Evanescent Plane Wave Approximation of Helmholtz Solutions in Spherical Domains

Abstract: The recent results presented in arXiv:2202.05608 have led to significant developments in achieving stable approximations of Helmholtz solutions by plane wave superposition. The study shows that the numerical instability and ill-conditioning inherent in plane wave-based Trefftz methods can be effectively overcome with regularization techniques, provided there exist accurate approximations in the form of expansions with bounded coefficients. Whenever the target solution contains high Fourier modes, propagative plane waves fail to yield stable approximations due to the exponential growth of the expansion coefficients. Conversely, evanescent plane waves, whose modal content covers high Fourier regimes, are able to provide both accurate and stable results. The developed numerical approach, which involves constructing evanescent plane wave approximation sets by sampling the parametric domain according to a probability density function, results in substantial improvements when compared to conventional propagative plane wave schemes. The following work extends this research to the three-dimensional setting, confirming the achieved results and introducing new ones. By generalizing the 3D Jacobi$-$Anger identity to complex-valued directions, we show that any Helmholtz solution in a ball can be represented as a continuous superposition of evanescent plane waves. This representation extends the classical Herglotz one and provides a relevant stability result that cannot be achieved with the use of propagative waves alone. The proposed numerical recipes have been tailored for the 3D setting and extended with new sampling strategies involving extremal systems of points. These methods are tested by numerical experiments, showing the desired accuracy and bounded-coefficient stability, in line with the two-dimensional case.