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A Multiscale Finite Element Method for Elliptic Problems in Composite Materials and Porous Media

Hou, Thomas Y. and Wu, Xiao-Hui (1997) A Multiscale Finite Element Method for Elliptic Problems in Composite Materials and Porous Media. Journal of Computational Physics, 134 (1). pp. 169-189. ISSN 0021-9991. https://resolver.caltech.edu/CaltechAUTHORS:20170707-134035089

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Abstract

In this paper, we study a multiscale finite element method for solving a class of elliptic problems arising from composite materials and flows in porous media, which contain many spatial scales. The method is designed to efficiently capture the large scale behavior of the solution without resolving all the small scale features. This is accomplished by constructing the multiscale finite element base functions that are adaptive to the local property of the differential operator. Our method is applicable to general multiple-scale problems without restrictive assumptions. The construction of the base functions is fully decoupled from element to element; thus, the method is perfectly parallel and is naturally adapted to massively parallel computers. For the same reason, the method has the ability to handle extremely large degrees of freedom due to highly heterogeneous media, which are intractable by conventional finite element (difference) methods. In contrast to some empirical numerical upscaling methods, the multiscale method is systematic and self- consistent, which makes it easier to analyze. We give a brief analysis of the method, with emphasis on the “resonant sampling” effect. Then, we propose an oversampling technique to remove the resonance effect. We demonstrate the accuracy and efficiency of our method through extensive numerical experiments, which include problems with random coefficients and problems with continuous scales. Parallel implementation and performance of the method are also addressed.


Item Type:Article
Related URLs:
URLURL TypeDescription
https://doi.org/10.1006/jcph.1997.5682DOIArticle
http://www.sciencedirect.com/science/article/pii/S0021999197956825PublisherArticle
Additional Information:© 1997 Academic Press. Received 5 August 1996. We thank Professor Bjorn Engquist and Mr. Yalchin Efendiev for many interesting and helpful discussions. This work is supported in part by ONR under the Grant N00014-94-0310 and DOE under the Grant DE-FG03-89ER25073.
Funders:
Funding AgencyGrant Number
Office of Naval Research (ONR)N00014-94-0310
Department of Energy (DOE)DE-FG03-89ER25073
Issue or Number:1
Record Number:CaltechAUTHORS:20170707-134035089
Persistent URL:https://resolver.caltech.edu/CaltechAUTHORS:20170707-134035089
Official Citation:Thomas Y. Hou, Xiao-Hui Wu, A Multiscale Finite Element Method for Elliptic Problems in Composite Materials and Porous Media, Journal of Computational Physics, Volume 134, Issue 1, 1997, Pages 169-189, ISSN 0021-9991, http://dx.doi.org/10.1006/jcph.1997.5682. (http://www.sciencedirect.com/science/article/pii/S0021999197956825)
Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:78863
Collection:CaltechAUTHORS
Deposited By: Tony Diaz
Deposited On:07 Jul 2017 21:02
Last Modified:03 Oct 2019 18:13

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