Strain accumulation and stress release along multiscale geological fault networks are fundamental mechanisms for earthquake and rupture processes in the lithosphere. Due to long periods of seismic quiescence, the scarcity of large earthquakes and incompleteness of paleoseismic, historical and instrumental record, there is a fundamental lack of insight into the multiscale, spatio-temporal nature of earthquake dynamics in fault networks. This thesis constitutes another step towards reliable earthquake prediction and quantitative hazard analysis. Its focus lies on developing a mathematical model for prototypical, layered fault networks on short time scales as well as their efficient numerical simulation.
This exposition begins by establishing a fault system consisting of layered bodies with viscoelastic Kelvin-Voigt rheology and non-intersecting faults featuring rate-and-state friction as proposed by Dieterich and Ruina. The individual bodies are assumed to experience small viscoelastic deformations, but possibly large relative tangential displacements. Thereafter, semi-discretization in time with the classical Newmark scheme of the variational formulation yields a sequence of continuous, nonsmooth, coupled, spatial minimization problems for the velocities and states in each time step, that are decoupled by means of a fixed point iteration. Subsequently, spatial discretization is based on linear and piecewise constant finite elements for the rate and state problems, respectively. A dual mortar discretization of the non-penetration constraints entails a hierarchical decomposition of the discrete solution space, that enables the localization of the non-penetration condition. Exploiting the resulting structure, an algebraic representation of the parametrized rate problem can be solved efficiently using a variant of the Truncated Nonsmooth Newton Multigrid (TNNMG) method. It is globally convergent due to nonlinear, block Gauß–Seidel type smoothing and employs nonsmooth Newton and multigrid ideas to enhance robustness and efficiency of the overall method. A key step in the TNNMG algorithm is the efficient computation of a correction obtained from a linearized, inexact Newton step.
The second part addresses the numerical homogenization of elliptic variational problems featuring fractal interface networks, that are structurally similar to the ones arising in the linearized correction step of the TNNMG method. Contrary to the previous setting, this model incorporates the full spatial complexity of geological fault networks in terms of truly multiscale fractal interface geometries. Here, the construction of projections from a fractal function space to finite element spaces with suitable approximation and stability properties constitutes the main contribution of this thesis. The existence of these projections enables the application of well-known approaches to numerical homogenization, such as localized orthogonal decomposition (LOD) for the construction of multiscale discretizations with optimal a priori error estimates or subspace correction methods, that lead to algebraic solvers with mesh- and scale-independent convergence rates.
Finally, numerical experiments with a single fault and the layered multiscale fault system illustrate
the properties of the mathematical model as well as the efficiency, reliability and scale-independence of the suggested algebraic solver
