Numerical simulation of crystalline curvature flow in 3D by interface diffusion

Evolution by mean curvature is recently attracting large attention especially when the underlying anisotropic structure degenerates to become crystalline. In such situation new phenomena must be taken into account: the evolution law becomes nonlocal and "hyperbolic" across facets; moreover events like face breaking or bending have to be considered especially in three dimensions. For this reason the ODE approach suggested by J. Taylor long ago cannot be used directly and the required modifications to the algorithm are not clear at the moment. The well known diffused interface approximation for the classical mean curvature flow, which leads to the Allen-Cahn equation, can be applied in this contex, resulting in a bistable reaction-diffusion equation with good convergence properties to the sharp interface evolution. This equation can then be discretized using finite elements in space and forward differences in time. Numerical simulations with the resulting scheme seem to recover the face breaking and face bending phenomena.