Hardy space infinite elements for time harmonic wave equations with phase and group velocities of different signs

We consider time harmonic wave equations in cylindrical wave-guides with physical solutions forwhich the signs of group andphasevelocities differ. Theperfectly matched layer methods select modes with positive phase velocity, and hence they yield stable, but unphysical solutions for such problems. We derive an infinite element method for a physically correct discretization of such wave-guide problems which is based on a Laplace transform in propagation direction. In the Laplace domain the space of transformed solutions can be separated into a sum of a space of incoming and a space of outgoing functions where both function spaces are Hardy spaces of a curved domain. The Hardy space is constructed such that it contains a simple and convenient Riesz basis with small condition numbers. In this paper the new method is only discussed for a one-dimensional fourth order model problem. Exponential Financial support by DFG in project HO 2551/5-1 is gratefully acknowledged. The first author acknowledges support from the Austrian Science Fund (FWF): W1245-N25. B Lothar Nannen [email protected] Martin Halla [email protected] Thorsten Hohage [email protected] Joachim Schöberl [email protected] 1 Institut für Analysis und Scientific Computing, Technische Universität Wien, 1040 Vienna, Austria 2 Institut für Numerische und Angewandte Mathematik, Georg-August Universität Göttingen, 37083 Göttingen, Germany