  Phase-field models of fracture introduce smeared cracks of width commensurate
with a regularisation length parameter $\epsilon$ and obeying a minimum energy
principle. Mesh adaptivity naturally suggests itself as a means of supplying
spatial resolution were needed while simultaneously keeping the computational
size of the model as small as possible. Here, a variational-based spatial
adaptivity is proposed for a phase-field model of fracture.
  The conventional phase-field model is generalised by allowing a spatial
variation of the regularisation length $\epsilon$ in the energy functional. The
optimal spatial variation of the regularisation length then follows by energy
minimisation in the same manner as the displacement and phase fields. The
extended phase-field model is utilised as a basis for an adaptive mesh
refinement strategy, whereby the mesh size is required to resolve the optimal
length parameter locally. The resulting solution procedure is implemented in
the framework of the finite element library FEniCS. Selected numerical tests
suggest that the spatially adaptive phase-field model exhibits the same
convergence rate as the conventional phase-field model, albeit with a vastly
superior constant, which results in considerable computational savings.
