Multiscale methods for parabolic equations with continuum spatial scales

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November 2007, 8(4): 833-859. doi: 10.3934/dcdsb.2007.8.833

Multiscale methods for parabolic equations with continuum spatial scales

Lijian Jiang ^1, , Yalchin Efendiev ^2, and Victor Ginting ^3,

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

Received March 2007 Revised June 2007 Published August 2007

Abstract
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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.

Keywords: parabolic equations, multiscale methods, global information, continuum scale..

Mathematics Subject Classification: Primary: 34E13, 34K28; Secondary: 33F0.

Citation: Lijian Jiang, Yalchin Efendiev, Victor Ginting. Multiscale methods for parabolic equations with continuum spatial scales. Discrete & Continuous Dynamical Systems - B, 2007, 8 (4) : 833-859. doi: 10.3934/dcdsb.2007.8.833

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