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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.
Chaumont-Frelet, T. 1 ; Nicaise, S. 1
@article{M2AN_2018__52_5_1803_0,
author = {Chaumont-Frelet, T. and Nicaise, S.},
title = {High-frequency behaviour of corner singularities in {Helmholtz} problems},
journal = {ESAIM: Mathematical Modelling and Numerical Analysis },
pages = {1803--1845},
publisher = {EDP-Sciences},
volume = {52},
number = {5},
year = {2018},
doi = {10.1051/m2an/2018031},
zbl = {1414.35053},
mrnumber = {3881571},
language = {en},
url = {http://geodesic.mathdoc.fr/articles/10.1051/m2an/2018031/}
}
TY - JOUR AU - Chaumont-Frelet, T. AU - Nicaise, S. TI - High-frequency behaviour of corner singularities in Helmholtz problems JO - ESAIM: Mathematical Modelling and Numerical Analysis PY - 2018 SP - 1803 EP - 1845 VL - 52 IS - 5 PB - EDP-Sciences UR - http://geodesic.mathdoc.fr/articles/10.1051/m2an/2018031/ DO - 10.1051/m2an/2018031 LA - en ID - M2AN_2018__52_5_1803_0 ER -
%0 Journal Article %A Chaumont-Frelet, T. %A Nicaise, S. %T High-frequency behaviour of corner singularities in Helmholtz problems %J ESAIM: Mathematical Modelling and Numerical Analysis %D 2018 %P 1803-1845 %V 52 %N 5 %I EDP-Sciences %U http://geodesic.mathdoc.fr/articles/10.1051/m2an/2018031/ %R 10.1051/m2an/2018031 %G en %F M2AN_2018__52_5_1803_0
Chaumont-Frelet, T.; Nicaise, S. High-frequency behaviour of corner singularities in Helmholtz problems. ESAIM: Mathematical Modelling and Numerical Analysis , Tome 52 (2018) no. 5, pp. 1803-1845. doi: 10.1051/m2an/2018031
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