The context of this paper is the simulation of parameter-dependent partial differential equations (PDEs). When the aim is to solve such PDEs for a large number of parameter values, Reduced Basis Methods (RBM) are often used to reduce computational costs of a classical high fidelity code based on Finite Element Method (FEM), Finite Volume (FVM) or Spectral methods. The efficient implementation of most of these RBM requires to modify this high fidelity code, which cannot be done, for example in an industrial context if the high fidelity code is only accessible as a "black-box" solver. The Non-Intrusive Reduced Basis (NIRB) method has been introduced in the context of finite elements as a good alternative to reduce the implementation costs of these parameter-dependent problems. The method is efficient in other contexts than the FEM one, like with finite volume schemes, which are more often used in an industrial environment. In this case, some adaptations need to be done as the degrees of freedom in FV methods have different meanings. At this time, error estimates have only been studied with FEM solvers. In this paper, we present a generalisation of the NIRB method to Finite Volume schemes and we show that estimates established for FEM solvers also hold in the FVM setting. We first prove our results for the hybrid-Mimetic Finite Difference method (hMFD), which is part the Hybrid Mixed Mimetic methods (HMM) family. Then, we explain how these results apply more generally to other FV schemes. Some of them are specified, such as the Two Point Flux Approximation (TPFA).
Keywords: Reduced basis method, finite volume method
@article{M2AN_2021__55_5_1941_0,
author = {Grosjean, Elise and Maday, Yvon},
title = {Error estimate of the non-intrusive reduced basis method with finite volume schemes},
journal = {ESAIM: Mathematical Modelling and Numerical Analysis },
pages = {1941--1961},
year = {2021},
publisher = {EDP-Sciences},
volume = {55},
number = {5},
doi = {10.1051/m2an/2021044},
mrnumber = {4315954},
zbl = {1477.65175},
language = {en},
url = {https://www.numdam.org/articles/10.1051/m2an/2021044/}
}
TY - JOUR AU - Grosjean, Elise AU - Maday, Yvon TI - Error estimate of the non-intrusive reduced basis method with finite volume schemes JO - ESAIM: Mathematical Modelling and Numerical Analysis PY - 2021 SP - 1941 EP - 1961 VL - 55 IS - 5 PB - EDP-Sciences UR - https://www.numdam.org/articles/10.1051/m2an/2021044/ DO - 10.1051/m2an/2021044 LA - en ID - M2AN_2021__55_5_1941_0 ER -
%0 Journal Article %A Grosjean, Elise %A Maday, Yvon %T Error estimate of the non-intrusive reduced basis method with finite volume schemes %J ESAIM: Mathematical Modelling and Numerical Analysis %D 2021 %P 1941-1961 %V 55 %N 5 %I EDP-Sciences %U https://www.numdam.org/articles/10.1051/m2an/2021044/ %R 10.1051/m2an/2021044 %G en %F M2AN_2021__55_5_1941_0
Grosjean, Elise; Maday, Yvon. Error estimate of the non-intrusive reduced basis method with finite volume schemes. ESAIM: Mathematical Modelling and Numerical Analysis , Tome 55 (2021) no. 5, pp. 1941-1961. doi: 10.1051/m2an/2021044
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