An adaptive edge element method with perfectly matched absorbing layers for wave scattering by biperiodic structures

An edge element adaptive strategy with error control is developed for wave scattering by biperiodic structures. The unbounded computational domain is truncated to a bounded one by a perfectly matched layer (PML) technique. The PML parameters, such as the thickness of the layer and the medium properties, are determined through sharp a posteriori error estimates. Numerical experiments are presented to illustrate the competitive behavior of the proposed adaptive method.

[1]  R. Hoppe,et al.  Residual based a posteriori error estimators for eddy current computation , 2000 .

[2]  L. R. Scott,et al.  Finite element interpolation of nonsmooth functions satisfying boundary conditions , 1990 .

[3]  Ricardo H. Nochetto,et al.  Data Oscillation and Convergence of Adaptive FEM , 2000, SIAM J. Numer. Anal..

[4]  Zhiming Chen,et al.  An adaptive perfectly matched layer technique for 3-D time-harmonic electromagnetic scattering problems , 2007, Math. Comput..

[5]  Ricardo H. Nochetto,et al.  Convergence of Adaptive Finite Element Methods , 2002, SIAM Rev..

[6]  J. Nédélec.  Mixed finite elements in ℝ3 , 1980 .

[7]  Zhiming Chen,et al.  On the Efficiency of Adaptive Finite Element Methods for Elliptic Problems with Discontinuous Coefficients , 2002, SIAM J. Sci. Comput..

[8]  J. Bérenger.  Three-Dimensional Perfectly Matched Layer for the Absorption of Electromagnetic Waves , 1996 .

[9]  F. Hu.  Absorbing Boundary Conditions , 2004 .

[10]  Joseph E. Pasciak,et al.  Analysis of a finite element PML approximation for the three dimensional time-harmonic Maxwell problem , 2008, Math. Comput..

[11]  W. Dörfler.  A convergent adaptive algorithm for Poisson's equation , 1996 .

[12]  David C. Dobson,et al.  On the scattering by a biperiodic structure , 2000 .

[13]  GANG BAO,et al.  Variational Approximation of Maxwell's Equations in Biperiodic Structures , 1997, SIAM J. Appl. Math..

[14]  Lifeng Li,et al.  New formulation of the Fourier modal method for crossed surface-relief gratings , 1997 .

[15]  Avner Friedman,et al.  The time-harmonic maxwell equations in a doubly periodic structure , 1992 .

[16]  S. Gedney.  An anisotropic perfectly matched layer-absorbing medium for the truncation of FDTD lattices , 1996 .

[17]  Yia-Chung Chang,et al.  Efficient finite-element, Green's function approach for critical-dimension metrology of three-dimensional gratings on multilayer films. , 2006, Journal of the Optical Society of America. A, Optics, image science, and vision.

[18]  A. Aziz.  The Mathematical Foundations of the Finite Element Method with Applications to Partial Differential Equations , 1972 .

[19]  Peter Schlattmann,et al.  Theory and Algorithms , 2009 .

[20]  K. Brown,et al.  Graduate Texts in Mathematics , 1982 .

[21]  Ricardo H. Nochetto,et al.  Convergence of Adaptive Finite Element Methods for General Second Order Linear Elliptic PDEs , 2005, SIAM J. Numer. Anal..

[22]  Douglas N. Arnold,et al.  Locally Adapted Tetrahedral Meshes Using Bisection , 2000, SIAM J. Sci. Comput..

[23]  T. Abboud.  Formulation variationnelle des équations de Maxwell dans un réseau bipériodique de R3 , 1993 .

[24]  H. Triebel,et al.  Topics in Fourier Analysis and Function Spaces , 1987 .

[25]  Gang Bao,et al.  Mathematical Modeling in Optical Science , 2001, Mathematical Modeling in Optical Science.

[26]  P. Monk,et al.  Optimizing the Perfectly Matched Layer , 1998 .

[27]  Rüdiger Verführt,et al.  A review of a posteriori error estimation and adaptive mesh-refinement techniques , 1996, Advances in numerical mathematics.

[28]  Endre Süli,et al.  The Adaptive Computation of Far-Field Patterns by A Posteriori Error Estimation of Linear Functionals , 1998 .

[29]  Gang Bao,et al.  Convergence Analysis of the Perfectly Matched Layer Problemsfor Time-Harmonic Maxwell's Equations , 2005, SIAM J. Numer. Anal..

[30]  Antonio Guarnieri,et al.  WITH THE COLLABORATION OF , 2009 .

[31]  Gang Bao,et al.  Analysis of finite dimensional approximations to a class of partial differential equations , 2004 .

[32]  Rob P. Stevenson,et al.  Optimality of a Standard Adaptive Finite Element Method , 2007, Found. Comput. Math..

[33]  Zhimin Zhang,et al.  Can We Have Superconvergent Gradient Recovery Under Adaptive Meshes? , 2007, SIAM J. Numer. Anal..

[34]  R. E. Edwards,et al.  Fourier series : a modern introduction , 1982 .

[35]  Roger Petit,et al.  Electromagnetic theory of gratings , 1980 .

[36]  W. Rheinboldt,et al.  Error Estimates for Adaptive Finite Element Computations , 1978 .

[37]  Habib Ammari,et al.  Low-Frequency Electromagnetic Scattering , 2000, SIAM J. Math. Anal..

[38]  Zhiming Chen,et al.  An Adaptive Perfectly Matched Layer Technique for Time-harmonic Scattering Problems , 2005 .

[39]  Kiyotoshi Yasumoto,et al.  Method of integral functionals for electromagnetic wave scattering from a double-periodic magnetodielectric layer. , 2007, Journal of the Optical Society of America. A, Optics, image science, and vision.

[40]  Matti Lassas,et al.  On the existence and convergence of the solution of PML equations , 1998, Computing.

[41]  M. Costabel,et al.  Maxwell and Lamé eigenvalues on polyhedra , 1999 .

[42]  Wolfgang Dahmen,et al.  Adaptive Finite Element Methods with convergence rates , 2004, Numerische Mathematik.

[43]  Habib Ammari,et al.  Maxwell's equations in periodic chiral structures , 2003 .

[44]  Haijun Wu,et al.  Adaptive finite-element method for diffraction gratings. , 2005, Journal of the Optical Society of America. A, Optics, image science, and vision.

[45]  Peter Monk,et al.  A posteriori error indicators for Maxwell's equations , 1998 .

[46]  J. Allen Cox,et al.  Mathematical studies in rigorous grating theory , 1995 .

[47]  Haijun Wu,et al.  An Adaptive Finite Element Method with Perfectly Matched Absorbing Layers for the Wave Scattering by Periodic Structures , 2003, SIAM J. Numer. Anal..

[48]  E. Turkel,et al.  Absorbing PML boundary layers for wave-like equations , 1998 .

[49]  Jean-Pierre Berenger,et al.  A perfectly matched layer for the absorption of electromagnetic waves , 1994 .

[50]  W. Chew,et al.  Systematic derivation of anisotropic PML absorbing media in cylindrical and spherical coordinates , 1997 .

[51]  Zhiming Chen,et al.  An Adaptive Perfectly Matched Layer Technique for Time-harmonic Scattering Problems , 2005, SIAM J. Numer. Anal..

[52]  Vivette Girault,et al.  Finite Element Methods for Navier-Stokes Equations - Theory and Algorithms , 1986, Springer Series in Computational Mathematics.

[53]  Joseph E. Pasciak,et al.  Analysis of a finite PML approximation for the three dimensional time-harmonic Maxwell and acoustic scattering problems , 2006, Math. Comput..