Efficient approximation of high-frequency Helmholtz solutions by Gaussian coherent states

T. Chaumont-Frelet (Corresponding author), V. Dolean, M. Ingremeau

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    Samenvatting

    We introduce new finite-dimensional spaces specifically designed to approximate the solutions to high-frequency Helmholtz problems with smooth variable coefficients in dimension d. These discretization spaces are spanned by Gaussian coherent states, that have the key property to be localised in phase space. We carefully select the Gaussian coherent states spanning the approximation space by exploiting the (known) micro-localisation properties of the solution. For a large class of source terms (including plane-wave scattering problems), this choice leads to discrete spaces that provide a uniform approximation error for all wavenumber k with a number of degrees of freedom scaling as kd-1/2, which we rigorously establish. In comparison, for discretization spaces based on (piecewise) polynomials, the number of degrees of freedom has to scale at least as kd to achieve the same property. These theoretical results are illustrated by one-dimensional numerical examples, where the proposed discretization spaces are coupled with a least-squares variational formulation.

    Originele taal-2Engels
    Pagina's (van-tot)1385-1426
    Aantal pagina's42
    TijdschriftNumerische Mathematik
    Volume156
    Nummer van het tijdschrift4
    DOI's
    StatusGepubliceerd - aug. 2024

    Bibliografische nota

    Publisher Copyright:
    © The Author(s), under exclusive licence to Springer-Verlag GmbH Germany, part of Springer Nature 2024.

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