A mixed multiscale finite element method for elliptic problems with oscillating coefficients

Authors:
Zhiming Chen and Thomas Y. Hou

Journal:
Math. Comp. **72** (2003), 541-576

MSC (2000):
Primary 65F10, 65F30

Published electronically:
June 28, 2002

MathSciNet review:
1954956

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Abstract: The recently introduced multiscale finite element method for solving elliptic equations with oscillating coefficients is designed to capture the large-scale structure of the solutions without resolving all the fine-scale structures. Motivated by the numerical simulation of flow transport in highly heterogeneous porous media, we propose a mixed multiscale finite element method with an over-sampling technique for solving second order elliptic equations with rapidly oscillating coefficients. The multiscale finite element bases are constructed by locally solving Neumann boundary value problems. We provide a detailed convergence analysis of the method under the assumption that the oscillating coefficients are locally periodic. While such a simplifying assumption is *not* required by our method, it allows us to use homogenization theory to obtain the asymptotic structure of the solutions. Numerical experiments are carried out for flow transport in a porous medium with a random log-normal relative permeability to demonstrate the efficiency and accuracy of the proposed method.

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Additional Information

**Zhiming Chen**

Affiliation:
LSEC, Institute of Computational Mathematics, Chinese Academy of Sciences, Beijing 100080, Peoples Republic of China

Email:
zmchen@lsec.cc.ac.cn

**Thomas Y. Hou**

Affiliation:
Applied Mathematics, California Institute of Technology, Pasadena, California 91125.

Email:
hou@acm.caltech.edu

DOI:
http://dx.doi.org/10.1090/S0025-5718-02-01441-2

Received by editor(s):
March 21, 2000

Received by editor(s) in revised form:
July 10, 2000, and May 29, 2001

Published electronically:
June 28, 2002

Additional Notes:
The first author was supported in part by China NSF under the grants 19771080 and 10025102 and by China MOS under the grant G1999032804.

The second author was supported in part by NSF under the grant DMS-0073916 and by ARO under the grant DAAD19-99-1-0141.

Article copyright:
© Copyright 2002
American Mathematical Society