Automatically adaptive finite element method via residual minimization onto dual norms for brittle and quasi-brittle fracture using phase fields

The accurate and efficient simulation of fracture is a fundamental challenge across many areas of science and engineering, from the design of reliable mechanical components to the study of geological formations and the prediction of environmental hazards. Traditional finite element approaches often require excessive refinement to resolve the small-scale structures associated with crack nucleation, propagation, and branching, resulting in prohibitive computational costs. In this work, we propose an adaptively stabilized finite element method that addresses these difficulties through a combination of discontinuous Galerkin (dG) spaces and a residual minimization onto the dG-induced dual norms. By recasting the problem in terms of a discretely stable saddle-point system, we exploit Galerkin orthogonality to obtain rigorous and physically relevant a posteriori error estimates. These error indicators serve as the backbone of a fully automated mesh-refinement strategy, capable of sharply localizing computational effort to regions of interest while avoiding redundant refinement elsewhere. We demonstrate the method’s effectiveness through five benchmark phase-field fracture problems, ranging from simple crack initiation to complex patterns of interaction and coalescence. Across all tests, our approach consistently achieves a balance between solution fidelity and computational efficiency, matching or surpassing the standard finite element performance on globally or locally refined meshes. Moreover, the methodology is naturally extensible to large-scale, multidimensional problems, making it a promising tool for advancing predictive modeling in both academic research and industrial practice.