# A Galerkin method with microlocalised shape functions to solve high-frequency Helmholtz problems

Théophile Chaumont-Frelet, Victorita Dolean, Maxime Ingremeau, Florentin Proust

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Théophile Chaumont-Frelet, Victorita Dolean, Maxime Ingremeau, Florentin Proust. A Galerkin method with microlocalised shape functions to solve high-frequency Helmholtz problems. Conference on Mathematics of Wave Phenomena 2025, Feb 2025, Karlsruhe, Germany. hal-05285954

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Submitted on 26 Sep 2025

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# A Galerkin Method with Microlocalised Shape Functions to
Solve High-frequency Helmholtz Problems

Théophile Chaumont-Frelet¹, Victorita Dolean², Maxime Ingremeau³, Florentin Proust⁴

¹Univ. Lille, Inria, CNRS, UMR 8524 - Laboratoire Paul Painlevé, F-59000 Lille, France

²Eindhoven University of Technology, Netherlands

³Université Côte d'Azur, CNRS, LJAD, France

⁴Université Côte d'Azur, Inria, CNRS, LJAD, France

High-frequency Helmholtz problems are very challenging to solve with the Finite Element Method. Indeed, the number of degrees of freedom needed to correctly approximate the solution grows at least as $k^d$, where $k$ is the wavenumber and $d$ the dimension.

In [1], it has been proposed to use the Galerkin or the Least Squares method with Gaussian coherent states (GCS) as basis functions. For a wide range of source terms, the Least Squares formulation with GCS provides an accurate solution with only about $k^{d-\frac{1}{2}}$ degrees of freedom. However, no similar result has been proved for the Galerkin formulation, and it has been observed that GCS can lead to ill-conditioned discretisation matrices.

In this work, we use as basis functions particular linear combinations of GCS which lead to well-conditioned discretised problems. We will show that with these new shape functions, both the Galerkin and the Least Squares formulations provide an accurate solution with about $k^{d-\frac{3}{2}}$ degrees of freedom. We will illustrate these results with 1D numerical examples.

## References

[1] T. Chaumont-Frelet, V. Dolean and M. Ingremeau, Efficient approximation of high-frequency Helmholtz solutions by Gaussian coherent states, *Numerische Mathematik*, 2024, vol. 156, no 4, pp. 1385-1426.