  We consider numerical solution of elliptic problems with heterogeneous
diffusion coefficients containing thin highly conductive structures. Such
problems arise e.g. in fractured porous media, reinforced materials, and
electric circuits. The main computational challenge is the high resolution
needed to resolve the data variation. We propose a multiscale method that
models the thin structures as interfaces and incorporate heterogeneities in
corrected shape functions. The construction results in an accurate upscaled
representation of the system that can be used to solve for several forcing
functions or to simulate evolution problems in an efficient way. By introducing
a novel interpolation operator, defining the fine scale of the problem, we
prove exponential decay of the shape functions which allows for a sparse
approximation of the upscaled representation. An a priori error bound is also
derived for the proposed method together with numerical examples that verify
the theoretical findings. Finally we present a numerical example to show how
the technique can be applied to evolution problems.
