[Une méthode de bases réduites « certifiée » utilisant la transformation de Laplace ; Application à lʼéquation de la chaleur et à lʼéquation des ondes]
On introduit une méthode de bases réduites « certifiée » pour lʼéquation de la chaleur et pour lʼéquation des ondes. Les outils sont les suivants : approximation en bases réduites « certifiée » de la transformée de Laplace, transformée de Laplace inverse pour lʼapproximation de la sortie en bases réduites pour la variable temps, estimations dʼerreurs rigoureuses, filtre en temps (de Butterworth) mettant en évidence la nécessité dʼune troncature « modale », décomposition en fonctions propres en bases réduites, intégrale de contour pour la décomposition « Offline–Online ». On donne des résultats numériques pour montrer lʼéfficacité et la précision de la méthode.
We present a certified reduced basis (RB) method for the heat equation and wave equation. The critical ingredients are certified RB approximation of the Laplace transform; the inverse Laplace transform to develop the time-domain RB output approximation and rigorous error bound; a (Butterworth) filter in time to effect the necessary “modal” truncation; RB eigenfunction decomposition and contour integration for Offline–Online decomposition. We present numerical results to demonstrate the accuracy and efficiency of the approach.
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@article{CRMATH_2011__349_7-8_401_0,
author = {Huynh, D.B. Phuong and Knezevic, David J. and Patera, Anthony T.},
title = {A {Laplace} transform certified reduced basis method; application to the heat equation and wave equation},
journal = {Comptes Rendus. Math\'ematique},
pages = {401--405},
publisher = {Elsevier},
volume = {349},
number = {7-8},
year = {2011},
doi = {10.1016/j.crma.2011.02.003},
language = {en},
url = {http://www.numdam.org/articles/10.1016/j.crma.2011.02.003/}
}
TY - JOUR AU - Huynh, D.B. Phuong AU - Knezevic, David J. AU - Patera, Anthony T. TI - A Laplace transform certified reduced basis method; application to the heat equation and wave equation JO - Comptes Rendus. Mathématique PY - 2011 SP - 401 EP - 405 VL - 349 IS - 7-8 PB - Elsevier UR - http://www.numdam.org/articles/10.1016/j.crma.2011.02.003/ DO - 10.1016/j.crma.2011.02.003 LA - en ID - CRMATH_2011__349_7-8_401_0 ER -
%0 Journal Article %A Huynh, D.B. Phuong %A Knezevic, David J. %A Patera, Anthony T. %T A Laplace transform certified reduced basis method; application to the heat equation and wave equation %J Comptes Rendus. Mathématique %D 2011 %P 401-405 %V 349 %N 7-8 %I Elsevier %U http://www.numdam.org/articles/10.1016/j.crma.2011.02.003/ %R 10.1016/j.crma.2011.02.003 %G en %F CRMATH_2011__349_7-8_401_0
Huynh, D.B. Phuong; Knezevic, David J.; Patera, Anthony T. A Laplace transform certified reduced basis method; application to the heat equation and wave equation. Comptes Rendus. Mathématique, Tome 349 (2011) no. 7-8, pp. 401-405. doi : 10.1016/j.crma.2011.02.003. http://www.numdam.org/articles/10.1016/j.crma.2011.02.003/
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