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M. Löhndorf, J. Melenk:
"Mapping properties of Helmholtz boundary integral operators and their application to the HP-BEM";
in: "ASC Report 34/2009", issued by: Institute for Analysis and Scientific Computing; Vienna University of Technology, Wien, 2009, ISBN: 978-3-902627-02-5.

For the Helmholtz equation (with wavenumber k) and analytic curves or surfaces
�� we analyze the mapping properties of the single layer, double layer as well combined potential
boundary integral operators. A k-explicit regularity theory for the single layer and double layer
potentials is developed, in which these operators are decomposed into three parts: the first part is
the single or double layer potential for the Laplace equation, the second part is an operator with
finite shift properties, and the third part is an operator that maps into a space of piecewise analytic
functions. For all parts, the k-dependence is made explicit. We also develop a k-explicit regularity
theory for the inverse of the combined potential operator A = ±1/2+K−i V and its adjoint, where
V and K are the single layer and double layer operators for the Helmholtz kernel and ∈ R is a
coupling parameter with | | ∼ |k|. Under the assumption that kA−1kL2(��)←L2(��) grows at most
polynomially in k, the inverse A−1 is decomposed into an operator A1 : L2(��) → L2(��) with bounds
independent of k and a smoothing operator A2 that maps into a space of analytic functions on ��.
The k-dependence of the mapping properties of A2 is made explicit. We show quasi-optimality (in
an L2(��)-setting) of the hp-version of the Galerkin BEM applied to A or A′ under the assumption
of scale resolution, i.e., the polynomial degree p is at least O(log k) and kh/p is bounded by a
number that is sufficiently small, but independent of k. Under this assumption, the constant in the
quasi-optimality estimate is independent of k. Numerical examples in 2D illustrate the theoretical
results.
1. introduction. Acoustic a

http://www.asc.tuwien.ac.at/preprint/2009/asc34x2009.pdf