  For the Poisson equation posed in a domain containing a large number of
polygonal perforations, we propose a low-dimensional coarse approximation space
based on a coarse polygonal partitioning of the domain. Similarly to other
multiscale numerical methods, this coarse space is spanned by locally discrete
harmonic basis functions. Along the subdomain boundaries, the basis functions
are piecewise polynomial. The main contribution of this article is an error
estimate regarding the H1-projection over the coarse space which depends only
on the regularity of the solution over the edges of the coarse partitioning.
For a specific edge refinement procedure, the error analysis establishes
superconvergence of the method even if the true solution has a low general
regularity. Combined with domain decomposition (DD) methods, the coarse space
leads to an efficient two-level iterative linear solver which reaches the
fine-scale finite element error in few iterations. It also bodes well as a
preconditioner for Krylov methods and provides scalability with respect to the
number of subdomains. Numerical experiments showcase the increased precision of
the coarse approximation as well as the efficiency and scalability of the
coarse space as a component of a DD algorithm.
