  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.
