In this work we are concerned with an efficient numerical solution of a perfectly matched layer (PML) system for a Maxwell scattering problem. The PML system is discretized by the edge finite element method, resulting in a symmetric but indefinite complex algebraic system. When the real and imaginary parts are considered independently, the complex algebraic system can be further transformed into a real generalized saddle-point system with some special structure. Based on an crucial observation to its Schur complement, we construct a symmetric and positive definite block diagonal preconditioner for the saddle-point system. Numerical experiments are presented to demonstrate the effectiveness and robustness of the new preconditioner.
DOI : 10.1051/m2an/2014058
Mots-clés : Maxwell scattering problem, edge finite elements, PML equations, Schur complement-type preconditioner
@article{M2AN_2015__49_3_839_0,
author = {Hu, Qiya and Liu, Chunmei and Shu, Shi and Zou, Jun},
title = {An effective preconditioner for a {PML} system for electromagnetic scattering problem},
journal = {ESAIM: Mathematical Modelling and Numerical Analysis },
pages = {839--854},
publisher = {EDP-Sciences},
volume = {49},
number = {3},
year = {2015},
doi = {10.1051/m2an/2014058},
mrnumber = {3342230},
zbl = {1318.35115},
language = {en},
url = {http://archive.numdam.org/articles/10.1051/m2an/2014058/}
}
TY - JOUR AU - Hu, Qiya AU - Liu, Chunmei AU - Shu, Shi AU - Zou, Jun TI - An effective preconditioner for a PML system for electromagnetic scattering problem JO - ESAIM: Mathematical Modelling and Numerical Analysis PY - 2015 SP - 839 EP - 854 VL - 49 IS - 3 PB - EDP-Sciences UR - http://archive.numdam.org/articles/10.1051/m2an/2014058/ DO - 10.1051/m2an/2014058 LA - en ID - M2AN_2015__49_3_839_0 ER -
%0 Journal Article %A Hu, Qiya %A Liu, Chunmei %A Shu, Shi %A Zou, Jun %T An effective preconditioner for a PML system for electromagnetic scattering problem %J ESAIM: Mathematical Modelling and Numerical Analysis %D 2015 %P 839-854 %V 49 %N 3 %I EDP-Sciences %U http://archive.numdam.org/articles/10.1051/m2an/2014058/ %R 10.1051/m2an/2014058 %G en %F M2AN_2015__49_3_839_0
Hu, Qiya; Liu, Chunmei; Shu, Shi; Zou, Jun. An effective preconditioner for a PML system for electromagnetic scattering problem. ESAIM: Mathematical Modelling and Numerical Analysis , Tome 49 (2015) no. 3, pp. 839-854. doi : 10.1051/m2an/2014058. http://archive.numdam.org/articles/10.1051/m2an/2014058/
and , On the analysis of Bérenger’s perfectly matched layers for Maxwell’s equations. ESAIM: M2AN 36 (2002) 87–119. | DOI | Numdam | MR | Zbl
, A perfectly matched layer for the absorption of electromagnetic waves. J. Comput. Phys. 114 (1994) 185-200. | DOI | MR | Zbl
J.-P. Bérenger, Perfectly Matched Layer (PML) for Computational Electromagnetics. Vol. 2 of Synthesis Lectures on Computational Electromagnetics. Morgan Claypool (2007).
and , Time domain electromagnetic scattering using finite elements and perfectly matched layers. Comput. Methods. Appl. Mech. Eng. 194 (2005) 149–168. | DOI | MR | Zbl
and , Convergence analysis of the perfectly matched layer problems for time harmonic Maxwell’s equations. SIAM J. Numer. Anal. 43 (2005) 2121–2143. | DOI | MR | Zbl
and , Preconditioned generalized minimal residual iterative scheme for perfectly matched layer terminated applications. IEEE Microw. Guided Wave Letters 9 (1999) 45–47. | DOI
and , Perfectly matched layer termination for finite-element meshed: Implementation and application, Microwave. Optim. Tech. Lett. 23 (1999) 166–172. | DOI
and , Analysis of a cartesian PML approximation to the three dimensional electromagnetic wave scattering problem, Int. J. Numer. Anal. Model. 9 (2012) 543–561. | MR | Zbl
and , An adaptive perfectly matched layer technique for 3-D time harmonic electromagnetic scattering problems. Math. Comput. 77 (2008) 673–698. | DOI | MR | Zbl
, , and , An Adaptive Finite Element Method for the Eddy Current Model with Circuit/Field Couplings, SIAM J. Sci. Comput. 32 (2010) 1020–1042. | DOI | MR | Zbl
, and , An adaptive anisotropic perfectly matched layer method for 3-D time harmonic electromagnetic scattering problems, Numer. Math. 125 (2013) 639–677. | DOI | MR | Zbl
and , A 3d perfectly matched medium for modified Maxwell’s equations with streched coordinates. Microwave Opt. Technol. Lett. 13 (1994) 599–604. | DOI
, Multigrid method for Maxwell’s equations. SIAM J. Numer. Anal. 36 (1998) 204–225. | DOI | MR | Zbl
and , Nodal auxiliary spaces preconditions in H(curl) and H(div) spaces. SIAM J. Numer. Anal. 45 (2007) 2483–2509. | DOI | MR | Zbl
and , Substructuring preconditioners for saddle-point problems arising from Maxwell’s equations in three dimensions. Math. Comput. 73 (2004) 35–61. | DOI | MR | Zbl
E. Post, Formal Structure of Electromagnetics: General Covariance and Electromagnetics. Dover (1997). | Zbl
and , Choice of the perfectly matched layer boundary condition for frequency-domain Maxwell’s equations solvers. J. Comput. Phys. 231 (2012) 3406–3431. | DOI | MR | Zbl
and , Differential forms, metrics, and the reflectionless absorption of electromagnetic waves. J. Electromagn. Waves Appl. 13 (1999) 665–686. | DOI | MR | Zbl
, Overlapping Schwarz methods for Maxwell’s equations in three dimensions. Numer. Math. 86 (2000) 733–752. | DOI | MR | Zbl
, and , An iterative substructuring method for Maxwell’s equations in two dimensions. Math. Comput. 70 (2001) 935–947. | DOI | MR | Zbl
, and , A sweeping preconditioner for time-harmonic Maxwell’s equations with finite elements. J. Comput. Phys. 231 (2012) 3770–3783. | DOI | MR | Zbl
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