论文标题
超定位的广义有限元法
A Super-Localized Generalized Finite Element Method
论文作者
论文摘要
本文提出了一种新型的多尺度方法，用于具有任意粗糙系数的椭圆形部分微分方程。本着数值同质化的精神，该方法构建了针对问题的ANSATZ空间，其代数近似速率均匀。与最近提出的超定位正交分解具有相同超指定定位属性的本地化基础函数可实现有效的实现。该方法的基础稳定性是使用Unity方法的分区实施的。提出了自然扩展到高阶，从而导致较高的近似率和增强的定位性能。我们进行了严格的先验和后验错误分析，并在一系列数值实验中确认了我们的理论发现。特别是，我们证明了该方法用于挑战高对比度通道系数的适用性。
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.
