A broadband stable and efficient addition theorem for the two-dimensional Helmholtz equation

Boundary integral equations are the principal tools for efficiently simulating electromagnetic fields in structures with piecewise constant material parameters. A specific trait of boundary integral equations is that the Green function of the Helmholtz equation appears as the integration kernel. Hence, discretizing a boundary integral equation leads to a dense linear system. Many such acceleration schemes exist, for example the renowned Multilevel Fast Multipole Algorithm (MLFMA).The MLFMA is based on an addition theorem that decomposes the Green function into a superposition of propagating plane waves. This decomposition is then used to evaluate the interactions between entire groups of basis functions, in contrast to evaluating the interactions between all the individual basis functions. Unfortunately, the MLFMA addition theorem is numerically unstable when the groups become significantly smaller than the wavelength. This phenomenon is called the low-frequency breakdown of the MLFMA and prevents the efficient simulation of structures with a lot of subwavelength geometrical detail.