一种超局部化广义有限元方法
A Super-Localized Generalized Finite Element Method
AI总结：
本文提出一种超局部化广义有限元方法，通过构造问题自适应基函数和单位分解保证稳定性，实现对任意粗糙系数椭圆方程的高效多尺度求解，并验证了其在高对比度通道化系数问题上的有效性。
摘要：
本文针对具有任意粗糙系数的椭圆型偏微分方程，提出了一种新颖的多尺度方法。秉承数值均匀化的思想，该方法构造了具有一致代数逼近率的问题自适应近似空间。局部化基函数具有与最近提出的超局部化正交分解相同的超指数局部化性质，从而能够实现高效计算。该方法的基稳定性通过单位分解方法加以保证。文中给出了向高阶的自然推广，从而获得更高的逼近率和更强的局部化性质。我们进行了严格的先验和后验误差分析，并通过一系列数值实验验证了理论结果。特别地，我们展示了该方法对于具有挑战性的高对比度通道化系数的适用性。
英文摘要：
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.
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