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Stable approximation of Helmholtz solutions in the 3D ball using evanescent plane waves
Nicola Galante1; Andrea Moiola2; Emile Parolin1
1 Sorbonne Université, Université Paris Cité, CNRS, INRIA, Laboratoire Jacques-Louis Lions, LJLL, EPC ALPINES, F-75005 Paris, France
2 Department of Mathematics “Felice Casorati”, University of Pavia, Pavia, Italy
The SMAI Journal of computational mathematics, Volume 11 (2025), pp. 435-472.

The goal of this paper is to show that evanescent plane waves are much better at numerically approximating Helmholtz solutions than classical propagative plane waves. By generalizing the Jacobi–Anger identity to complex-valued directions, we first prove that any solution of the Helmholtz equation on a three dimensional ball can be written as a continuous superposition of evanescent plane waves in a stable way. We then propose a practical numerical recipe to select discrete approximation sets of evanescent plane waves, which exhibits considerable improvements over standard propagative plane wave schemes in numerical experiments. We show that all this is not possible for propagative plane waves: they cannot stably represent general Helmholtz solutions, and any approximation based on discrete sets of propagative plane waves is doomed to have exponentially large coefficients and thus to be numerically unstable. This paper is motivated by applications to Trefftz-type Galerkin schemes and extends the recent results in [33] from two to three space dimensions.

Published online:
DOI: 10.5802/smai-jcm.130
Classification: 35J05, 41A30, 42C15, 44A15
Keywords: Helmholtz equation, Plane wave, Evanescent plane wave, Trefftz method, Stable approximation, Sampling, Herglotz representation, Jacobi–Anger identity

Nicola Galante 1; Andrea Moiola 2; Emile Parolin 1

1 Sorbonne Université, Université Paris Cité, CNRS, INRIA, Laboratoire Jacques-Louis Lions, LJLL, EPC ALPINES, F-75005 Paris, France
2 Department of Mathematics “Felice Casorati”, University of Pavia, Pavia, Italy 
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     title = {Stable approximation of {Helmholtz} solutions in the {3D} ball using evanescent plane waves},
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     pages = {435--472},
     publisher = {Soci\'et\'e de Math\'ematiques Appliqu\'ees et Industrielles},
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Nicola Galante; Andrea Moiola; Emile Parolin. Stable approximation of Helmholtz solutions in the 3D ball using evanescent plane waves. The SMAI Journal of computational mathematics, Volume 11 (2025), pp. 435-472. doi : 10.5802/smai-jcm.130. https://smai-jcm.centre-mersenne.org/articles/10.5802/smai-jcm.130/

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