A Variational Multigrid Framework for Multiscale Fault Systems with Rate- and State-Dependent Friction: Convergence Analysis and Adaptive Solution Strategies

Gul Agha Jan  Assar; Mohammad Khalid  Storai

doi:10.65065/jvcdt034

Authors

Gul Agha Jan Assar Department of Applied Mathematics, Faculty of Mathematics, Kabul University Author
Mohammad Khalid Storai Department of Applied Mathematics, Faculty of Mathematics, Kabul University Author

DOI:

https://doi.org/10.65065/jvcdt034

Keywords:

Rate- and State-Dependent Friction; Multibody Contact; Mortar Methods; Non-Smooth Multigrid; Variational Inequalities; Adaptive Time-Stepping

Abstract

We present a comprehensive variational framework for the numerical simulation of multiscale geological fault systems governed by rate- and state-dependent friction laws. Extending the variational approach for subduction zones to layered fault networks with multiple non-intersecting interfaces, we derive a mathematical model that encompasses both Dieterich- and Ruina-type friction as special cases while accommodating large tangential displacements. Semi-discretization in time via an energy-conserving Newmark scheme yields a coupled system of nonsmooth convex minimization problems at each time step. Spatial discretization employs a dual-mortar finite element method with a piecewise-constant state approximation, enabling efficient decoupling via fixed-point iteration. For the resulting rate problems, we establish convergence of truncated nonsmooth Newton multigrid (TNNMG) methods and prove stability of the fully discrete scheme under standard regularity assumptions. Numerical experiments on spring-slider and layered five-body configurations demonstrate the robustness of the solver with respect to the number of faults, with adaptive time-stepping capturing slip events across multiple orders of magnitude in velocity variation

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