| Issue |
ESAIM: M2AN
Volume 57, Number 5, September-October 2023
|
|
|---|---|---|
| Page(s) | 2775 - 2802 | |
| DOI | https://doi.org/10.1051/m2an/2023013 | |
| Published online | 14 September 2023 | |
Reduced order modeling for elliptic problems with high contrast diffusion coefficients
1
Laboratoire Jacques-Louis Lions, Sorbonne Université, 4 place Jussieu, 75005 Paris, France
2
Department of Mathematics, University of South Carolina, 1523 Greene St, Columbia, SC 29208, USA
* Corresponding author: albert.cohen@sorbonne-universite.fr
Received:
31
January
2022
Accepted:
8
February
2023
We consider a parametric elliptic PDE with a scalar piecewise constant diffusion coefficient taking arbitrary positive values on fixed subdomains. This problem is not uniformly elliptic, as the contrast can be arbitrarily high, contrary to the Uniform Ellipticity Assumption (UEA) that is commonly made on parametric elliptic PDEs.We construct reduced model spaces that approximate uniformly well all solutions with estimates in relative error that are independent of the contrast level. These estimates are sub-exponential in the reduced model dimension, yet exhibiting the curse of dimensionality as the number of subdomains grows. Similar estimates are obtained for the Galerkin projection, as well as for the state estimation and parameter estimation inverse problems. A key ingredient in our construction and analysis is the study of the convergence towards limit solutions of stiff problems when diffusion tends to infinity in certain domains.
Mathematics Subject Classification: 35A35 / 35J70 / 65N12 / 65N99
Key words: Reduced bases / high contrast elliptic PDEs / piecewise polynomial approximations / state and parameter estimation
© The authors. Published by EDP Sciences, SMAI 2023
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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