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For the Poisson equation posed in a domain containing a large number ofpolygonal perforations, we propose a low-dimensional coarse approximation spacebased on a coarse polygonal partitioning of the domain. Similarly to othermultiscale numerical methods, this coarse space is spanned by locally discreteharmonic basis functions. Along the subdomain boundaries, the basis functionsare piecewise polynomial. The main contribution of this article is an errorestimate regarding the H1-projection over the coarse space which depends onlyon the regularity of the solution over the edges of the coarse partitioning.For a specific edge refinement procedure, the error analysis establishessuperconvergence of the method even if the true solution has a low generalregularity. Combined with domain decomposition (DD) methods, the coarse spaceleads to an efficient two-level iterative linear solver which reaches thefine-scale finite element error in few iterations. It also bodes well as apreconditioner for Krylov methods and provides scalability with respect to thenumber of subdomains. Numerical experiments showcase the increased precision ofthe coarse approximation as well as the efficiency and scalability of thecoarse space as a component of a DD algorithm.

Robust Methods for Multiscale Coarse Approximations of Diffusion Models in Perforated Domains