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.

翻译：本文介绍了一种新颖的多尺度方法,用于使用任意粗略系数的椭圆部分差异方程式。 本着数字同质化精神,该方法构建了经问题调适的肛门空间,具有统一的代数近似率。 本地化基础功能与最近提议的超本地化正弦形分解功能具有相同的超扩展本地化特性,可以有效地实施。 该方法的基础稳定性通过统一方法的分割得以实现。 该方法被自然扩展至更高的顺序,导致近似率提高,本地化特性增强。 我们进行了严格的先验和后验误差分析,并在一系列数字实验中确认了我们的理论结论。 特别是,我们展示了该方法在挑战高争议通道系数方面的适用性。