Additive Schwarz Method for Scattering Problems Using the PML Method at Interfaces

  • Achim Schädle
  • Lin Zschiedrich
Part of the Lecture Notes in Computational Science and Engineering book series (LNCSE, volume 55)


The exterior Helmholtz problem is a basic model for wave propagation in the frequency domain on unbounded domains. As a rule of thumb, 10–20 grid points per wavelength are required. Hence if the modeling structures are a multiple of wavelengths in size, a discretization with finite elements results in large sparse indefinite and unsymmetric problems. There are no well established solvers, or preconditioners for these linear systems as there are for positive definite elliptic problems.


Helmholtz Equation Perfectly Match Layer Domain Decomposition Method Iteration Cycle Schwarz Method 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    J.-D. Benamou and B. Després, A domain decomposition method for the Helmholtz equation and related optimal control, J. Comp. Phys., 136 (1997), pp. 68–82.zbMATHCrossRefGoogle Scholar
  2. 2.
    F. Collino, S. Ghanemi, and P. Joly, Domain decomposition method for harmonic wave propagation: A general presentation, Comput. Methods Appl. Mech. Eng., 184 (2000), pp. 171–211.zbMATHCrossRefMathSciNetGoogle Scholar
  3. 3.
    F. Collino and P. Monk, The perfectly matched layer in curvilinear coordinates, SIAM J. Sci. Comput., 19 (1998), pp. 2061–2090.zbMATHCrossRefMathSciNetGoogle Scholar
  4. 4.
    M. J. Gander, F. Magoulès, and F. Nataf, Optimized Schwarz methods without overlap for the Helmholtz equation, SIAM J. Sci. Comput., 24 (2002), pp. 38–60.zbMATHCrossRefMathSciNetGoogle Scholar
  5. 5.
    D. Givoli, Non-reflecting boundary conditions, J. Comput. Phys., 94 (1991), pp. 1–29.zbMATHCrossRefMathSciNetGoogle Scholar
  6. 6.
    T. Hagstrom, Radiation boundary conditions for numerical simulation of waves, Acta Numerica, 8 (1999), pp. 47–106.CrossRefMathSciNetGoogle Scholar
  7. 7.
    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), pp. 547–560.zbMATHCrossRefMathSciNetGoogle Scholar
  8. 8.
    M. Lassas and E. Somersalo, On the existence and convergence of the solution of PML equations, Computing, 60 (1998), pp. 229–241.zbMATHMathSciNetGoogle Scholar
  9. 9.
    V. V. Shaidurov and E. I. Ogorodnikov, Some numerical method of solving Helmholtz wave equation, in Mathematical and numerical aspects of wave propagation phenomena, G. Cohen, L. Halpern, and P. Joly, eds., SIAM, 1991, pp. 73–79.Google Scholar
  10. 10.
    A. Toselli, Some results on overlapping Schwarz methods for the Helmholtz equation employing perfectly matched layers, Tech. Rep. 765, Courant Institute of Mathematical Sciences, New York University, New York, June 1998.Google Scholar
  11. 11.
    S. Tsynkov, Numerical solution of problems on unbounded domains. a review, Appl. Numer. Math., 27 (1998), pp. 465–532.zbMATHCrossRefMathSciNetGoogle Scholar
  12. 12.
    L. Zschiedrich, R. Klose, A. Schdle, and F. Schmidt, A new finite element realization of the perfectly matched layer method for helmholtz scattering problems on polygonal domains in 2D, Tech. Rep. 03–44, Konrad-Zuse-Zentrum fur Informationstechnik Berlin, December 2003.Google Scholar

Copyright information

© Springer 2007

Authors and Affiliations

  • Achim Schädle
    • 1
  • Lin Zschiedrich
    • 2
  1. 1.Zuse InstituteTakustr
  2. 2.Zuse InstituteTakustr

Personalised recommendations