Volume 52, Number 5, September–October 2018
|Page(s)||1803 - 1845|
|Published online||30 November 2018|
High-frequency behaviour of corner singularities in Helmholtz problems★
BCAM – Basque Center for Applied Mathematics, Alameda Mazarredo, 14,
2 Univ. Valenciennes, EA 4015, LAMAV – Laboratoire de Mathématiques et leurs Applications de Valenciennes, FR CNRS 2956, 59313 Valenciennes, France.
* Corresponding author: firstname.lastname@example.org
Accepted: 30 April 2018
We analyze the singular behaviour of the Helmholtz equation set in a non-convex polygon. Classically, the solution of the problem is split into a regular part and one singular function for each re-entrant corner. The originality of our work is that the “amplitude” of the singular parts is bounded explicitly in terms of frequency. We show that for high frequency problems, the “dominant” part of the solution is the regular part. As an application, we derive sharp error estimates for finite element discretizations. These error estimates show that the “pollution effect” is not changed by the presence of singularities. Furthermore, a consequence of our theory is that locally refined meshes are not needed for high-frequency problems, unless a very accurate solution is required. These results are illustrated with numerical examples that are in accordance with the developed theory.
Mathematics Subject Classification: 35J05 / 35J75 / 65N30 / 78A45
Key words: Helmholtz problems / corner singularities / finite elements / pollution effect
T. Chaumont-Frelet has received funding from the European Union's Horizon 2020 research and innovation programme under the Marie Sklodowska-Curie grant agreement No. 644602, the Projects of the Spanish Ministry of Economy and Competitiveness with reference MTM2016-76329-R (AEI/FEDER, EU), and MTM2016-81697-ERC/AEI, the BCAM “Severo Ochoa” accreditation of excellence SEV-2013-0323, and the Basque Government through the BERC 2014-2017 program, and the Consolidated Research Group Grant IT649-13 on “Mathematical Modeling, Simulation, and Industrial Applications (M2SI)”.
© EDP Sciences, SMAI 2018
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