Reduced-Contrast Approximations for High-Contrast Multiscale Flow Problems

Type
Article

Authors
Chung, Eric T.
Efendiev, Yalchin R.

KAUST Grant Number
KUS-C1-016-04

Date
2010-01

Abstract
In this paper, we study multiscale methods for high-contrast elliptic problems where the media properties change dramatically. The disparity in the media properties (also referred to as high contrast in the paper) introduces an additional scale that needs to be resolved in multiscale simulations. First, we present a construction that uses an integral equation to represent the highcontrast component of the solution. This representation involves solving an integral equation along the interface where the coefficients are discontinuous. The integral representation suggests some multiscale approaches that are discussed in the paper. One of these approaches entails the use of interface functions in addition to multiscale basis functions representing the heterogeneities without high contrast. In this paper, we propose an approximation for the solution of the integral equation using the interface problems in reduced-contrast media. Reduced-contrast media are obtained by lowering the variance of the coefficients. We also propose a similar approach for the solution of the elliptic equation without using an integral representation. This approach is simpler to use in the computations because it does not involve setting up integral equations. The main idea of this approach is to approximate the solution of the high-contrast problem by the solutions of the problems formulated in reduced-contrast media. In this approach, a rapidly converging sequence is proposed where only problems with lower contrast are solved. It was shown that this sequence possesses the convergence rate that is inversely proportional to the reduced contrast. This approximation allows choosing the reduced-contrast problem based on the coarse-mesh size as discussed in this paper. We present a simple application of this approach to homogenization of elliptic equations with high-contrast coefficients. The presented approaches are limited to the cases where there are sharp changes in the contrast (i.e., the high contrast can be represented by piecewise constant functions with disparate values). We present analysis for the proposed approaches and the estimates for the approximations used in multiscale algorithms. Numerical examples are presented. © 2010 Society for Industrial and Applied Mathematics.

Citation
Chung ET, Efendiev Y (2010) Reduced-Contrast Approximations for High-Contrast Multiscale Flow Problems. Multiscale Model Simul 8: 1128–1153. Available: http://dx.doi.org/10.1137/09075620X.

Acknowledgements
Department of Mathematics, The Chinese University of Hong Kong, Shatin, Hong Kong (tschung@math.cuhk.edu.hk). This author's research was partially supported by the RGC general research fund (project number 400609) and by the Direct Grant for Research of CUHK.Department of Mathematics and Institute of Scientific Computation, Texas A&M University, College Station, TX 77843-3368 (efendiev@math.tamu.edu). This author's research was partially supported by the DOE, the NSF (DMS 0934837, DMS 0902552, and DMS 0811180), and KAUST (award KUS-C1-016-04).

Publisher
Society for Industrial & Applied Mathematics (SIAM)

Journal
Multiscale Modeling & Simulation

DOI
10.1137/09075620X

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