A space–time quasi-Trefftz DG method for the wave equation with piecewise-smooth coefficients
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- by Lise-Marie Imbert-Gérard, Andrea Moiola and Paul Stocker;
- Math. Comp. 92 (2023), 1211-1249
- DOI: https://doi.org/10.1090/mcom/3786
- Published electronically: November 22, 2022
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Abstract:
Trefftz methods are high-order Galerkin schemes in which all discrete functions are elementwise solution of the PDE to be approximated. They are viable only when the PDE is linear and its coefficients are piecewise-constant. We introduce a “quasi-Trefftz” discontinuous Galerkin (DG) method for the discretisation of the acoustic wave equation with piecewise-smooth material parameters: the discrete functions are elementwise approximate PDE solutions. We show that the new discretisation enjoys the same excellent approximation properties as the classical Trefftz one, and prove stability and high-order convergence of the DG scheme. We introduce polynomial basis functions for the new discrete spaces and describe a simple algorithm to compute them. The technique we propose is inspired by the generalised plane waves previously developed for time-harmonic problems with variable coefficients; it turns out that in the case of the time-domain wave equation under consideration the quasi-Trefftz approach allows for polynomial basis functions.References
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Bibliographic Information
- Lise-Marie Imbert-Gérard
- Affiliation: Department of Mathematics, University of Arizona, 617 N. Santa Rita Ave., Tuscon, Arizona 85721-0066
- ORCID: 0000-0001-7754-8582
- Email: lmig@math.arizona.edu
- Andrea Moiola
- Affiliation: Department of Mathematics, University of Pavia, Via Ferrata 5, 27100 Pavia, Italy
- MR Author ID: 931770
- ORCID: 0000-0002-6251-4440
- Email: andrea.moiola@unipv.it
- Paul Stocker
- Affiliation: Institute for Numerical and Applied Mathematics, University of Göttingen, Lotzestr. 16-18. 37038 Göttingen, Germany
- MR Author ID: 1377412
- ORCID: 0000-0001-5073-3366
- Email: p.stocker@math.uni-goettingen.de
- Received by editor(s): November 9, 2020
- Received by editor(s) in revised form: September 17, 2021, January 23, 2022, July 11, 2022, and August 24, 2022
- Published electronically: November 22, 2022
- Additional Notes: The first author was supported by the US National Science Foundation: this work was supported by the NSF under Grants No. DMS-1818747 and DMS-2105487. The second author was supported by GNCS–INDAM, by PRIN project “NA_FROM-PDEs” and by MIUR through the “Dipartimenti di Eccellenza” Programme (2018–2022)—Dept. of Mathematics, University of Pavia. The third author was supported by the Austrian Science Fund (FWF) through the projects F 65 and W 1245.
- © Copyright 2022 American Mathematical Society
- Journal: Math. Comp. 92 (2023), 1211-1249
- MSC (2020): Primary 65M60, 65M15, 35L05, 41A10, 41A25
- DOI: https://doi.org/10.1090/mcom/3786
- MathSciNet review: 4550324