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.

翻译：我们针对ASM和RAS预处理器提出了一种新的粗空间，用于求解穿孔域上的椭圆型偏微分方程，其中大量多边形穿孔代表城市数据中的墙体、建筑物等结构。该研究以非线性扩散波方程模拟城市洪水的最终目标为牵引，重点关注穿孔域上线性问题的求解。所提出的粗空间采用多边形子区域划分，由Trefftz型基函数张成，这些基函数在子区域边界上分段线性且在内部调和。该粗空间基于考虑穿孔与子区域边界交点的节点自由度。作为参照，我们在相同子区域划分条件下将该粗空间与经过充分研究的Nicolaides粗空间进行对比。已知Nicolaides空间在子区域不连通时无法阻止收敛停滞，我们通过将每个子区域按不连通分量进行分离来解决该问题。基于真实城市地形的多组数据集测试了可扩展性与鲁棒性。数值结果表明，与Nicolaides方法相比，新粗空间具有极强的鲁棒性，且无论数据复杂度如何，都能有效加速Krylov迭代次数。