Well-Posedness and homogenization applied to a one-dimensional elliptic problem
Boa postura e homogeneização aplicada a um problema elíptico unidimensional
DOI:
https://doi.org/10.5902/2179460X94345Keywords:
One-dimensional problem with rapidly oscillating coefficient, Well-Posedness, Asymptotic homogenization method, Effective coefficientAbstract
This paper studies the mathematical homogenization of a one-dimensional elliptic problem with a rapidly oscillating coefficient. The analysis focuses on the well-posedness of the problem, ensuring existence, uniqueness, and stability of the solution through the Lax–Milgram theorem and the coercivity and continuity of the associated bilinear form. Using the asymptotic homogenization method, an asymptotic expansion in two scales is derived, and the convergence between the exact and homogenized solutions is discussed, demonstrating O(ε) convergence order. An example with analytical solutions is provided to illustrate the theoretical results.
Downloads
References
Bakhvalov, N. & Panasenko, G. (1989). Homogenisation: Averaging Processes in Periodic Media: Mathematical Problems in the Mechanics of Composite Materials. Springer Netherlands, Dordrecht, 1 edition.
Bensoussan, A., Lions, J., & Papanicolau, G. (1978). Asymptotic Analysis for Periodic Structures. North Holland, New York, 1 edition.
Brezis, H. (2011). Functional Analysis, Sobolev Spaces and Partial Differential Equations. Springer, New York, 1 edition.
Cioranescu, D., Donato, P., & Roque, M. P. (2017). An Introduction to Second Order Partial Differential Equations: Classical and Variational Solutions. World Scientific Publishing Co. Pte. Ltd., New Jersey ; London ; Singapore, 1 edition.
Ciouranescu, D. & Donato, P. (1999). An Introduction to Homogenization. Oxford University Press, New York, 1 edition.
Costa, C. M. & dos Santos, R. W. (2010). Limitations of the homogenized cardiac monodomain model for the case of low gap junctional coupling. In 2010 Annual International Conference of the IEEE Engineering in Medicine and Biology, pages 228–231.
Dimitrienko, Y. (1997). Heat mass transport and thermal stresses in porous charring materials. Transport in Porous Media, 27:143–170.
Larsson, S. & Thom´ee, V. (2009). Partial Differential Equations with Numerical Methods, volume 45 of Texts in Applied Mathematics. Springer, Berlin, Heidelberg, 1 edition.
Rudin, W. (1991). Functional analysis. Mc-Graw-Hill, New York, 2 edition.
Tartar, L. (2009). The General Theory of Homogenization: A Personalized Introduction. Springer, New York, 1 edition.
Downloads
Published
Issue
Section
License
Copyright (c) 2026 Ciência e Natura
This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International License.
To access the DECLARATION AND TRANSFER OF COPYRIGHT AUTHOR’S DECLARATION AND COPYRIGHT LICENSE click here.
Ethical Guidelines for Journal Publication
The Ciência e Natura journal is committed to ensuring ethics in publication and quality of articles.
Conformance to standards of ethical behavior is therefore expected of all parties involved: Authors, Editors, Reviewers, and the Publisher.
In particular,
Authors: Authors should present an objective discussion of the significance of research work as well as sufficient detail and references to permit others to replicate the experiments. Fraudulent or knowingly inaccurate statements constitute unethical behavior and are unacceptable. Review Articles should also be objective, comprehensive, and accurate accounts of the state of the art. The Authors should ensure that their work is entirely original works, and if the work and/or words of others have been used, this has been appropriately acknowledged. Plagiarism in all its forms constitutes unethical publishing behavior and is unacceptable. Submitting the same manuscript to more than one journal concurrently constitutes unethical publishing behavior and is unacceptable. Authors should not submit articles describing essentially the same research to more than one journal. The corresponding Author should ensure that there is a full consensus of all Co-authors in approving the final version of the paper and its submission for publication.
Editors: Editors should evaluate manuscripts exclusively on the basis of their academic merit. An Editor must not use unpublished information in the editor's own research without the express written consent of the Author. Editors should take reasonable responsive measures when ethical complaints have been presented concerning a submitted manuscript or published paper.
Reviewers: Any manuscripts received for review must be treated as confidential documents. Privileged information or ideas obtained through peer review must be kept confidential and not used for personal advantage. Reviewers should be conducted objectively, and observations should be formulated clearly with supporting arguments, so that Authors can use them for improving the paper. Any selected Reviewer who feels unqualified to review the research reported in a manuscript or knows that its prompt review will be impossible should notify the Editor and excuse himself from the review process. Reviewers should not consider manuscripts in which they have conflicts of interest resulting from competitive, collaborative, or other relationships or connections with any of the authors, companies, or institutions connected to the papers.
