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Analysis of the heterogeneous multiscale method for elliptic homogenization problems


Authors: Weinan E, Pingbing Ming and Pingwen Zhang
Journal: J. Amer. Math. Soc. 18 (2005), 121-156
MSC (2000): Primary 65N30, 74Q05; Secondary 74Q15, 65C30
DOI: https://doi.org/10.1090/S0894-0347-04-00469-2
Published electronically: September 16, 2004
MathSciNet review: 2114818
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Abstract: A comprehensive analysis is presented for the heterogeneous multiscale method (HMM for short) applied to various elliptic homogenization problems. These problems can be either linear or nonlinear, with deterministic or random coefficients. In most cases considered, optimal estimates are proved for the error between the HMM solutions and the homogenized solutions. Strategies for retrieving the microstructural information from the HMM solutions are discussed and analyzed.


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Additional Information

Weinan E
Affiliation: Department of Mathematics and PACM, Princeton University, Princeton, New Jersey 08544 and School of Mathematical Sciences, Peking University, Beijing 100871, People’s Republic of China
Email: weinan@Princeton.EDU

Pingbing Ming
Affiliation: No. 55, Zhong-Guan-Cun East Road, Institute of Computational Mathematics, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing 100080, People’s Republic of China
Email: mpb@lsec.cc.ac.cn

Pingwen Zhang
Affiliation: School of Mathematical Sciences, Peking University, Beijing 100871, People’s Republic of China
Email: pzhang@pku.edu.cn

DOI: https://doi.org/10.1090/S0894-0347-04-00469-2
Keywords: Heterogeneous multiscale method, homogenization problems.
Received by editor(s): January 2, 2003
Published electronically: September 16, 2004
Additional Notes: The work of the first author was partially supported by ONR grant N00014-01-1-0674 and the National Natural Science Foundation of China through a Class B Award for Distinguished Young Scholars 10128102.
The work of the second author was partially supported by the Special Funds for the Major State Basic Research Projects G1999032804 and was also supported by the National Natural Science Foundation of China 10201033.
The work of the third author was partially supported by the Special Funds for the Major State Research Projects G1999032804 and the National Natural Science Foundation of China for Distinguished Young Scholars 10225103.
We thank Bjorn Engquist for inspiring discussions on the topic studied here.
Article copyright: © Copyright 2004 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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