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 -projection over the coarse space; this error estimate 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. Additionally, this contribution numerically explores the combination of the coarse space with domain decomposition (DD) methods. This combination 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.

中文翻译：

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穿孔域中扩散模型多尺度粗略近似的鲁棒方法

对于包含大量多边形穿孔的域中提出的泊松方程，我们提出了基于域的粗多边形划分的低维粗近似空间。与其他多尺度数值方法类似，该粗空间由局部离散调和基函数跨越。沿着子域边界，基函数是分段多项式。本文的主要贡献是关于粗空间上的 投影的误差估计；该误差估计仅取决于粗划分边缘上的解的规律性。对于特定的边缘细化过程，即使真实解具有较低的一般规律性，误差分析也会建立该方法的超收敛性。此外，该贡献在数值上探索了粗糙空间与域分解（DD）方法的结合。这种组合产生了一个高效的两级迭代线性求解器，只需几次迭代即可达到精细尺度的有限元误差。它也是 Krylov 方法的预处理器，并提供了子域数量的可扩展性。