Asymptotic expansions for high-contrast elliptic equations

Abstract

In this paper, we present a high-order expansion for elliptic equations in high-contrast media. The background conductivity is taken to be one and we assume the medium contains high (or low) conductivity inclusions. We derive an asymptotic expansion with respect to the contrast and provide a procedure to compute the terms in the expansion. The computation of the expansion does not depend on the contrast which is important for simulations. The latter allows avoiding increased mesh resolution around high conductivity features. This work is partly motivated by our earlier work in \cite{ge09_1} where we design efficient numerical procedures for solving high-contrast problems. These multiscale approaches require local solutions and our proposed high-order expansion can be used to approximate these local solutions inexpensively. In the case of a large-number of inclusions, the proposed analysis can help to design localization techniques for computing the terms in the expansion. In the paper, we present a rigorous analysis of the proposed high-order expansion and estimate the remainder of it. We consider both high and low conductivity inclusions.

References in corpus (2)

• Localized bases for finite dimensional homogenization approximations with non-separated scales and high-contrast
• Optimal Local Approximation Spaces for Generalized Finite Element Methods with Application to Multiscale Problems

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• A Robust Preconditioner for High-Contrast Problems
• Asymptotic analysis of the stress concentration between two adjacent stiff inclusions in all dimensions
• Asymptotic Expansions for High-Contrast Linear Elasticity
• Asymptotic Approximation of the Dirichlet to Neumann Map of High Contrast Conductive Media