Analysis of radial complex scaling methods: scalar resonance problems

We consider radial complex scaling/perfectly matched layer methods for scalar resonance problems in homogeneous exterior domains. We introduce a new abstract framework to analyze the convergence of domain truncations and discretizations. Our theory requires rather minimal assumptions on the scaling profile and includes affin, smooth and also unbounded profiles. We report a swift technique to analyze the convergence of domain truncations and a more technical one for approximations through simultaneaous truncation and discretization. We adapt the latter technique to cover also so-called exact methods which do not require a domain truncation. Our established results include convergence rates of eigenvalues and eigenfunctions. The introduced framework is based on the ideas to interpret the domain truncation as Galerkin approximation, to apply theory on holomorphic Fredholm operator eigenvalue approximation theory to a linear eigenvalue problem, to employ the notion of weak T-coercivity and T-compatible approximations, to construct a suitable T-operator as multiplicatin operator, to smooth its symbol and to apply the discrete commutator technique.

[1]  Anne-Sophie Bonnet-Ben Dhia,et al.  Perfectly Matched Layers for Time-Harmonic Acoustics in the Presence of a Uniform Flow , 2006, SIAM J. Numer. Anal..

[2]  Patrick Joly,et al.  Stability of perfectly matched layers, group velocities and anisotropic waves , 2003 .

[3]  Lucas Chesnel,et al.  On the use of Perfectly Matched Layers at corners for scattering problems with sign-changing coefficients , 2015, J. Comput. Phys..

[4]  Daniele Boffi,et al.  Finite element approximation of eigenvalue problems , 2010, Acta Numerica.

[5]  Peter Monk,et al.  The Perfectly Matched Layer in Curvilinear Coordinates , 1998, SIAM J. Sci. Comput..

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

[7]  Morten Hjorth-Jensen.  Eigenvalue Problems , 2021, Explorations in Numerical Analysis.

[8]  Joseph E. Pasciak,et al.  Analysis of a finite PML approximation to the three dimensional elastic wave scattering problem , 2010, Math. Comput..

[9]  Salvatore Caorsi,et al.  On the Convergence of Galerkin Finite Element Approximations of Electromagnetic Eigenproblems , 2000, SIAM J. Numer. Anal..

[10]  N. Moiseyev,et al.  Quantum theory of resonances: calculating energies, widths and cross-sections by complex scaling , 1998 .

[11]  Lothar Nannen,et al.  Complex scaled infinite elements for exterior Helmholtz problems , 2019, ArXiv.

[12]  Otto Karma,et al.  Approximation in eigenvalue problems for holomorphic fredholm operator functions I , 1996 .

[13]  Martin Halla,et al.  Galerkin approximation of holomorphic eigenvalue problems: weak T-coercivity and T-compatibility , 2019, Numerische Mathematik.

[14]  Barry Simon,et al.  Resonances and complex scaling: a rigorous overview , 1978 .

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

[16]  Lexing Ying,et al.  Sweeping Preconditioner for the Helmholtz Equation: Moving Perfectly Matched Layers , 2010, Multiscale Model. Simul..

[17]  É. Bécache,et al.  Stable perfectly matched layers for a class of anisotropic dispersive models. Part I: Necessary and sufficient conditions of stability , 2017 .

[18]  D. Arnold,et al.  Finite element exterior calculus: From hodge theory to numerical stability , 2009, 0906.4325.

[19]  Alfredo Bermúdez,et al.  An Exact Bounded Perfectly Matched Layer for Time-Harmonic Scattering Problems , 2007, SIAM J. Sci. Comput..

[20]  Israel Michael Sigal,et al.  Introduction to Spectral Theory , 1996 .

[21]  J. Rappaz,et al.  On spectral approximation. Part 1. The problem of convergence , 1978 .

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

[23]  Xiaohui Zhang,et al.  Convergence of the PML method for elastic wave scattering problems , 2016, Math. Comput..

[24]  Anne-Sophie Bonnet-Ben Dhia,et al.  Perfectly Matched Layers for the Convected Helmholtz Equation , 2004, SIAM J. Numer. Anal..

[25]  Joachim Schöberl,et al.  Hardy space infinite elements for time harmonic wave equations with phase and group velocities of different signs , 2016, Numerische Mathematik.

[26]  Stefan Hein,et al.  Acoustic resonances in a high-lift configuration , 2007, Journal of Fluid Mechanics.

[27]  L. R. Scott,et al.  The Mathematical Theory of Finite Element Methods , 1994 .

[28]  Julien Diaz,et al.  A time domain analysis of PML models in acoustics , 2006 .

[29]  W. Koch,et al.  On resonances in open systems , 2004, Journal of Fluid Mechanics.

[30]  M. Halla,et al.  Hardy space infinite elements for time-harmonic two-dimensional elastic waveguide problems , 2015, 1506.04781.

[31]  Joseph E. Pasciak,et al.  The computation of resonances in open systems using a perfectly matched layer , 2009, Math. Comput..

[32]  Alfredo Bermúdez,et al.  An exact bounded PML for the Helmholtz equation , 2004 .

[33]  Silvia Bertoluzza,et al.  The discrete commutator property of approximation spaces , 1999 .

[34]  Lothar Nannen,et al.  Two scale Hardy space infinite elements for scalar waveguide problems , 2018, Adv. Comput. Math..

[35]  Thorsten Hohage,et al.  Hardy Space Infinite Elements for Scattering and Resonance Problems , 2009, SIAM J. Numer. Anal..

[36]  Olaf Steinbach,et al.  A boundary element method for the Dirichlet eigenvalue problem of the Laplace operator , 2009, Numerische Mathematik.

[37]  Martin Halla.  Convergence of Hardy Space Infinite Elements for Helmholtz Scattering and Resonance Problems , 2016, SIAM J. Numer. Anal..

[38]  Frank Schmidt,et al.  Solving Time-Harmonic Scattering Problems Based on the Pole Condition II: Convergence of the PML Method , 2003, SIAM J. Math. Anal..

[39]  Weng Cho Chew,et al.  A 3D perfectly matched medium from modified maxwell's equations with stretched coordinates , 1994 .

[40]  T. Hagstrom.  Radiation boundary conditions for the numerical simulation of waves , 1999, Acta Numerica.

[41]  Patrick Joly,et al.  Mathematical models for dispersive electromagnetic waves: An overview , 2017, Comput. Math. Appl..

[42]  D. Givoli.  Numerical Methods for Problems in Infinite Domains , 1992 .

[43]  Philippe Lalanne,et al.  Perfectly matched layers as nonlinear coordinate transforms: a generalized formalization. , 2005, Journal of the Optical Society of America. A, Optics, image science, and vision.

[44]  T. Hohage,et al.  Convergence of infinite element methods for scalar waveguide problems , 2014, 1409.6450.

[45]  E. A. Skelton,et al.  Guided elastic waves and perfectly matched layers , 2007 .

[46]  Lexing Ying,et al.  Additive Sweeping Preconditioner for the Helmholtz Equation , 2015, Multiscale Model. Simul..

[47]  Annalisa Buffa,et al.  Remarks on the Discretization of Some Noncoercive Operator with Applications to Heterogeneous Maxwell Equations , 2005, SIAM J. Numer. Anal..

[48]  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..

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

[50]  Ragnar Winther,et al.  On variational eigenvalue approximation of semidefinite operators , 2010, 1005.2059.

[51]  Seungil Kim,et al.  Analysis of a pml method applied to computation of resonances in open systems and acoustic scattering problems , 2009 .

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