We propose a multiscale method for elliptic problems on complex domains, e.g.
domains with cracks or complicated boundary. For local singularities this paper
also offers a discrete alternative to enrichment techniques such as XFEM. We
construct corrected coarse test and trail spaces which takes the fine scale
features of the computational domain into account. The corrections only need to
be computed in regions surrounding fine scale geometric features. We achieve
linear convergence rate in energy norm for the multiscale solution. Moreover,
the conditioning of the resulting matrices is not affected by the way the
domain boundary cuts the coarse elements in the background mesh. The analytical
findings are verified in a series of numerical experiments
