The present work concerns the approximation of the solution map S associated to the parametric Helmholtz boundary value problem, i.e., the map which associates to each (real) wavenumber belonging to a given interval of interest the corresponding solution of the Helmholtz equation. We introduce a least squares rational Padé-type approximation technique applicable to any meromorphic Hilbert space-valued univariate map, and we prove the uniform convergence of the Padé approximation error on any compact subset of the interval of interest that excludes any pole. This general result is then applied to the Helmholtz solution map S, which is proven to be meromorphic in ℂ, with a pole of order one in every (single or multiple) eigenvalue of the Laplace operator with the considered boundary conditions. Numerical tests are provided that confirm the theoretical upper bound on the Padé approximation error for the Helmholtz solution map.
Accepté le :
DOI : 10.1051/m2an/2017050
Mots-clés : Hilbert space-valued meromorphic maps, Padé approximants, convergence of Padé approximants, parametric PDEs, Helmholtz equation
@article{M2AN_2018__52_4_1261_0,
author = {Bonizzoni, Francesca and Nobile, Fabio and Perugia, Ilaria},
title = {Convergence analysis of {Pad\'e} approximations for {Helmholtz} frequency response problems},
journal = {ESAIM: Mathematical Modelling and Numerical Analysis },
pages = {1261--1284},
publisher = {EDP-Sciences},
volume = {52},
number = {4},
year = {2018},
doi = {10.1051/m2an/2017050},
mrnumber = {3875286},
zbl = {1411.35079},
language = {en},
url = {http://www.numdam.org/articles/10.1051/m2an/2017050/}
}
TY - JOUR AU - Bonizzoni, Francesca AU - Nobile, Fabio AU - Perugia, Ilaria TI - Convergence analysis of Padé approximations for Helmholtz frequency response problems JO - ESAIM: Mathematical Modelling and Numerical Analysis PY - 2018 SP - 1261 EP - 1284 VL - 52 IS - 4 PB - EDP-Sciences UR - http://www.numdam.org/articles/10.1051/m2an/2017050/ DO - 10.1051/m2an/2017050 LA - en ID - M2AN_2018__52_4_1261_0 ER -
%0 Journal Article %A Bonizzoni, Francesca %A Nobile, Fabio %A Perugia, Ilaria %T Convergence analysis of Padé approximations for Helmholtz frequency response problems %J ESAIM: Mathematical Modelling and Numerical Analysis %D 2018 %P 1261-1284 %V 52 %N 4 %I EDP-Sciences %U http://www.numdam.org/articles/10.1051/m2an/2017050/ %R 10.1051/m2an/2017050 %G en %F M2AN_2018__52_4_1261_0
Bonizzoni, Francesca; Nobile, Fabio; Perugia, Ilaria. Convergence analysis of Padé approximations for Helmholtz frequency response problems. ESAIM: Mathematical Modelling and Numerical Analysis , Tome 52 (2018) no. 4, pp. 1261-1284. doi : 10.1051/m2an/2017050. http://www.numdam.org/articles/10.1051/m2an/2017050/
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