For the Helmholtz equation (with wavenumber k) and analytic curves or surfaces Γ 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 quasioptimality 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.

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We employ a data-sparse, recursive matrix representation, so-called H 2 -matrices, for the efficient treatment of discretized integral operators. We obtain this format using local tensor product interpolants of the kernel function and replacing high-order approximations with piecewise lowerorder ones. The scheme has optimal, i.e., linear, complexity in the memory requirement and time for the matrix-vector multiplication. We present an error analysis for integral operators of order zero. In particular, we show that the optimal convergence O(h) is retained for the classical double layer potential discretized with piecewise constant functions.

We present a simple, PDE-based proof of the result [17] by M. Johnson that the error estimates of J. Duchon [11] for thin plate spline interpolation can be improved by h 1/2 . We illustrate that H -matrix techniques can successfully be employed to solve very large thin plate spline interpolation problems.

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