Convergence of a multiscale finite element method for elliptic problems with rapidly oscillating coefficients

Authors:
Thomas Y. Hou, Xiao-Hui Wu and Zhiqiang Cai

Journal:
Math. Comp. **68** (1999), 913-943

MSC (1991):
Primary 65F10, 65F30

DOI:
https://doi.org/10.1090/S0025-5718-99-01077-7

Published electronically:
March 3, 1999

MathSciNet review:
1642758

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Abstract: We propose a multiscale finite element method for solving second order elliptic equations with rapidly oscillating coefficients. The main purpose is to design a numerical method which is capable of correctly capturing the large scale components of the solution on a coarse grid without accurately resolving all the small scale features in the solution. This is accomplished by incorporating the local microstructures of the differential operator into the finite element base functions. As a consequence, the base functions are *adapted* to the local properties of the differential operator. In this paper, we provide a detailed convergence analysis of our method under the assumption that the oscillating coefficient is of two scales and is periodic in the fast scale. While such a simplifying assumption is *not* required by our method, it allows us to use homogenization theory to obtain a useful asymptotic solution structure. The issue of boundary conditions for the base functions will be discussed. Our numerical experiments demonstrate convincingly that our multiscale method indeed converges to the correct solution, independently of the small scale in the homogenization limit. Application of our method to problems with continuous scales is also considered.

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Additional Information

**Thomas Y. Hou**

Affiliation:
Applied Mathematics, 217-50 California Institute of Technology Pasadena, CA 91125

Email:
hou@ama.caltech.edu

**Xiao-Hui Wu**

Affiliation:
Applied Mathematics, 217-50, California Institute of Technology, Pasadena, CA 91125

Address at time of publication:
Exxon Production Research Company, P. O. Box 2189, Houston, TX 77252

Email:
xwu@ama.caltech.edu

**Zhiqiang Cai**

Affiliation:
Department of Mathematics, Purdue University, West Lafayette, IN 47907-1395

Email:
zcai@math.purdue.edu

DOI:
https://doi.org/10.1090/S0025-5718-99-01077-7

Keywords:
Multiscale base functions,
finite element,
homogenization,
oscillating coefficients

Received by editor(s):
August 5, 1996

Received by editor(s) in revised form:
November 10, 1997

Published electronically:
March 3, 1999

Additional Notes:
This work is supported in part by ONR under the grant N00014-94-0310, by DOE under the grant DE-FG03-89ER25073, and by NSF under the grant DMS-9704976.

Article copyright:
© Copyright 1999
American Mathematical Society