no. 5
Convergence of the uniaxial PML method for time-domain electromagnetic scattering problems
ESAIM: Mathematical Modelling and Numerical Analysis , Tome 55 (2021) no. 5, pp. 2421-2443

In this paper, we propose and study the uniaxial perfectly matched layer (PML) method for three-dimensional time-domain electromagnetic scattering problems, which has a great advantage over the spherical one in dealing with problems involving anisotropic scatterers. The truncated uniaxial PML problem is proved to be well-posed and stable, based on the Laplace transform technique and the energy method. Moreover, the L2-norm and L∞-norm error estimates in time are given between the solutions of the original scattering problem and the truncated PML problem, leading to the exponential convergence of the time-domain uniaxial PML method in terms of the thickness and absorbing parameters of the PML layer. The proof depends on the error analysis between the EtM operators for the original scattering problem and the truncated PML problem, which is different from our previous work (Wei et al. [SIAM J. Numer. Anal. 58 (2020) 1918–1940]).

DOI : 10.1051/m2an/2021064
Classification : 65N30, 65N50, 78A25
Keywords: Well-posedness, stability, time-domain electromagnetic scattering, uniaxial PML, exponential convergence 
@article{M2AN_2021__55_5_2421_0,
     author = {Wei, Changkun and Yang, Jiaqing and Zhang, Bo},
     title = {Convergence of the uniaxial {PML} method for time-domain electromagnetic scattering problems},
     journal = {ESAIM: Mathematical Modelling and Numerical Analysis },
     pages = {2421--2443},
     year = {2021},
     publisher = {EDP-Sciences},
     volume = {55},
     number = {5},
     doi = {10.1051/m2an/2021064},
     mrnumber = {4330731},
     language = {en},
     url = {https://www.numdam.org/articles/10.1051/m2an/2021064/}
}
TY  - JOUR
AU  - Wei, Changkun
AU  - Yang, Jiaqing
AU  - Zhang, Bo
TI  - Convergence of the uniaxial PML method for time-domain electromagnetic scattering problems
JO  - ESAIM: Mathematical Modelling and Numerical Analysis 
PY  - 2021
SP  - 2421
EP  - 2443
VL  - 55
IS  - 5
PB  - EDP-Sciences
UR  - https://www.numdam.org/articles/10.1051/m2an/2021064/
DO  - 10.1051/m2an/2021064
LA  - en
ID  - M2AN_2021__55_5_2421_0
ER  - 
%0 Journal Article
%A Wei, Changkun
%A Yang, Jiaqing
%A Zhang, Bo
%T Convergence of the uniaxial PML method for time-domain electromagnetic scattering problems
%J ESAIM: Mathematical Modelling and Numerical Analysis 
%D 2021
%P 2421-2443
%V 55
%N 5
%I EDP-Sciences
%U https://www.numdam.org/articles/10.1051/m2an/2021064/
%R 10.1051/m2an/2021064
%G en
%F M2AN_2021__55_5_2421_0
Wei, Changkun; Yang, Jiaqing; Zhang, Bo. Convergence of the uniaxial PML method for time-domain electromagnetic scattering problems. ESAIM: Mathematical Modelling and Numerical Analysis , Tome 55 (2021) no. 5, pp. 2421-2443. doi: 10.1051/m2an/2021064

[1] G. Bao and H. Wu, Convergence analysis of the perfectly matched layer problems for time-harmonic Maxwell’s equations. SIAM. J. Numer. Anal. 43 (2005) 2121–2143. | MR | Zbl | DOI

[2] G. Bao, Y. Gao and P. Li, Time-domain analysis of an acoustic-elastic interaction problem. Arch. Ration. Mech. Anal. 229 (2018) 835–884. | MR | DOI

[3] J. P. Bérenger, A perfectly matched layer for the absorption of electromagnetic waves. J. Comput. Phys. 114 (1994) 185–200. | MR | Zbl | DOI

[4] J. H. Bramble and J. E. Pasciak, Analysis of a finite PML approximation for the three dimensional time-harmonic Maxwell and acoustic scattering problems. Math. Comp. 76 (2007) 597–614. | MR | Zbl | DOI

[5] J. H. Bramble and J. E. Pasciak, Analysis of a finite element PML approximation for the three dimensional time-harmonic Maxwell problem. Math. Comput. 77 (2008) 1–10. | MR | Zbl | DOI

[6] J. H. Bramble and J. E. Pasciak, 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

[7] J. H. Bramble and J. E. Pasciak, Analysis of a Cartesian PML approximation to acoustic scattering problems in ℝ2 and ℝ3. J. Comput. Appl. Math. 247 (2013) 209–230. | MR | Zbl | DOI

[8] J. H. Bramble, J. E. Pasciak and D. Trenev, Analysis of a finite PML approximation to the three dimensional elastic wave scattering problem. Math. Comput. 79 (2010) 2079–2101. | MR | Zbl | DOI

[9] A. Buffa, M. Costabel and D. Sheen, On traces for H ( curl , Ω ) in Lipschitz domains. J. Math. Anal. Appl. 276 (2002) 845–867. | MR | Zbl | DOI

[10] Z. Chen, Convergence of the time-domain perfectly matched layer method for acoustic scattering problems. Int. J. Numer. Anal. Model. 6 (2009) 124–146. | MR | Zbl

[11] J. Chen and Z. Chen, An adaptive perfectly matched layer technique for 3-D time-harmonic electromagnetic scattering problems. Math. Comput. 77 (2007) 673–698. | MR | Zbl | DOI

[12] Z. Chen and X. Liu, An adaptive perfectly matched layer technique for time-harmonic scattering problems. SIAM J. Numer. Anal. 43 (2005) 645–671. | MR | Zbl | DOI

