Volume 52, Number 4, July-August 2018
|Page(s)||1261 - 1284|
|Published online||28 September 2018|
Convergence analysis of Padé approximations for Helmholtz frequency response problems★
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
3 Faculty of Mathematics, University of Vienna, Oskar-Morgenstern-Platz 1, 1090 Wien, Austria
* Corresponding author: firstname.lastname@example.org
Accepted: 28 September 2017
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.
Mathematics Subject Classification: 30D30 / 41A21 / 41A25 / 35J05 / 65N30
Key words: Hilbert space-valued meromorphic maps / Padé approximants / convergence of Padé approximants / parametric PDEs / Helmholtz equation
© EDP Sciences, SMAI 2018
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