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When is the error in the \(h\)-BEM for solving the Helmholtz equation bounded independently of \(k\)?

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Abstract

We consider solving the sound-soft scattering problem for the Helmholtz equation with the \(h\)-version of the boundary element method using the standard second-kind combined-field integral equations. We obtain sufficient conditions for the relative best approximation error to be bounded independently of \(k\). For certain geometries, these rigorously justify the commonly-held belief that a fixed number of degrees of freedom per wavelength is sufficient to keep the relative best approximation error bounded independently of \(k\). We then obtain sufficient conditions for the Galerkin method to be quasi-optimal, with the constant of quasi-optimality independent of \(k\). Numerical experiments indicate that, while these conditions for quasi-optimality are sufficient, they are not necessary for many geometries.

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Notes

  1. Of course, if the goal is to compute the solution in (a subset of) the domain, then after using the \(h\)-BEM one must evaluate the integrals in Green’s integral representation (1.7) or the ansatz (1.12). This adds to the computational cost of the \(h\)-BEM, but the question “which of the \(h\)-BEM and \(h\)-FEM achieves the goal of computing the solution with the least cost?” is independent of the question “to what extent does each method suffers from the pollution effect?”.

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Acknowledgments

I.G.G and E.A.S were supported by EPSRC Grant EP/F06795X/1 and E.A.S by EPSRC Grant EP/1025995/1.

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Authors and Affiliations

  1. Department of Mathematical Sciences, University of Bath, Bath, BA27, AY, UK

    I. G. Graham & E. A. Spence

  2. Kapsch TrafficCom, Am Europlatz 2, 1120 , Wien, Austria

    M. Löhndorf

  3. Institut für Analysis und Scientific Computing, Technische Universität Wien, 1040 , Wien, Austria

    J. M. Melenk

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  1. I. G. Graham
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  2. M. Löhndorf
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  3. J. M. Melenk
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  4. E. A. Spence
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Correspondence to E. A. Spence.

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Communicated by Ralf Hiptmair.

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Graham, I.G., Löhndorf, M., Melenk, J.M. et al. When is the error in the \(h\)-BEM for solving the Helmholtz equation bounded independently of \(k\)?. Bit Numer Math 55, 171–214 (2015). https://doi.org/10.1007/s10543-014-0501-5

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