[13] Q. Chen and P. Monk, Discretization of the time domain CFIE for acoustic scattering problems using convolution quadrature. SIAM J. Math. Anal. 46 (2014) 3107–3130. | MR | Zbl | DOI

[14] Z. Chen and J. C. Nédélec, On Maxwell equations with the transparent boundary condition. J. Comput. Math. 26 (2008) 284–296. | MR | Zbl

[15] Z. Chen and X. Wu, An adaptive uniaxial perfectly matched layer method for time-harmonic scattering problems. Numer. Math. TMA 1 (2008) 113–137. | MR | Zbl

[16] Z. Chen and X. Wu, Long-time stability and convergence of the uniaxial perfectly matched layer method for time-domain acoustic scattering problems. SIAM J. Numer. Anal. 50 (2012) 2632–2655. | MR | Zbl | DOI

[17] Z. Chen and W. Zheng, Convergence of the uniaxial perfectly matched layer method for time-harmonic scattering problems in two-layered media. SIAM J. Numer. Anal. 48 (2010) 2158–2185. | MR | Zbl | DOI

[18] Z. Chen and W. Zheng, PML method for electromagnetic scattering problem in a two-layer medium. SIAM. J. Numer. Anal. 55 (2017) 2050–2084. | MR | DOI

[19] Z. Chen, T. Cui and L. Zhang, An adaptive anisotropic perfectly matched layer method for 3-D time harmonic electromagnetic scattering problems. Numer. Math. 125 (2013) 639–677. | MR | Zbl | DOI

[20] Z. Chen, X. Xiang and X. Zhang, Convergence of the PML method for elastic wave scattering problems. Math. Comp. 85 (2016) 2687–2714. | MR | DOI

[21] W. C. Chew and W. H. Weedon, A 3D perfectly matched medium from modified Maxwell’s equations with stretched coordinates. Microw. Opt. Technol. Lett. 7 (1994) 599–604. | DOI

[22] A. M. Cohen, Numerical Methods for Laplace Transform Inversion. Springer (2007). | MR | Zbl

[23] F. Collino and P. Monk, The perfectly matched layer in curvilinear coordinates. SIAM J. Sci. Comput. 19 (1998) 2061–2090. | MR | Zbl | DOI

[24] A. T. De Hoop, P. M. Van Den Berg and R. F. Remis, Absorbing boundary conditions and perfectly matched layers: an analytic time-domain performance analysis. IEEE Trans. Magn. 38 (2002) 657–660. | DOI

[25] J. Diaz and P. Joly, A time domain analysis of PML models in acoustics. Comput. Methods Appl. Mech. Eng. 195 (2006) 3820–3853. | MR | Zbl | DOI

[26] Y. Gao and P. Li, Analysis of time-domain scattering by periodic structures. J. Differ. Equ. 261 (2016) 5094–5118. | MR | DOI

[27] Y. Gao and P. Li, Electromagnetic scattering for time-domain Maxwell’s equations in an unbounded structure. Math. Models Methods Appl. Sci. 27 (2017) 1843–1870. | MR | DOI

[28] Y. Gao, P. Li and B. Zhang, Analysis of transient acoustic-elastic interaction in an unbounded structure. SIAM J. Math. Anal. 49 (2017) 3951–3972. | MR | DOI

[29] T. Hagstrom, Radiation boundary conditions for the numerical simulation of waves. Acta Numer. 8 (1999) 47–106. | MR | Zbl | DOI

[30] T. Hohage, F. Schmidt and L. Zschiedrich, Solving time-harmonic scattering problems based on the pole condition II: convergence of the PML method. SIAM J. Math. Anal. 35 (2003) 547–560. | MR | Zbl | DOI

[31] G. C. Hsiao, F. J. Sayas and R. J. Weinacht, Time-dependent fluid-structure interaction. Math. Method. Appl. Sci. 40 (2017) 486–500. | MR | DOI

[32] M. Lassas and E. Somersalo, On the existence and convergence of the solution of PML equations. Computing 60 (1998) 229–241. | MR | Zbl | DOI

[33] P. Li, L. Wang and A. Wood, Analysis of transient electromagnetic scattering from a three-dimensional open cavity. SIAM J. Appl. Math. 75 (2015) 1675–1699. | MR | DOI

[34] P. Monk, Finite Element Methods for Maxwell’s Equations. Oxford Univ. Press, New York (2003). | MR | Zbl | DOI

[35] F. L. Teixeira and W. C. Chew, Advances in the theory of perfectly matched layers, edited by W. C. Chew et al. In: Fast and Efficient Algorithms in Computational Electromagnetics. Artech House, Boston (2001) 283–346.

[36] F. Trèves, Basic Linear Partial Differential Equations. Academic Press, New York (1975). | MR | Zbl

[37] C. Wei and J. Yang, Analysis of a time-dependent fluid-solid interaction problem above a local rough surface. Sci. Chin. Math. 63 (2020) 887–906. | MR | DOI

[38] C. Wei, J. Yang and B. Zhang, Convergence of the perfectly matched layer method for transient acoustic-elastic interaction above an unbounded rough surfacePreprint: (2019). | arXiv

[39] C. Wei, J. Yang and B. Zhang, A time-dependent interaction problem between an electromagnetic field and an elastic body. Acta Math. Appl. Sin. Engl. Ser. 36 (2020) 95–118. | MR | DOI

[40] C. Wei, J. Yang and B. Zhang, Convergence analysis of the PML method for time-domain electromagnetic scattering problems. SIAM J. Numer. Anal. 58 (2020) 1918–1940. | MR | DOI

Cité par Sources :