theory will be presented.

KW - Complete radiation boundary

KW - Condition

KW - Helmholtz equation

KW - Nonoverlapping Schwarz method

KW - Waveguide

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Optimized Schwarz method with complete radiation transmission conditions for the Helmholtz equation in waveguides

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Keywords

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