Fractal homogenization of multiscale interface problems

TL;DR

This paper introduces fractal homogenization for multiscale interface problems inspired by geological faults, providing a new way to analyze the asymptotic behavior of solutions as the number of scales tends to infinity.

Contribution

It proposes a novel fractal homogenization method for multiscale interface problems where classical techniques fail, and characterizes the limit solutions in terms of generalized jumps and gradients.

Findings

Existence and uniqueness of the fractal limit solution.

Exponential convergence of finite-scale solutions.

Numerical experiments support theoretical results.

Abstract

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 non-separating 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 L2 and Hs, 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.