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 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.

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We consider the numerical homogenization of a class of fractal elliptic interface problems inspired by related mechanical contact problems from the geosciences. A particular feature is that the solution space depends on the actual fractal geometry. Our main results concern the construction of projection operators with suitable stability and approximation properties. The existence of such projections then allows for the application of existing concepts from localized orthogonal decomposition (LOD) and successive subspace correction to construct first multiscale discretizations and iterative algebraic solvers with scale-independent convergence behavior for this class of problems.

&lt;p&gt;The release of elastic energy along an active fault is accommodated by a wide range of slip modes. It ranges from long-term slow slip events (SSEs) and creep to short-term tremors and earthquakes. They vary not only in their characteristic duration but also in their magnitude, spatial extend and slip velocities. As all slip modes are related to earthquakes, the understanding of the relationships between the different slip modes and the underlying mechanisms is crucial to assess earthquake hazards in various regions. The exact relation is unclear, as in some regions many slip modes occur simultaneously (e.g. Tohoku-Oki) and in others certain slip modes are completely absent (e.g. Cascadia).&lt;/p&gt;&lt;p&gt;One of the driving factors in the generation of this large variety of slip modes is the interplay of fault heterogeneity and geometrical complexity of the fault system. Using a scaled physical model we test various settings in terms of fault heterogeneity and geometrical complexity. The experimental results are then validated and benchmarked using multi-scale numerical simulations. We describe the system using the rate-and-state frictional framework and introduce the on-fault heterogeneity with variable frictional properties. All properties are the same for analogue and simulation as far as they can be determined or realized experimentally (a-b, v&lt;sub&gt;load&lt;/sub&gt;, S&lt;sub&gt;hmax&lt;/sub&gt;, S&lt;sub&gt;hmin&lt;/sub&gt;, etc...). As analogue material we use segmented, decimetre sized neoprene foam blocks in multiple configurations (e.g. biaxial shear at forces &lt;1 kN) to simulate the elastic upper crust. The contact surfaces are spray-painted with acrylic paint to generate velocity weakening characteristics in between the blocks. The major advantage of using neoprene over other materials, such as gelatine or polyurethane foams, is that it has closed pores and thus exhibits a more favourable Poisson&amp;#8217;s ratio in comparison with rocks and shows better elastic strain propagation in the block. Furthermore, all used materials are inert and do not change their properties over time.&lt;/p&gt;&lt;p&gt;We are able to reliably generate frequent stick-slip events of variable size and recurrence intervals. The slip characteristics, such as slip distribution, are in good agreement with analytical solutions of fault slip in elastic media. In this contribution we will highlight the material properties, experimental results and used methodologies to monitor and process the experimental data. Additionally, we are going to give an outlook on the interaction behaviour of multiple faults in dependence of their geometric configuration and the generation of power-law type magnitude scaling relations.&lt;/p&gt;