An adaptive finite element method is presented for the elastic scattering of a time-harmonic plane wave by a periodic surface. First, the unbounded physical domain is truncated into a bounded computational domain by introducing the perfectly matched layer (PML) technique. The well-posedness and exponential convergence of the solution are established for the truncated PML problem by developing an equivalent transparent boundary condition. Second, an a posteriori error estimate is deduced for the discrete problem and is used to determine the finite elements for refinements and to determine the PML parameters. Numerical experiments are included to demonstrate the competitive behavior of the proposed adaptive method.
Accepté le :
DOI : 10.1051/m2an/2017018
Mots-clés : Elastic wave equation, adaptive finite element, perfectly matched layer, a posteriori error estimate
@article{M2AN_2017__51_5_2017_0,
author = {Jiang, Xue and Li, Peijun and Lv, Junliang and Zheng, Weiying},
title = {An adaptive finite element {PML} method for the elastic wave scattering problem in periodic structures},
journal = {ESAIM: Mathematical Modelling and Numerical Analysis },
pages = {2017--2047},
publisher = {EDP-Sciences},
volume = {51},
number = {5},
year = {2017},
doi = {10.1051/m2an/2017018},
mrnumber = {3731558},
zbl = {1408.74048},
language = {en},
url = {http://www.numdam.org/articles/10.1051/m2an/2017018/}
}
TY - JOUR AU - Jiang, Xue AU - Li, Peijun AU - Lv, Junliang AU - Zheng, Weiying TI - An adaptive finite element PML method for the elastic wave scattering problem in periodic structures JO - ESAIM: Mathematical Modelling and Numerical Analysis PY - 2017 SP - 2017 EP - 2047 VL - 51 IS - 5 PB - EDP-Sciences UR - http://www.numdam.org/articles/10.1051/m2an/2017018/ DO - 10.1051/m2an/2017018 LA - en ID - M2AN_2017__51_5_2017_0 ER -
%0 Journal Article %A Jiang, Xue %A Li, Peijun %A Lv, Junliang %A Zheng, Weiying %T An adaptive finite element PML method for the elastic wave scattering problem in periodic structures %J ESAIM: Mathematical Modelling and Numerical Analysis %D 2017 %P 2017-2047 %V 51 %N 5 %I EDP-Sciences %U http://www.numdam.org/articles/10.1051/m2an/2017018/ %R 10.1051/m2an/2017018 %G en %F M2AN_2017__51_5_2017_0
Jiang, Xue; Li, Peijun; Lv, Junliang; Zheng, Weiying. An adaptive finite element PML method for the elastic wave scattering problem in periodic structures. ESAIM: Mathematical Modelling and Numerical Analysis , Tome 51 (2017) no. 5, pp. 2017-2047. doi : 10.1051/m2an/2017018. http://www.numdam.org/articles/10.1051/m2an/2017018/
, The scattering of plane elastic waves by a one-dimensional periodic surface. Math. Methods Appl. Sci. 22 (1999) 55–72. | DOI | MR | Zbl
, A new integral equation formulation for the scattering of plane elastic waves by diffraction gratings. J. Int. Equ. Appl. 11 (1999) 275–297. | MR | Zbl
I. Babuška and A. Aziz, Survey Lectures on Mathematical Foundations of the Finite Element Method, in The Mathematical Foundations of the Finite Element Method with Application to the Partial Differential Equations, edited by A. Aziz. Academic Press, New York (1973) 5–359. | MR | Zbl
, Finite element approximation of time harmonic waves in periodic structures. SIAM J. Numer. Anal. 32 (1995) 1155–1169. | DOI | MR | Zbl
, Variational approximation of Maxwell’s equations in biperiodic structures. SIAM J. Appl. Math. 57 (1997) 364–381. | DOI | MR | Zbl
, and , Adaptive finite element method for diffraction gratings. J. Opt. Soc. Amer. A 22 (2005) 1106–1114. | DOI | MR
, and , Mathematical studies in rigorous grating theory. J. Opt. Soc. Amer. A 12 (1995) 1029–1042. | DOI | MR
