An Adaptive Finite Element Approximation of a Variational Model of Brittle Fracture

Abstract

The energy of the Francfort-Marigo model of brittle fracture can be approximated, in the sense of $\Gamma$-convergence, by the Ambrosio-Tortorelli functional. In this work, we formulate and analyze two adaptive finite element algorithms for the computation of its (local) minimizers. For each algorithm, we combine a Newton-type method with residual-driven adaptive mesh refinement. We present two theoretical results which demonstrate convergence of our algorithms to local minimizers of the Ambrosio-Tortorelli functional.