Adaptive Matrix‐Free Simulations of Fluid‐Filled Phase‐Field Fractures With Fixed‐Stress Coupling and Fracture‐Width Computation

ABSTRACT Fluid‐filled phase‐field fracture simulations require robust, scalable solvers that can handle strongly nonlinear, non‐smooth mechanics and tightly coupled flow on locally refined meshes. In this work, we develop an adaptive finite element framework for quasi‐static, fluid‐filled phase‐field fractures that combines semi‐smooth Newton methods, fixed‐stress iterative coupling, and matrix‐free geometric multigrid preconditioning. The geomechanics subproblem, with displacement and phase field as primary variables, is solved by a semi‐smooth combined Newton method based on a primal‐dual active set formulation. The resulting linear systems are treated with the generalized minimal residual method and matrix‐free geometric multigrid preconditioners on locally refined meshes, with a local smoothing approach and careful treatment in the active set updates. The pressure equation is coupled to the geomechanics system via a fixed‐stress iterative scheme which iterates until both subproblem residuals are sufficiently minimized, and each flow subproblem is likewise preconditioned by matrix‐free geometric multigrid on the same adaptive meshes. In addition, we introduce an enhanced fracture‐width computation, inspired by and extending existing phase‐field aperture formulas, to obtain more accurate and mesh‐robust width fields consistent with the regularized fracture profile. The overall computational framework is demonstrated on a set of two‐ and three‐dimensional benchmark problems, highlighting its robustness, efficiency, and accuracy for simulating fluid‐filled phase‐field fractures with local mesh refinement.