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M. Löhndorf, J. Melenk:
"Wavenumber-explicit hp-BEM for high frequency scattering";
SIAM Journal on Numerical Analysis, 49 (2011).

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 $L^2$-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 two dimensions 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.

high order boundary element method, high frequency scattering, combined field integral equation, Helmholtz equation

http://dx.doi.org/10.1137/100786034