Using weak forms to derive asymptotic expansions of elliptic equations with high-contrast cofficients

In this work we review some recent results on approximation of solutions of elliptic problems with high-contrast coefficients. In particular, we detail the derivation of asymptotic expansions for the solution in terms of the high-contrast of the cofficients. We consider the case of high-contrast elliptic equations and we present the case of only one high-contrast inclusion. The case of more inclusions follows similarly. In order to simplify the presentation we consider first the one dimensional case so no further complicated requirements regarding the computation of two or higher dimensional integrals are needed. We review the case of two-dimensional problems and give some numerical examples.