Geodesic
Convergence analysis of Padé approximations for Helmholtz frequency response problems
Bonizzoni, Francesca1 ; Nobile, Fabio2  ; Perugia, Ilaria1 
1 Faculty of Mathematics, University of Vienna, Oskar-Morgenstern-Platz 1, 1090 Wien, Austria
2 CSQI – Calcul Scientifique et Quantification de l’Incertitude, MATHICSE, École Polytechnique Fédérale de Lausanne, Station 8, CH-1015 Lausanne, Switzerland
ESAIM: Mathematical Modelling and Numerical Analysis , Tome 52 (2018) no. 4, pp. 1261-1284

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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.

Reçu le :
Accepté le :
DOI : 10.1051/m2an/2017050
Classification : 30D30, 41A21, 41A25, 35J05, 65N30
Keywords: Hilbert space-valued meromorphic maps, Padé approximants, convergence of Padé approximants, parametric PDEs, Helmholtz equation

Bonizzoni, Francesca 1 ; Nobile, Fabio 2 ; Perugia, Ilaria 1

1 Faculty of Mathematics, University of Vienna, Oskar-Morgenstern-Platz 1, 1090 Wien, Austria
2 CSQI – Calcul Scientifique et Quantification de l’Incertitude, MATHICSE, École Polytechnique Fédérale de Lausanne, Station 8, CH-1015 Lausanne, Switzerland 
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     title = {Convergence analysis of {Pad\'e} approximations for {Helmholtz} frequency response problems},
     journal = {ESAIM: Mathematical Modelling and Numerical Analysis },
     pages = {1261--1284},
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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

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