We consider numerical solution of elliptic problems with heterogeneous diffusion coefficients containing thin highly conductive structures. Such problems arise e.g. in fractured porous media, reinforced materials, and electric circuits. The main computational challenge is the high resolution needed to resolve the data variation. We propose a multiscale method that models the thin structures as interfaces and incorporate heterogeneities in corrected shape functions. The construction results in an accurate upscaled representation of the system that can be used to solve for several forcing functions or to simulate evolution problems in an efficient way. By introducing a novel interpolation operator, defining the fine scale of the problem, we prove exponential decay of the shape functions which allows for a sparse approximation of the upscaled representation. An a priori error bound is also derived for the proposed method together with numerical examples that verify the theoretical findings. Finally we present a numerical example to show how the technique can be applied to evolution problems.
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Publié le :
DOI : 10.1051/m2an/2020061
Keywords: Generalized finite element method, localized orthogonal decomposition, porous media, fracture, Darcy flow
@article{M2AN_2021__55_S1_S761_0,
author = {Hellman, Fredrik and M\r{a}lqvist, Axel and Wang, Siyang},
title = {Numerical upscaling for heterogeneous materials in fractured domains},
journal = {ESAIM: Mathematical Modelling and Numerical Analysis },
pages = {S761--S784},
year = {2021},
publisher = {EDP-Sciences},
volume = {55},
number = {Suppl\'ement},
doi = {10.1051/m2an/2020061},
mrnumber = {4221307},
zbl = {1491.65138},
language = {en},
url = {https://www.numdam.org/articles/10.1051/m2an/2020061/}
}
TY - JOUR AU - Hellman, Fredrik AU - Målqvist, Axel AU - Wang, Siyang TI - Numerical upscaling for heterogeneous materials in fractured domains JO - ESAIM: Mathematical Modelling and Numerical Analysis PY - 2021 SP - S761 EP - S784 VL - 55 IS - Supplément PB - EDP-Sciences UR - https://www.numdam.org/articles/10.1051/m2an/2020061/ DO - 10.1051/m2an/2020061 LA - en ID - M2AN_2021__55_S1_S761_0 ER -
%0 Journal Article %A Hellman, Fredrik %A Målqvist, Axel %A Wang, Siyang %T Numerical upscaling for heterogeneous materials in fractured domains %J ESAIM: Mathematical Modelling and Numerical Analysis %D 2021 %P S761-S784 %V 55 %N Supplément %I EDP-Sciences %U https://www.numdam.org/articles/10.1051/m2an/2020061/ %R 10.1051/m2an/2020061 %G en %F M2AN_2021__55_S1_S761_0
Hellman, Fredrik; Målqvist, Axel; Wang, Siyang. Numerical upscaling for heterogeneous materials in fractured domains. ESAIM: Mathematical Modelling and Numerical Analysis , Tome 55 (2021), pp. S761-S784. doi: 10.1051/m2an/2020061
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