  In this manuscript we review some recent results about 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 coefficients and we consider some interesting
applications. We use the Finite Element Method, which is applied in the
numerical computation of terms of the asymptotic expansion. We also present an
application to Multiscale Finite Elements, in particular, we numerically design
approximation for the term $u_0$ with local harmonic characteristic functions.
We also show the case of the linear elasticity problem, where we study the
asymptotic problem with one inelastic inclusion and we provide the convergence
for this expansion problem. Finally, we state some conclusions and final
comments.
