I. Graham, M. Löhndorf, J. Melenk, E. Spence:

"When is the error in the h-BEM for solving the Helmholtz equation bounded independently of k?";

in: "ASC Report 28/2013", issued by: Institute for Analysis and Scientific Computing; Vienna University of Technology, Wien, 2013, ISBN: 978-3-902627-06-3, 1 - 31.

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 eld integral equations. We obtain su cient 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 xed number of degrees of freedom per wavelength is su cient to keep the relative

best approximation error bounded independently of k. We then obtain su cient 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 su cient, they are not necessary for many geometries.

Helmholtz equation, high frequency, boundary integral equation, boundary element method, pollution e ffect

http://www.asc.tuwien.ac.at/preprint/2013/asc28x2013.pdf

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