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.

翻译：针对包含大量多边形穿孔的域中的泊松方程，我们提出了一种基于域粗多边形划分的低维粗逼近空间。与其它多尺度数值方法类似，该粗空间由局部离散调和基函数张成。沿子域边界，基函数为分片多项式。本文的主要贡献是给出了关于粗空间上H1投影的误差估计，该估计仅依赖于粗划分边上解的正则性。对于特定的边加密过程，即使真实解具有较低的一般正则性，误差分析也证明了方法的超收敛性。结合区域分解(DD)方法，该粗空间可构建高效的两层迭代线性求解器，能够在少量迭代内达到细尺度有限元误差。该粗空间还适用于作为Krylov方法的预处理器，并提供了对子域数量的可扩展性。数值实验展示了粗逼近的更高精度，以及粗空间作为DD算法组件的效率和可扩展性。