M. Löhndorf, J. Melenk:

"Wavenumber-explicit HP-BEM for high frequency scattering";

in: "ASC Report 02/2010", issued by: Institute for Analysis and Scientific Computing; Vienna University of Technology, Wien, 2010, ISBN: 978-3-902627-03-2.

For the Helmholtz equation (with wavenumber $k$) and analytic curves or surfaces $\Gamma$

we analyze the Galerkin discretization of classical combined field integral equations

in an $L2$-setting.

We give abstract conditions on the approximation properties of the ansatz space that ensure

stability and quasi-optimality of the Galerkin method. Special attention

is paid to the $hp$-version of the boundary element method ($hp$-BEM). Under the assumption

of polynomial growth of the solution operator we show stability and quasi-optimality of the

$hp$-BEM if the following scale resolution condition is satisfied:

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 and even suggest that in many cases

quasi-optimality is given under the weaker condition that $kh/p$ is sufficiently small.

http://www.asc.tuwien.ac.at/preprint/2010/asc02x2010.pdf

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