An efficient algorithm for computing anisotropic evolution by mean curvature

A number of numerical simulations of surfaces evolving by mean curvature in anisotropic materials are presented and discussed. We approximate an evolution law based on an appropriate definition of anisotropic mean curvature. This is first regularized by means of a relaxation procedure that produces a reaction-diffusion equation similar to the model proposed by Allen and Cahn for the evolution of antiphase boundaries. The successive discretization process is based on linear finite elements in space and forward differences in time. The final algorithm also exploits the localized behaviour of the solution using a dynamically maintained mesh that spans the sole transition region of the relaxed solution, corresponding to a thin strip around the moving interface.