Numerical simulations of crystalline motion by mean curvature with Allen-Cahn relaxation

In this paper we present some numerical simulations of motion by mean curvature associated to an underlying anisotropy of crystalline type. Such anisotropies are characterized by a Frank diagram (and consequently a Wulff shape) of polygonal type. Our simulations are based on an Allen-Cahn type regularization, performed in the context of a Finsler geometry. The choice of a nonregular double well potential leads to a double obstacle formulation that allows us to exploit the dynamic mesh algorithm, thus reducing considerably the computational cost. A number of simulations show the robustness of our discretization process, which is capable to deal with situations (presence of a generic forcing term) that are critical for other known numerical techniques.