| Issue |
ESAIM: M2AN
Volume 58, Number 6, November-December 2024
Special issue - To commemorate Assyr Abdulle
|
|
|---|---|---|
| Page(s) | 2187 - 2223 | |
| DOI | https://doi.org/10.1051/m2an/2024013 | |
| Published online | 04 December 2024 | |
An optimization-based method for sign-changing elliptic PDEs*
1
ANMC, Institute of Mathematics, École Polytechnique Fédérale de Lausanne, 1015 Lausanne, Switzerland
2
Inria, Univ. Lille, CNRS, UMR 8524 – Laboratoire Paul Painlevé, 59000 Lille, France
** Corresponding author: simon.lemaire@inria.fr
Received:
25
June
2023
Accepted:
27
February
2024
We study the numerical approximation of sign-shifting problems of elliptic type. We fully analyze and assess the method briefly introduced in [A. Abdulle, M.E. Huber and S. Lemaire, C. R. Math. 355 (2017) 472–478]. Our method is based on domain decomposition and optimization. Upon assuming extra integrability on the exact normal flux trace along the sign-changing interface, our method is proved to be convergent as soon as, for a given loading, the PDE admits a unique solution of finite energy. Departing from the T-coercivity approach, which relies on the use of geometrically fitted mesh families, our method works for arbitrary (interface-compliant) mesh sequences. Moreover, it is shown convergent for a class of problems for which T-coercivity is not applicable. A comprehensive set of test-cases complements our analysis.
Mathematics Subject Classification: 65N30 / 65N55 / 78A48
Key words: Sign-shifting PDEs / metamaterials / finite elements / domain decomposition / optimization
© The authors. Published by EDP Sciences, SMAI 2024
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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