  We consider a new coarse space for the ASM and RAS preconditioners to solve
elliptic partial differential equations on perforated domains, where the
numerous polygonal perforations represent structures such as walls and
buildings in urban data. With the eventual goal of modelling urban floods by
means of the nonlinear Diffusive Wave equation, this contribution focuses on
the solution of linear problems on perforated domains. Our coarse space uses a
polygonal subdomain partitioning and is spanned by Trefftz-like basis functions
that are piecewise linear on the boundary of a subdomain and harmonic inside
it. It is based on nodal degrees of freedom that account for the intersection
between the perforations and the subdomain boundaries. As a reference, we
compare this coarse space to the well-studied Nicolaides coarse space with the
same subdomain partitioning. It is known that the Nicolaides space is unable to
prevent stagnation in convergence when the subdomains are not connected; we
work around this issue by separating each subdomain by disconnected component.
Scalability and robustness are tested for multiple data sets based on realistic
urban topography. Numerical results show that the new coarse space is very
robust and accelerates the number of Krylov iterations when compared to
Nicolaides, independent of the complexity of the data.
