A popular method for treating Helmholtz scattering and resonance problems is the so-called complexscaling. The idea of this method is to introduce an artificial damping of the waves outside of a chosencomputational interior domain in a way that no additional reflections are induced. When so-called perfectlymatched layers are used, the exterior domain (i.e., the part of the domain where the damping is introduced)is truncated to a bounded layer and discretized using, for instance, finite elements. In [1] we presented anumber of modifications and improvements to the method described above.To obtain a larger number of equally-well approximated resonances, we use method parameters thatdepend on the unknown resonance frequency. This approach leads to non-linear eigenvalue problems, insteadof linear ones.For the discretization of the problem, we use a method based on the decomposition of a wave into apropagating radial and an oscillating transversal part. The discrete ansatz functions for the propagatingpart are functions with unbounded support and are closely related to the ansatz functions of Hardy spaceinfinite element methods and spectral element methods. Due to the use of these functions, we avoid theartificial truncation of the exterior domain and obtain super-algebraic approximation properties. Moreover,this decomposition makes it straightforward to adapt the method to the specific geometry of the givenproblem.Lastly, we present an efficient method to approximate the eigenvalues of the resulting discrete, non-lineareigenvalue problems, which requires no significant extra computational effort, compared to similar methodsfor linear eigenvalue problems.The main focus of this talk will be on the analysis of the resulting non-linear eigenvalue problems andthe corresponding numerical methods.References.[1] Wess, M. (2020),Frequency-Dependent Complex-Scaled Infinite Elements for Exterior Helmholtz Reso-nance Problems, PhD Thesis, TU Wien.

[2] Nannen, L., Wess, M. (2018),Computing scattering resonances using perfectly matched layers withfrequency dependent scaling functions,BIT Numerical Mathematics.

[3] Hohage, T., Nannen, L. (2009),Hardy space infinite elements for scattering and resonance problems,SIAM J. Numer. Anal. 47.