Volume 55, Number 1, January-February 2021
|Page(s)||37 - 55|
|Published online||18 February 2021|
An iterative method for elliptic problems with rapidly oscillating coefficients
Courant Institute of Mathematical Sciences, New York University, New York, USA
2 Department of Mathematics and Systems Analysis, Aalto University, Espoo, Finland
3 Department of Mathematics and Statistics, University of Helsinki, Helsinki, Finland
4 DMA, Ecole normale supérieure, CNRS, PSL Research University, Paris, France
* Corresponding author: email@example.com
Accepted: 19 November 2020
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.
Mathematics Subject Classification: 65N55 / 35B27
Key words: Multiscale method / multigrid method / homogenization
© The authors. Published by EDP Sciences, SMAI 2021
This is an Open Access article distributed under the terms of the Creative Commons Attribution License (https://creativecommons.org/licenses/by/4.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
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