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-2 | Engels |
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Pagina's (van-tot) | 1385-1426 |
Aantal pagina's | 42 |
Tijdschrift | Numerische Mathematik |
Volume | 156 |
Nummer van het tijdschrift | 4 |
DOI's | |
Status | Gepubliceerd - aug. 2024 |
Bibliografische nota
Publisher Copyright:© The Author(s), under exclusive licence to Springer-Verlag GmbH Germany, part of Springer Nature 2024.