We introduce a new iterative method for computing solutions of elliptic equations with random rapidly oscillating coefficients. Similarly to a multigrid method, each step of the iteration involves different computations meant to address different length scales. However, we use here the homogenized equation on all scales larger than a fixed multiple of the scale of oscillation of the coefficients. While the performance of standard multigrid methods degrades rapidly under the regime of large scale separation that we consider here, we show an explicit estimate on the contraction factor of our method which is independent of the size of the domain. We also present numerical experiments which confirm the effectiveness of the method, with openly available source code.
Keywords: Multiscale method, multigrid method, homogenization
@article{M2AN_2021__55_1_37_0,
author = {Armstrong, Scott and Hannukainen, Antti and Kuusi, Tuomo and Mourrat, Jean-Christophe},
title = {An iterative method for elliptic problems with rapidly oscillating coefficients},
journal = {ESAIM: Mathematical Modelling and Numerical Analysis },
pages = {37--55},
year = {2021},
publisher = {EDP-Sciences},
volume = {55},
number = {1},
doi = {10.1051/m2an/2020080},
mrnumber = {4216834},
zbl = {1481.65251},
language = {en},
url = {https://www.numdam.org/articles/10.1051/m2an/2020080/}
}
TY - JOUR AU - Armstrong, Scott AU - Hannukainen, Antti AU - Kuusi, Tuomo AU - Mourrat, Jean-Christophe TI - An iterative method for elliptic problems with rapidly oscillating coefficients JO - ESAIM: Mathematical Modelling and Numerical Analysis PY - 2021 SP - 37 EP - 55 VL - 55 IS - 1 PB - EDP-Sciences UR - https://www.numdam.org/articles/10.1051/m2an/2020080/ DO - 10.1051/m2an/2020080 LA - en ID - M2AN_2021__55_1_37_0 ER -
%0 Journal Article %A Armstrong, Scott %A Hannukainen, Antti %A Kuusi, Tuomo %A Mourrat, Jean-Christophe %T An iterative method for elliptic problems with rapidly oscillating coefficients %J ESAIM: Mathematical Modelling and Numerical Analysis %D 2021 %P 37-55 %V 55 %N 1 %I EDP-Sciences %U https://www.numdam.org/articles/10.1051/m2an/2020080/ %R 10.1051/m2an/2020080 %G en %F M2AN_2021__55_1_37_0
Armstrong, Scott; Hannukainen, Antti; Kuusi, Tuomo; Mourrat, Jean-Christophe. An iterative method for elliptic problems with rapidly oscillating coefficients. ESAIM: Mathematical Modelling and Numerical Analysis , Tome 55 (2021) no. 1, pp. 37-55. doi: 10.1051/m2an/2020080
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