Multiscale methods for parabolic equations with continuum spatial scales

Lijian Jiang; Yalchin Efendiev; Victor Ginting

doi:10.3934/dcdsb.2007.8.833

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2007, Volume 8, Issue 4: 833-859. Doi: 10.3934/dcdsb.2007.8.833

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Multiscale methods for parabolic equations with continuum spatial scales

1.
Department of Mathematics, Texas A&M University, College Station, TX 77843-3368
2.
Department of Mathematics & ISC, Texas A&M University, 3404 TAMU, College Station, TX 77843-3404
3.
Department of Mathematics, Colorado State University, Fort Collins, CO 80523-1874

Received: February 28, 2007

Revised: May 31, 2007

Published: October 2007

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Abstract

In this paper, we consider multiscale approaches for solving parabolic equations with heterogeneous coefficients. Our interest stems from porous media applications and we assume that there is no scale separation with respect to spatial variables. To compute the solution of these multiscale problems on a coarse grid, we define global fields such that the solution smoothly depends on these fields. We present various finite element discretization techniques and provide analyses of these methods. A few representative numerical examples are presented using heterogeneous fields with strong non-local features. These numerical results demonstrate that the solution can be captured more accurately on the coarse grid when some type of limited global information is used.

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Mathematics Subject Classification: Primary: 34E13, 34K28; Secondary: 33F05.

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