Communications in Mathematical Sciences

Numerical averaging of non-divergence structure elliptic operators

Brittany D. Froese and Adam M. Oberman

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Abstract

Many important equations in science and engineering contain rapidly varying operators that cannot be practically sufficiently resolved for accurate solutions. In some cases it is possible to obtain approximate solutions by replacing the rapidly varying operator with an appropri- ately averaged operator. In this paper we use formal asymptotic techniques to recover a formula for the averaged form of a second order, non-divergence structure, linear elliptic operator. For several special cases the averaged operator is obtained analytically. For genuinely multi-dimensional cases, the averaged operator is also obtained numerically using finite difference method, which also has a probabilistic interpretation.

Article information

Source
Commun. Math. Sci., Volume 7, Number 4 (2009), 785-804.

Dates
First available in Project Euclid: 25 January 2010

Permanent link to this document
https://projecteuclid.org/euclid.cms/1264434133

Mathematical Reviews number (MathSciNet)
MR2604620

Zentralblatt MATH identifier
1183.35094

Subjects
Primary: 35J15: Second-order elliptic equations 35B27: Homogenization; equations in media with periodic structure [See also 74Qxx, 76M50] 65L12: Finite difference methods

Keywords
Homogenization partial differential equations elliptic partial differential equations diffusions finite difference methods

Citation

Froese, Brittany D.; Oberman, Adam M. Numerical averaging of non-divergence structure elliptic operators. Commun. Math. Sci. 7 (2009), no. 4, 785--804. https://projecteuclid.org/euclid.cms/1264434133


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