This paper presents a novel multi-scale method for elliptic partial differential equations with arbitrarily rough coefficients. In the spirit of numerical homogenization, the method constructs problem-adapted ansatz spaces with uniform algebraic approximation rates. Localized basis functions with the same super-exponential localization properties as the recently proposed Super-Localized Orthogonal Decomposition enable an efficient implementation. The method’s basis stability is enforced using a partition of unity approach. A natural extension to higher order is presented, resulting in higher approximation rates and enhanced localization properties. We perform a rigorous a priori and a posteriori error analysis and confirm our theoretical findings in a series of numerical experiments. In particular, we demonstrate the method’s applicability for challenging high-contrast channeled coefficients.

本文提出了一种新颖的多尺度方法，用于求解具有任意粗糙系数的椭圆偏微分方程。本着数值均质化的精神，该方法构建了具有统一代数近似率的适应问题的 ansatz 空间。具有与最近提出的超局部正交分解相同的超指数局部化特性的局部基函数可以实现高效的实现。该方法的基础稳定性是使用单位划分方法来增强的。提出了向更高阶的自然扩展，从而获得更高的逼近率和增强的定位特性。我们进行了严格的先验和后验误差分析，并在一系列数值实验中证实了我们的理论发现。特别是，我们证明了该方法对于挑战高对比度通道系数的适用性。