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Numerische Mathematik

, Volume 137, Issue 1, pp 1–34 | Cite as

Approximation of the high-frequency Helmholtz kernel by nested directional interpolation: error analysis

  • Steffen Börm
  • Jens M. Melenk
Article

Abstract

We present and analyze an approximation scheme for a class of highly oscillatory kernel functions, taking the 2D and 3D Helmholtz kernels as examples. The scheme is based on polynomial interpolation combined with suitable pre- and post-multiplication by plane waves. It is shown to converge exponentially in the polynomial degree and supports multilevel approximation techniques. Our convergence analysis may be employed to establish exponential convergence of certain classes of fast methods for discretizations of the Helmholtz integral operator that feature polylogarithmic-linear complexity.

Mathematics Subject Classification

35J05 65D05 65N38 41A10 65N12 

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Copyright information

© Springer-Verlag Berlin Heidelberg 2017

Authors and Affiliations

  1. 1.Institut für InformatikChristian-Albrechts-Universität zu Kiel, D-24118KielGermany
  2. 2.Institut für Analysis und Scientific ComputingTechnische Universität WienViennaAustria

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