In this paper, we propose an offline-online strategy based on the Localized Orthogonal Decomposition (LOD) method for elliptic multiscale problems with randomly perturbed diffusion coefficient. We consider a periodic deterministic coefficient with local defects that occur with probability p. The offline phase pre-computes entries to global LOD stiffness matrices on a single reference element (exploiting the periodicity) for a selection of defect configurations. Given a sample of the perturbed diffusion the corresponding LOD stiffness matrix is then computed by taking linear combinations of the pre-computed entries, in the online phase. Our computable error estimates show that this yields a good approximation of the solution for small p, which is illustrated by extensive numerical experiments. This makes the proposed technique attractive already for moderate sample sizes in a Monte Carlo simulation.
Keywords: Numerical homogenization, multiscale method, finite elements, random perturbations
@article{M2AN_2022__56_1_237_0,
author = {M\r{a}lqvist, Axel and Verf\"urth, Barbara},
title = {An offline-online strategy for multiscale problems with random defects},
journal = {ESAIM: Mathematical Modelling and Numerical Analysis },
pages = {237--260},
year = {2022},
publisher = {EDP-Sciences},
volume = {56},
number = {1},
doi = {10.1051/m2an/2022006},
mrnumber = {4378546},
zbl = {1484.65302},
language = {en},
url = {https://www.numdam.org/articles/10.1051/m2an/2022006/}
}
TY - JOUR AU - Målqvist, Axel AU - Verfürth, Barbara TI - An offline-online strategy for multiscale problems with random defects JO - ESAIM: Mathematical Modelling and Numerical Analysis PY - 2022 SP - 237 EP - 260 VL - 56 IS - 1 PB - EDP-Sciences UR - https://www.numdam.org/articles/10.1051/m2an/2022006/ DO - 10.1051/m2an/2022006 LA - en ID - M2AN_2022__56_1_237_0 ER -
%0 Journal Article %A Målqvist, Axel %A Verfürth, Barbara %T An offline-online strategy for multiscale problems with random defects %J ESAIM: Mathematical Modelling and Numerical Analysis %D 2022 %P 237-260 %V 56 %N 1 %I EDP-Sciences %U https://www.numdam.org/articles/10.1051/m2an/2022006/ %R 10.1051/m2an/2022006 %G en %F M2AN_2022__56_1_237_0
Målqvist, Axel; Verfürth, Barbara. An offline-online strategy for multiscale problems with random defects. ESAIM: Mathematical Modelling and Numerical Analysis , Tome 56 (2022) no. 1, pp. 237-260. doi: 10.1051/m2an/2022006
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