For approximating parametric problem solutions, Reduced Basis Methods (RBMs) are frequently proposed. They intend to reduce the computational costs of High Fidelity (HF) codes while maintaining the HF accuracy. They generally require an offline/online decomposition and a significant modification of the HF code in order for the online computation to be performed in short (or even real) time. We focus on the Non-Intrusive Reduced Basis (NIRB) two-grid method in this paper. Its main feature is the use of the HF code as a “black-box” on a coarse mesh for a new parameter during the online process, followed by an accuracy improvement based on the reduced basis paradigm that is realized in a very short time. Unlike other more intrusive methods, this approach does not necessitate code modification. As a result, it costs significantly less than an HF evaluation. This method was developed for elliptic equations with finite elements and has since then been extended to finite volume schemes.
In this paper, we extend the NIRB two-grid method to parabolic equations. We recover optimal estimates in natural norms, using the heat equation as a model problem and present several numerical results on the heat equation and on the Brusselator problem.
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Keywords: Non-intrusive reduced basis, parabolic equations, finite elements method
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@article{SMAI-JCM_2023__9__227_0,
author = {Elise Grosjean and Yvon Maday},
title = {Error estimate of the {Non-Intrusive} {Reduced} {Basis} {(NIRB)} two-grid method with parabolic equations},
journal = {The SMAI Journal of computational mathematics},
pages = {227--256},
publisher = {Soci\'et\'e de Math\'ematiques Appliqu\'ees et Industrielles},
volume = {9},
year = {2023},
doi = {10.5802/smai-jcm.100},
language = {en},
url = {https://smai-jcm.centre-mersenne.org/articles/10.5802/smai-jcm.100/}
}
TY - JOUR AU - Elise Grosjean AU - Yvon Maday TI - Error estimate of the Non-Intrusive Reduced Basis (NIRB) two-grid method with parabolic equations JO - The SMAI Journal of computational mathematics PY - 2023 SP - 227 EP - 256 VL - 9 PB - Société de Mathématiques Appliquées et Industrielles UR - https://smai-jcm.centre-mersenne.org/articles/10.5802/smai-jcm.100/ DO - 10.5802/smai-jcm.100 LA - en ID - SMAI-JCM_2023__9__227_0 ER -
%0 Journal Article %A Elise Grosjean %A Yvon Maday %T Error estimate of the Non-Intrusive Reduced Basis (NIRB) two-grid method with parabolic equations %J The SMAI Journal of computational mathematics %D 2023 %P 227-256 %V 9 %I Société de Mathématiques Appliquées et Industrielles %U https://smai-jcm.centre-mersenne.org/articles/10.5802/smai-jcm.100/ %R 10.5802/smai-jcm.100 %G en %F SMAI-JCM_2023__9__227_0
Elise Grosjean; Yvon Maday. Error estimate of the Non-Intrusive Reduced Basis (NIRB) two-grid method with parabolic equations. The SMAI Journal of computational mathematics, Volume 9 (2023), pp. 227-256. doi : 10.5802/smai-jcm.100. https://smai-jcm.centre-mersenne.org/articles/10.5802/smai-jcm.100/
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