Numerische Mathematik

, Volume 135, Issue 1, pp 27–71 | Cite as

Frequency-adapted Galerkin boundary element methods for convex scattering problems

Article

Abstract

We introduce a class of hybrid boundary element methods for the solution of sound soft scattering problems in the exterior of two-dimensional smooth convex obstacles. To facilitate the applicability of our algorithms throughout the entire frequency spectrum, we have enriched our Galerkin approximation spaces, through incorporation of oscillations in the incident field of radiation, into the algebraic and trigonometric polynomial approximation spaces. The resulting methodologies have three distinctive properties. Indeed, from a theoretical point of view (1) they can be tuned to demand only an\(\mathcal {O}(k^{\varepsilon })\)increase (for any \(\varepsilon >0)\)in the number of degrees of freedom to maintain a fixed accuracy with increasing wavenumberk, owing to the optimal adaptation of approximation spaces to asymptotic stretching (shrinking) of illuminated and deep shadow regions (shadow boundaries), and (2) they are convergent for each fixed wavenumberk, thanks to the additional approximation spaces in the deep shadow region. Perhaps more importantly, from a practical point of view (3) they give rise to linear systems with significantly enhanced condition numbers and this, in turn, allows for more accurate solutions if desired.

Mathematics Subject Classification

65N38 78M15 35P25 65N12 

Notes

Acknowledgments

The first author would like to thank Simon Chandler-Wilde (Reading), Víctor Domínguez (Tudela), Ivan G. Graham (Bath), and Valery Smyshlyaev (London) for useful discussions.

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Copyright information

© Springer-Verlag Berlin Heidelberg 2016

Authors and Affiliations

  1. 1.Department of MathematicsBoğaziçi UniversityIstanbulTurkey
  2. 2.Applied Physics and Applied Mathematics Department, The Fu Foundation School of Engineering and Applied ScienceColumbia UniversityNew YorkUSA

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