  We solve elliptic systems of equations posed on highly heterogeneous
materials. Examples of this class of problems are composite structures and
geological processes. We focus on a model problem which is a second-order
elliptic equation with discontinuous coefficients. These coefficients represent
the conductivity of a composite material. We assume a background with low
conductivity that contains inclusions with different thermal properties. Under
this scenario we design a multiscale finite element method to efficiently
approximate solutions. The method is based on an asymptotic expansions of the
solution in terms of the ratio between the conductivities. The resulting method
constructs (locally) finite element basis functions (one for each inclusion).
These bases that generate the multiscale finite element space where the
approximation of the solution is computed. Numerical experiments show the good
performance of the proposed methodology.
