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

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

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Théophile Chaumont-Frelet, Victorita Dolean, M Ingremeau, Florentin Proust. A Galerkin method with microlocalised shape functions to solve high-frequency Helmholtz problems. ECCOMAS CONGRESS 2024 - 9th European Congress on Computational Methods in Applied Sciences and Engineering, Jun 2024, Lisbonne, Portugal. hal-04892429

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https://inria.hal.science/hal-04892429v1

Submitted on 16 Jan 2025

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# A GALERKIN METHOD WITH MICROLOCALISED SHAPE FUNCTIONS TO SOLVE HIGH-FREQUENCY HELMHOLTZ PROBLEMS

T. Chaumont-Frelet<fnref n="1" />, V. Dolean<fnref n="2" />, M. Ingremeau<fnref n="3" /> and F. Proust*<fnref n="4" />

**Keywords:** Galerkin method, Helmholtz equation, high-frequency problems, semi-classical analysis, time-frequency analysis, Gabor frames, Wilson basis, convergence analysis

High-frequency wave propagation problems, often modeled by the Helmholtz equation, are very challenging to solve numerically. For instance, when using the Finite Element Method (FEM), the number of degrees of freedom needed to approximate the solution correctly grows at least as $k^d$, where $k$ is the wavenumber and $d$ the dimension (see e.g. [3]).

An alternative has been recently proposed in [1]. There, Gaussian coherent states are used as basis functions instead of the piecewise polynomials of the FEM. For a wide range of source terms (such as those arising in plane-wave scattering), it is proved in [1] that Gaussian coherent states provide an accurate solution with only about $k^{d-\frac{1}{2}}$ degrees of freedom, provided they are combined with a least squares formulation. However, this approach is currently limited in two ways. On the one hand, for Galerkin formulations (instead of least squares), no similar result has been proved. On the other hand, it has been numerically observed that Gaussian coherent states can lead to ill-conditioned discretisation matrices.

In this work, we improve on this. We use as basis functions particular linear combinations of Gaussian coherent states which form what is called a *Wilson basis* [2], and which lead to well-conditioned discretised problems. We will show that with these new shape functions, both the least squares and the Galerkin formulations provide an accurate representation of the solution with a number of degrees of freedom scaling roughly as $k^{d-\frac{1}{2}}$. Furthermore, we will illustrate these theoretical findings with one-dimensional numerical examples.

## REFERENCES

[1] Théophile Chaumont-Frelet, Victorita Dolean, and Maxime Ingremeau. "Efficient approximation of high-frequency Helmholtz solutions by Gaussian coherent states". In: *arXiv preprint arXiv:2208.04851* (2022).

[2] Karlheinz Gröchenig. *Foundations of time-frequency analysis*. Springer Science & Business Media, 2001.

[3] J.M. Melenk and S. Sauter. "Convergence analysis for finite element discretizations of the Helmholtz equation with Dirichlet-to-Neumann boundary conditions". In: *Math. Comp.* 79.272 (2010), pp. 1871–1914.

<footnote n="1">¹ Univ. Lille, Inria, CNRS, UMR 8524 - Laboratoire Paul Painlevé, F-59000 Lille, France, theophile.chaumont@inria.fr</footnote>

<footnote n="2">² Université Côte d'Azur, CNRS, LJAD, France, victorita.dolean@univ-cotedazur.fr</footnote>

<footnote n="3">³ Université Côte d'Azur, CNRS, LJAD, France, maxime.ingremeau@univ-cotedazur.fr</footnote>

<footnote n="4">⁴ Université Côte d'Azur, Inria, CNRS, LJAD, France, florentin.proust@inria.fr</footnote>