Multiscale Modeling &Simulation, Volume 18, Issue 1, Page 294-314, January 2020.
Inspired by continuum mechanical contact problems with geological fault networks, we consider elliptic second order differential equations with jump conditions on a sequence of multiscale networks of interfaces with a finite number of nonseparating scales. Our aim is to derive and analyze a description of the asymptotic limit of infinitely many scales in order to quantify the effect of resolving the network only up to some finite number of interfaces, and to consider all further effects as homogeneous. As classical homogenization techniques are not suited for this kind of geometrical setting, we suggest a new concept, called fractal homogenization, to derive and analyze an asymptotic limit problem from a corresponding sequence of finite-scale interface problems. We provide an intuitive characterization of the corresponding fractal solution space in terms of generalized jumps and gradients together with continuous embeddings into $L^2$ and $H^s$, $s<1/2$. We show existence and uniqueness of the solution of the asymptotic limit problem and exponential convergence of the approximating finite-scale solutions. Computational experiments involving a related numerical homogenization technique illustrate our theoretical findings.

中文翻译：

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多尺度界面问题的分形均质化

2020年1月，《多尺度建模与仿真》，第18卷，第1期，第294-314页。
受地质断层网络的连续机械接触问题的启发，我们考虑了具有有限数量的非分离尺度的一系列多尺度接口网络上具有跳跃条件的椭圆二阶微分方程。我们的目的是推导和分析无限多个尺度的渐近极限的描述，以便量化仅在有限数量的接口处解析网络的效果，并将所有进一步的效果视为同质的。由于经典的均质化技术不适用于这种几何设置，因此我们提出了一种称为分形均质化的新概念，该概念可以从有限级接口问题的相应序列中导出和分析渐近极限问题。我们根据广义跃迁和梯度以及连续嵌入到$ L ^ 2 $和$ H ^ s $，$ s <1/2 $中，提供了对应分形解空间的直观表征。我们证明了渐近极限问题的解的存在性和唯一性以及逼近的有限尺度解的指数收敛性。涉及相关的数值均化技术的计算实验说明了我们的理论发现。