Adaptive Finite Element Approximation of a Variational Model of Fracture

The energy of the Francfort-Marigo model of brittle fracture can be approximated, in the sense of $\Gamma$-convergence, by the Ambrosio-Tortorelli functional. We formulate and analyse an adaptive finite element algorithm for the computation of its (local) minimisers. In the algorithm, we combine a Newton-type method with a residual-driven adaptive mesh refinement. We present two theoretical results which demonstrate convergence of the algorithm to local minimisers of the Ambrosio-Tortorelli functional. The talk is based on joint work with Siobhan Burke (University of Oxford) and Christoph Ortner (University of Warwick).