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arXiv · 2607.19950

An efficient Galerkin method for high-frequency scattering problems using Wilson bases

T. Chaumont-Frelet·M. Ingremeau·F. Proust
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Abstract

We propose a new Galerkin discretization scheme for wave scattering problems that is based on microlocalised basis functions. We show that the proposed method can be made uniformly accurate for large wavenumbers $k$ with a number of degrees of freedom only scaling as $k^{d-1/2}$, while leading to an essentially sparse linear system. In contrast, finite element methods are known to require a number of degrees of freedom scaling at least as $k^d$ to achieve the same property. A similar method based on a Gabor frame was previously introduced by two of the authors, but it was suffering from severe conditioning issues. In the present work, by replacing the Gabor frame by a Wilson basis, we completely alleviate this problem. We rigorously establish error estimates and condition number bounds for the proposed method, and we provide one-dimensional numerical examples illustrating our theoretical findings.

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T. Chaumont-Frelet, M. Ingremeau, F. Proust. 2026-07-22. An efficient Galerkin method for high-frequency scattering problems using Wilson bases. https://arxiv.org/abs/2607.19950

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