Journal of Scientific Computing

, Volume 72, Issue 3, pp 936–956 | Cite as

An Adaptive Finite Element Method for the Wave Scattering with Transparent Boundary Condition

Article
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Abstract

Consider the acoustic wave scattering by an impenetrable obstacle in two dimensions. The model is formulated as a boundary value problem for the Helmholtz equation with a transparent boundary condition. Based on a duality argument technique, an a posteriori error estimate is derived for the finite element method with the truncated Dirichlet-to-Neumann boundary operator. The a posteriori error estimate consists of the finite element approximation error and the truncation error of boundary operator which decays exponentially with respect to the truncation parameter. A new adaptive finite element algorithm is proposed for solving the acoustic obstacle scattering problem, where the truncation parameter is determined through the truncation error and the mesh elements for local refinements are marked through the finite element discretization error. Numerical experiments are presented to illustrate the competitive behavior of the proposed adaptive method.

Keywords

Acoustic scattering problem Adaptive finite element method Transparent boundary condition A posteriori error estimate 

Mathematics Subject Classification

65M30 78A45 35Q60 
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Notes

Acknowledgements

The research of X.J. was supported in part by China NSF Grant 11401040 and by the Fundamental Research Funds for the Central Universities 24820152015RC17. The research of P.L. was supported in part by the NSF Grant DMS-1151308. The research of J.L. was partially supported by the China NSF Grants 11126040 and 11301214. The author of W.Z. was supported in part by China NSF 91430215, by the Funds for Creative Research Groups of China (Grant 11321061), and by the National Magnetic Confinement Fusion Science Program (2015GB110003).

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Copyright information

© Springer Science+Business Media New York 2017

Authors and Affiliations

  • Xue Jiang
    • 1
  • Peijun Li
    • 2
  • Junliang Lv
    • 3
  • Weiying Zheng
    • 4
  1. 1.School of ScienceBeijing University of Posts and TelecommunicationsBeijingChina
  2. 2.Department of MathematicsPurdue UniversityWest LafayetteUSA
  3. 3.School of MathematicsJilin UniversityChangchunChina
  4. 4.NCMIS, LSEC, ICMSEC, Academy of Mathematics and System SciencesChinese Academy of SciencesBeijingChina

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