G. Bao, L. Cowsar and W. Masters, Mathematical Modeling in Optical Science. Vol. 22 of Front. Appl. Math. SIAM, Philadelphia (2001). | MR | Zbl
, and , An adaptive edge element method with perfectly matched absorbing layers for wave scattering by periodic structures. Math. Comput. 79 (2010) 1–34. | DOI | MR | Zbl
and , On the convergence of the solutions of PML equations for Maxwell’s equations. SIAM J. Numer. Anal. 43 (2005) 2121–2143. | MR | Zbl
, A perfectly matched layer for the absorption of electromagnetic waves. J. Comput. Phys. 114 (1994) 185–200. | DOI | MR | Zbl
and , Analysis of a finite PML approximation for the three dimensional time-harmonic Maxwell and acoustic scattering problems. Math. Comput. 76 (2007) 597–614. | DOI | MR | Zbl
, and , Analysis of a finite PML approximation to the three dimensional elastic wave scattering problem. Math. Comput. 79 (2010) 2079–2101. | DOI | 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 , Maxwell’s equations in a periodic structure. Trans. Amer. Math. Soc. 323 (1991) 4650–507. | MR | Zbl
and , An adaptive finite element method with perfectly matched absorbing layers for the wave scattering by periodic structures. SIAM J. Numer. Anal. 41 (2003) 799–826. | DOI | MR | Zbl
and , An adaptive perfectly matched layer technique for time-harmonic scattering problems. SIAM J. Numer. Anal. 43 (2005) 645–671. | DOI | MR | Zbl
, and , Convergence of the PML method for elastic wave scattering problems. Math. Comput. 85 (2016) 2687–2714. | DOI | MR | Zbl
and , The perfectly matched layer in curvilinear coordinates. SIAM J. Sci. Comput. 19 (1998) 2061–1090. | DOI | MR | Zbl
and , Application of the perfectly matched absorbing layer model to the linear elastodynamic problem in anisotropic heterogeneous media. Geophys. 66 (2001) 294–307. | DOI
and , A 3D perfectly matched medium for modified Maxwell’s equations with stretched coordinates. Microwave Opt. Technol. Lett. 13 (1994) 599–604. | DOI
and , The time-harmonic Maxwell equations in a doubly periodic structure. J. Math. Anal. Appl. 166 (1992) 507–528. | DOI | MR | Zbl
and , Variational approach to scattering of plane elastic waves by diffraction gratings. Math. Meth. Appl. Sci. 33 (2010) 1924–1941. | MR | Zbl
and , Scattering of plane elastic waves by three-dimensional diffraction gratings. Math. Models Methods Appl. Sci. 22 (2012) 1150019. | DOI | MR | Zbl
, and , Application of the perfectly matched layer (PML) absorbing boundary condition to elastic wave propagation. J. Acoust. Soc. Am. 100 (1996) 3061–3069. | DOI
, New development in FreeFem++. J. Numer. Math. 20 (2012) 251–265. | DOI | MR | Zbl
, and , Solving time-harmonic scattering problems based on the pole condition. II: Convergence of the PML method. SIAM J. Math. Anal. 35 (2003) 547–560. | DOI | MR | Zbl
and , On the existence and convergence of the solution of PML equations. Computing 60 (1998) 229–241. | DOI | MR | Zbl
, and , Inverse elastic surface scattering with near-field data. Inverse Probl. 31 (2015) 035009. | DOI | MR | Zbl
, An observation concerning Ritz-Galerkin methods with indefinite bilinear forms. Math. Comput. 28 (1974) 959–962. | DOI | MR | Zbl
and , Finite element interpolation of nonsmooth functions satisfying boundary conditions. Math. Comput. 54 (1990) 483-493. | DOI | MR | Zbl
and , Absorbing PML boundary layers for wave-like equations. Appl. Numer. Math. 27 (1998) 533–557. | DOI | MR | Zbl
, , , and , An adaptive finite element method for the diffraction grating problem with transparent boundary condition. SIAM J. Numer. Anal. 53 (2015) 1585–1607. | DOI | MR | Zbl
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