# Mathematics of Computation

Published by the American Mathematical Society since 1960 (published as Mathematical Tables and other Aids to Computation 1943-1959), Mathematics of Computation is devoted to research articles of the highest quality in computational mathematics.

ISSN 1088-6842 (online) ISSN 0025-5718 (print)

The 2020 MCQ for Mathematics of Computation is 1.78.

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## Convergence of a multiscale finite element method for elliptic problems with rapidly oscillating coefficientsHTML articles powered by AMS MathViewer

by Thomas Y. Hou, Xiao-Hui Wu and Zhiqiang Cai
Math. Comp. 68 (1999), 913-943 Request permission

## 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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• 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
• MR Author ID: 235961
• Email: zcai@math.purdue.edu
• 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.
• © Copyright 1999 American Mathematical Society
• Journal: Math. Comp. 68 (1999), 913-943
• MSC (1991): Primary 65F10, 65F30
• DOI: https://doi.org/10.1090/S0025-5718-99-01077-7
• MathSciNet review: 1642758