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A multi-scale patch approximation for Poisson problems with a small inhomogeneous inclusion
Saber Amdouni1; Mohamed Khaled Gdoura2; Arnaud Heibig3; Thomas Homolle4; Nidhal Mannai1; Adrien Petrov3; Yves Renard3
1 Laboratory For Mathematical And Numerical Modeling In Engineering Science, LAMSIN, LR99ES20, University of Tunis El Manar, National Engineering School of Tunis, Tunis, Tunisia
2 University of Carthage, National Institute of Applied Science and Technology, Centre Urbain Nord BP 676-1080 Tunis Cedex, Tunisia
3 INSA Lyon, ICJ UMR5208, CNRS, Ecole Centrale de Lyon, Universite Claude Bernard Lyon 1, Université Jean Monnet, 69621 Villeurbanne, France
4 Numerical Modeling and Simulation (FEA) Department of Simulation and Data-Science, Manufacture Française des Pneumatiques Michelin Center of Technologies, Ladoux, Clermont-Ferrand, France
The SMAI Journal of computational mathematics, Volume 11 (2025), pp. 139-163.

The paper deals with the multi-scale approximation of the influence of a small inhomogeneity of arbitrary shape in an elastic medium. A new multi-scale patch method is introduced, whose caracteristic is to deal with a large scale problem without inclusion, a small-scale problem on a patch surrounding the inclusion defining a corrector and an iterative procedure between these two problems. Theoretical results of convergence of the iterations, a posteriori error estimate and comparison of the corrector with the asymptotic expansion are provided. The finite element approximation is also addressed together with some numerical tests.

Published online:
DOI: 10.5802/smai-jcm.121
Classification: 65N12, 65N30, 74Q05, 74Q15
Keywords: Patch method, Asymptotic expansion, finite element method, multi-scale analysis, transmission problem

Saber Amdouni 1; Mohamed Khaled Gdoura 2; Arnaud Heibig 3; Thomas Homolle 4; Nidhal Mannai 1; Adrien Petrov 3; Yves Renard 3

1 Laboratory For Mathematical And Numerical Modeling In Engineering Science, LAMSIN, LR99ES20, University of Tunis El Manar, National Engineering School of Tunis, Tunis, Tunisia
2 University of Carthage, National Institute of Applied Science and Technology, Centre Urbain Nord BP 676-1080 Tunis Cedex, Tunisia
3 INSA Lyon, ICJ UMR5208, CNRS, Ecole Centrale de Lyon, Universite Claude Bernard Lyon 1, Université Jean Monnet, 69621 Villeurbanne, France
4 Numerical Modeling and Simulation (FEA) Department of Simulation and Data-Science, Manufacture Française des Pneumatiques Michelin Center of Technologies, Ladoux, Clermont-Ferrand, France
License: CC-BY 4.0
Copyrights: The authors retain unrestricted copyrights and publishing rights 
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     author = {Saber Amdouni and Mohamed Khaled Gdoura and Arnaud Heibig and Thomas Homolle and Nidhal Mannai and Adrien Petrov and Yves Renard},
     title = {A multi-scale patch approximation for {Poisson} problems with a small inhomogeneous inclusion},
     journal = {The SMAI Journal of computational mathematics},
     pages = {139--163},
     publisher = {Soci\'et\'e de Math\'ematiques Appliqu\'ees et Industrielles},
     volume = {11},
     year = {2025},
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Saber Amdouni; Mohamed Khaled Gdoura; Arnaud Heibig; Thomas Homolle; Nidhal Mannai; Adrien Petrov; Yves Renard. A multi-scale patch approximation for Poisson problems with a small inhomogeneous inclusion. The SMAI Journal of computational mathematics, Volume 11 (2025), pp. 139-163. doi : 10.5802/smai-jcm.121. https://smai-jcm.centre-mersenne.org/articles/10.5802/smai-jcm.121/

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