Asymptotic expansions for high-contrast elliptic equations

TL;DR

This paper develops a high-order asymptotic expansion for elliptic equations in media with high or low conductivity inclusions, enabling efficient simulations without increased mesh resolution around inclusions.

Contribution

It introduces a contrast-independent expansion method for high-contrast elliptic equations, facilitating local solution approximation and multiscale computations.

Findings

Derived a high-order asymptotic expansion with respect to contrast.

Provided a procedure to compute expansion terms efficiently.

Analyzed the remainder and applicability to multiple inclusions.

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.